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data of pierre auger observatory show a proton - dominated chemical composition of ultrahigh - energy cosmic rays spectrum at ( @xmath0 ) eev and a steadily heavier composition with energy increasing . in order to explain this feature we assume that ( @xmath0 ) eev protons are extragalactic and derive their maximum acceleration energy , @xmath1 eev , compatible with both the spectrum and the composition . we also assume the rigidity - dependent acceleration mechanism of heavier nuclei , @xmath2 . the proposed model has rather disappointing consequences : i ) no pion photo - production on cmb photons in extragalactic space and hence ii ) no high - energy cosmogenic neutrino fluxes ; iii ) no gzk - cutoff in the spectrum ; iv ) no correlation with nearby sources due to nuclei deflection in the galactic magnetic fields up to highest energies . spectra and chemical compositions of ultrahigh - energy ( @xmath3 eev ) cosmic rays ( uhecr ) measured by two largest detectors , high resolution fly s eye ( hires ) @xcite and pierre auger observatory ( pao ) @xcite , are significantly different . the hires data show pure _ proton _ composition @xcite , confirming such signatures of their propagation through cmbr as the gzk cutoff @xcite and the pair - production dip @xcite . the pao data , on the contrary , strongly favor the nuclei composition getting progressively heavier at @xmath4 eev . this feature , in terms of energy dependence of eas development maximum in atmosphere , @xmath5 , and r.m.s . of this observable , rms(@xmath6 ) , is clearly seen in fig . [ fig3 ] . the data also suggest that the nucleus charge number @xmath7 changes smoothly in sources . here we demonstrate that the simple , but disappointing for future experiments , model @xcite can naturally explain both energy spectrum and mass composition observed by the pao . the basic assumption of the model is the _ proton composition _ of uhecr spectrum at @xmath8 eev , the feature supported both by pao and hires . two more assumptions are that these protons are _ extragalactic _ and that acceleration of primary nuclei in sources is _ rigidity - dependent _ , i.e. that @xmath9 , where @xmath10 is a universal energy to be determined from data ; @xmath7 is a nucleus charge number . in order to determine the maximum acceleration energy of protons , @xmath11 , let us calculate the extragalactic diffuse proton flux , assuming the power - law generation spectrum @xmath12 with @xmath13 , and normalize it by the pao flux at ( @xmath0 ) eev . varying @xmath14 in the range @xmath15 , the maximum value of @xmath10 allowed by the pao mass composition ( see fig . [ fig3 ] ) and energy spectrum ( see fig . [ fig4 ] ) may be obtained . in our calculations a homogeneous distribution of sources with no cosmological evolution ( @xmath16 ) was assumed ; the highest redshift of sources @xmath17 . as a criterion of contradiction an excess of calculated proton flux at @xmath18 eev was chosen . the contradiction has different character for different values of @xmath14 . for steep source generation functions with @xmath19 the shape and flux of the pao spectrum may be described by @xmath20 ev ; the contradiction occurs only in data on mass composition . the extreme case , given by @xmath21 , is displayed in the left panel of fig . [ fig4 ] . for flat generation spectra ( see the extreme case of @xmath22 in the right panel of fig . [ fig4 ] ) the contradiction is very pronounced . for @xmath23 eev the calculated proton flux exceeds the observed one even at @xmath24 eev . it is clear that with some redundancy @xmath25 eev for all @xmath26 . an influence of possible intergalactic magnetic fields on proton spectrum calculated in a diffusive model is shown in the left panel of fig . [ fig5 ] . here @xmath27 , which might be the case for acceleration by relativistic shocks . the kolmogorov diffusion in turbulent magnetic field with basic scales @xmath28 was assumed ( see @xcite ) and distances between sources were @xmath29 mpc . the analysis of proton maximum energy of acceleration gives again @xmath30 eev , in a rough agreement with the analysis made for homogeneous distribution of sources . the account for diffusion brings to the flattening of the proton spectrum at @xmath31 eev , seen in fig . [ fig5 ] as a diffusive cutoff , which provides a transition from the steep galactic spectrum , most probably composed of iron , to the flat spectrum of extragalactic protons . the basic feature of the pao mass composition , the progressively heavier composition with energy increasing , is guaranteed in our model by the rigidity - dependent maximum energy of acceleration : at energy higher than @xmath32 nuclei with charge @xmath33 disappear , while heavier nuclei with larger @xmath7 survive . starting from @xmath34 eev , the higher energies are accessible only for nuclei with progressively larger values of @xmath7 . let us now consider a two - component model , with only protons and iron nuclei being produced in sources with generation index @xmath22 and the maximum acceleration energy @xmath35 eev , shown in the right panel of fig . [ fig5 ] . the primary iron nuclei spectrum is calculated as in @xcite for homogeneous distribution of sources . one may notice that the calculated spectrum of iron describes well the cutoff in the pao spectrum . this steepening is caused by the photo - disintegration of iron nuclei . to agree with the mass composition of pao , the iron spectrum in fig . [ fig5 ] must have a low - energy cutoff at @xmath36 eev . most naturally it is produced as a diffusive cutoff which appears in models with lattice - located sources due to _ magnetic horizon _ @xcite . such cutoffs are shown in fig . [ fig6 ] for three different sets of parameters @xmath37 . the beginning of this cutoff @xmath38 for iron nuclei is @xmath39 times higher than for protons , i.e. @xmath40 ev , which has a reasonable physical meaning . the gap between @xmath41 eev and @xmath42 eev is expected to be filled by intermediate nuclei . to provide a smooth rms(@xmath6 ) curve seen in fig . [ fig3 ] , there are many free parameters , e.g. arbitrary fractions of nuclei accelerated in distant sources . with the proton composition ( the 6th low - energy bin of the pao data in fig . [ fig3 ] ) . if this energy increases , @xmath43 increases , too . the model collapses when the allowed @xmath43 reaches e.g. ( @xmath44 ) eev . another case is given by the mass composition beinglight nuclei starting right from 1 eev @xcite . the cosmological evolution of sources are not included in our calculations ; since this effect slightly decreases @xmath45 , it is not needed to be taken into account . in principle , it is also possible that the eev protons detected by pao are secondary ones , i.e. those produced in photo - dissociation of primary nuclei in collisions with cmbr and extragalactic ir / uv photons . however , in fact , as it was demonstrated in @xcite , the flux of secondary protons in the eev range is always smaller than the sum of primary and secondary nuclei fluxes . the predictions of our model are very disappointing for the future detectors . really , the maximum acceleration energy @xmath46 eev for iron nuclei implies the energy per nucleon @xmath47 , well below the gzk cutoff for epochs with @xmath48 . therefore , practically no cosmogenic neutrinos can be produced in collisions of protons and nuclei with cmb photons . correlation with uhecr sources also is absent due to deflection of nuclei in the galactic magnetic fields . the lack of correlation in the model is strengthened by the dependence of the maximum energy on @xmath7 . the signatures of the disappointing model for the pao detector are the mass - energy relation , already seen in the elongation curve @xmath5 , and transition from galactic to extragalactic cosmic rays below the characteristic energy @xmath49 eev . there are some uncertainties in the model presented above . the most important one relates to estimates of @xmath43 . it is determined by the lowest energy where pao data become inconsistent the work of a. gazizov was supported by a contract with gran sasso center for astroparticle physics ( cfa ) funded by european union and regione abruzzo under the contract p.o . fse abruzzo 2007 - 2013 , ob . cro .

the study of dwarf galaxies has important implications for our current understanding of processes governing the formation of galaxies , stars , and large scale structure in general . the generally accepted models for structure formation via hierarchical clustering predict the existence of a large number of dwarf galaxies and that these galaxies are expected to be the sites of the earliest star formation ( see white & frenk 1991 ) . although dwarf galaxies are the most numerous type of galaxies in the nearby universe , their numbers are far fewer than theoretical predictions @xcite . furthermore , the observations also seem to suggest that the smallest galaxies are among the youngest rather than the oldest , and this phenomena appears to be independent of environment . for example , many of the local group dwarf spheroidals and dwarf irregular galaxies appear to have formed a significant fraction of their stars in starbursts at @xmath1 ( see , for example , van den bergh 1994 and tolstoy 1999 ) . deep imaging studies of intermediate redshift ( @xmath2 ) clusters ( e.g. dressler et al . 1994 ; couch et al . 1994 , 1998 ) have found that most of the blue galaxies responsible for the butcher - oemler effect are small late - type spirals or irregular galaxies . it has been suggested that these galaxies are starburst dwarf galaxies ( koo et al . 1997 ) that eventually fade away or star - forming remnants of ` harrassed ' small galaxies that will eventually evolve into the cluster dwarf spheroidal population ( moore et al . 1996 ; moore , lake , & katz 1998 ) . similarly , detailed analyses of the faint blue galaxies that numerically dominate the field galaxy population at intermediate redshifts ( @xmath2 ) indicates that a significant fraction of the very faint blue galaxies are small ( and therefore , dwarf ) galaxies that are actively forming stars at a relatively recent epoch ( see babul & ferguson 1996 and references therein ; campos 1997 ; driver & fernandez - soto 1998 ; fioc & rocca - volmerange 1999 ) . the discrepancy between the theoretical expectation that small galaxies ought to be among the oldest and the observational evidence to the contrary can either be due to the fact that hierarchical clustering model does not provide a complete description of structure formation particularly on the scales of interest , or that the astrophysics underlying the formation of dwarfs galaxies is not well understood , with the latter being the more accepted of the two possibilities . for example , it has long been recognized that dwarf galaxies are rather fragile systems that their formation and subsequent evolution , in fact their very character , is likely to be strongly affected by both internal and external conditions . the ` galaxy harrassment ' model of moore et al . ( 1996 ; 1998 ) , the models advocating catastrophic mass loss following supernovae explosions ( larson 1974 , saito 1979 , vader 1986 , dekel & silk 1986 ) , and models that argue that star formation in these systems is strongly modulated by the internal and external uv radiation field ( e.g. babul & rees 1992 ; efstathiou 1992 ; kepner , babul & spergel 1997 ; norman & spaans 1997 ; spaans & norman 1997 ; corbelli , galli & palla 1997 ) are all based on the fragility of the dwarf systems . the ` galaxy harrassment ' mechanism of moore et al . ( 1996 ; 1998 ) operates only in high density enviroments . based on results of numerical simulations , moore et al . have proposed that multiple , high - speed encounters between small disk galaxies and the large massive galaxies in the cluster enviroment can cause multiple starbursts in , as well as significant mass loss from , the lower mass galaxies . the interactions would rearrange the internal structure of the smaller galaxies , converting them into much more resilient , compact dwarf elliptical - like galaxies . moore et al . also suggested that the continued harrassment of the debris tails material torn from the original galaxies will create tidal shocks that will promote further condensations and formation of dwarf galaxies . in this scenario , the most numerous class of galaxies in clusters have formed at moderate redshifts . with regard to the impact of uv radiation on the star formation in dwarf galaxies , babul & rees ( 1992 ) and efstathiou ( 1992 ) have discussed in detail how an ionizing flux of an intense metagalactic uv background , such as that established by quasars and early starbursts , can prevent the gas in halos with shallow potential wells from cooling and forming stars until @xmath3 , in spite of the fact that the halos themselves may have formed at some earlier epoch , by initially keeping the gas ionized and latter , by suppressing the formation of molecular hydrogen , the only coolant available to metal - poor gas ( see kepner , babul & spergel 1997 ) . however , analytic calculations ( rees 1986 ; ikeuchi 1986 ) and subsequent numerical studies ( katz , weinberg & hernquist 1996 ; thoul & weinberg 1996 ; quinn , katz , & efstathiou 1996 ; navarro & steinmetz 1997 ; forcada - miro 1997 ) show that photoheating most strongly affects halos with circular velocities @xmath4 . larger halos are largely unaffected . these limits , however , should be treated with some caution because none of these studies considered the impact of uv radiation from the very first stars on further star formation . in addition , all numerical studies except that of kepner et al . ( 1997 ) ignored the fact that before metal - poor gas could cool and form stars , the ambient conditions must allow for the formation of neutral hydrogen , and even kepner et al . ( 1997 ) only explored models involving spherical collapse . subsequent studies by corbelli , galli & palla ( 1997 ) , norman & spaans ( 1997 ) , and spaans & norman ( 1997 ) have gone much further . the latter authors , for example , considered halos with total masses spanning the range @xmath5 and found that even if a halo does not get ` hung - up ' by the background uv radition , the star formation in disk - like protogalaxies will initially proceed very slowly because the backreaction of star formation on the ionization and chemical equilibrium can greatly impact upon the abundance of @xmath6 . the tight coupling between the radiation field and star formation rate is reduced only when the gas is sufficiently enriched . consequently , the onset of massive starbursts , if any , even in magellanic - type systems will be delayed until @xmath7 . once stars do form , there appears to a consensus , at least among the theorists , that the dwarf galaxies , due to their low escape velocities , will suffer supernova - driven outflows ( eg . larson 1974 , saito 1979 , vader 1986 , dekel & silk 1986 ) . whether or not such outflows result in catastrophic loss of the interstellar medium is likely to depend on a variety of factors , such as the energy input in the interstellar medium ( ism ) , the ellipticity of the ism distribution , etc . ( see de young & heckman 1994 ) . however , it is worth noting that the catastrophic loss of the gas for a galaxy and the subsequent quenching of the star formation provide a natural explanation for the low surface brightness and low metal abundances of dwarf galaxies ( eg . dekel & silk 1986 , de young & gallagher 1990 ) . it also provides an explanation for the apparently rapid disappearance of the large numbers of small blue galaxies ( low mass starbursting systems ) that dominate the galaxy number counts at very faint magnitudes ( see babul & rees 1992 , babul & ferguson 1996 ) . although supernova - driven outflow is an important aspect of any theoretical discussion of dwarf galaxy evolution , there has been , until recently , quite limited observational evidence for the existence of starburst - driven mass loss . the most detailed and convincing evidence was first reported by meurer et al . ( 1992 ) , who found a kpc - scale ` superbubble ' of ionised gas expanding at @xmath8 km / s in the core of the post - starburst dwarf galaxy ngc1705 . this has now been followed up by marlowe et al . ( 1995 ) , who have studied a large sample of starbursting dwarf amorphous galaxies and found evidence for outflows in approximately half of them . also , heckman et al . ( 1995 ) have argued that x - ray emission from dwarf galaxy ngc1569 is a signature of starburst - driven outflow . generally , it is assumed that if the starburst imparts sufficient energy to the affected fraction of the ism to unbind it from the galaxy s gravitational potential well , then this material will be ejected from the galaxy . the situation is not as simple . the supernova rate needs to be high enough so that the remnants percolate the galaxy s interstellar medium in a time short compared to the remnant s radiative timescale ( see , for example , dekel & silk 1986 ) . in addition , babul & rees ( 1992 ) have noted that whether or not the outflow actually escapes from the low mass galaxy also depends on the state of the local intergalactic medium ( igm ) . the thermal pressure due to the igm will resist the flow of material out of the galaxy . they argued that in regions of high pressure , such as in clusters of galaxies , the outflow would not be able to expand beyond the extent of the galaxy s dark halo and hence , will eventually accrete back onto the galaxy , allowing the galaxy to engage further star formation . in regions of low thermal pressure , the outflow would escape unhindered . one consequence of the thermal - confinement picture is that there ought to be correlations between the properties of dwarf galaxies and their environment . for example , the low mass galaxies in high pressure environments , by virtue of being able to retain a greater fraction of their gas , ought to be more luminous and more metal - rich than comparable galaxies in the field . in clusters of galaxies , one would expect the dwarf galaxy population to exhibit a radial luminosity and metallicity gradients , with the galaxies closer to cluster center being , on the average , brighter and more metal - rich ( babul & rees 1992 ) . recent observations seem to suggest that such trends do exist . for example , secker ( 1996 ) finds significant color gradient in the radial distribution of de galaxies in coma , with redder galaxies tending to be closer to the cluster center . secker interprets the color gradient as indicative of a metallicity gradient . similar radial color gradients have also been detected in other clusters ( o. lopez - cruz , private communications ) . secker , harris & plummer ( 1997 ) also find that redder de galaxies are also more luminous . more generally , dwarf elliptical galaxies are found only in high - density environments such as clusters of galaxies or clustered around giant galaxies ( vader & sandage 1991 ; ferguson 1992 ; vader & chaboyer 1993 ) . according to the the thermal confinement scenario , this is because low mass galaxies in high density environments are more luminous and have higher surface brightness , and are therefore easier to detect . ferguson & sandage ( 1991 ) and vader & sandage ( 1991 ) have argued that the number ratio of early - type dwarfs to early - type giant galaxies is correlated with cluster richness with the ratio being larger for richer clusters . the bright nucleated de galaxies have only been sighted in clusters and even in this environment , they tend to be much more centrally concentrated than the non - nucleated des ( ferguson & binggeli , 1994 ) . we also be note that if the intrinsic mass function of the galaxies is steeply rising towards the low mass end , as predicted by most hierarchical galaxy formation theories , then this steep end ought to be easiest to observe in high density envrionments since the galaxies of a given mass in such environments will tend to be relatively brighter . luminosity functions of cluster galaxies with steep rise at the low luminosity end have been observed by , for example , trentham ( 1988 ) . in this first of a series of studies , we investigate the idea of pressure - confined outflows put forth by babul & rees ( 1992 ) . specifically , we use a 2d hydrodynamical code to study the influence of the local igm on outflows from spherically symmetric dwarf galaxies , such as the local group dwarf spheroidals and cluster dwarf ellipticals , in various different environments . admittedly , spherically symmetric dwarf galaxies are likely to lose the greatest amount of ism ; however , in the present study , we are not so much concerned with the degree of mass loss as with the interaction of the outflow with the local intergalactic environment . in addition , dwarf galaxies are not static objects . in high - density cluster environments , for example , the dwarfs are likely to be ploughing through the local igm at relatively high velocities . in such cases , they will also be subject to ram pressure and contrary to the action of thermal pressure , ram pressure acts to strip the ism from the galaxy ( gunn & gott 1972 ; gisler 1976 ; lea & de young 1976 ; toyama & ikeuchi 1980 ; fabian , schwarz , & forman 1980 ; takeda , nulsen , & fabian 1984 ; gaetz , salpeter , & shaviv 1987 ; portnoy , pistinner & shaviv 1993 ; balsara , livio , & odea 1994 ) . we also investigate the combined impact of ram pressure and thermal pressure under different conditions . in 2 , we define our model for the dwarf galaxy and outline the methods used in this investigation . in 3.1 , we discuss the evolution of outflows subject only to thermal pressure and in 3.2 , we consider the combined impact of thermal and ram pressure . in 4 , we discuss the possible observational consequences of the interaction between the outflows from dwarf galaxies and the surrounding intergalactic medium . finally , we summarize the our findings in 5 . since we are interested in low - mass galaxies , we will adopt as a fiducial galaxy , a system whose dark halo mass is @xmath9 and the circular velocity characterizing the gravitational potential of the system is @xmath10 km / s . for simplicity , we assume that the virialized dark halos have spherically symmetric density profiles @xmath11 where @xmath12 is the core radius set by the numerical resolution of our simulation , which is either @xmath13 , @xmath14 , or @xmath15 kpc . the corresponding one - dimensional velocity dispersion of the galaxy is @xmath16 km / s . this velocity dispersion is somewhat larger than that of a typical local group dwarf spheroidal ( @xmath17 km / s ) ; on the other hand , the general characteristics of our system ( i.e. low mass and shallow potential well ) are comparable to those of the dwarf ellipticals ( peterson and caldwell 1993 ) found in clusters of galaxies . we assume a mass of @xmath18 for the gas in the halo . in order to be susceptible to star formation , the gas must be at least marginally self - gravitating ( eg . mathews 1972 ) . to satisfy this constraint , we require all the gas to be concentrated within a central @xmath19 kpc of the halo and identify the centrally condensed baryonic system in the halo as the galaxy . all stellar activity , from star formation to supernova explosions , takes place in the galaxy . the ` natural ' rate for star formation in a self - gravitating gas cloud is @xmath20 where @xmath21 is the instantaneous mass of gas cloud and @xmath22 is the free - fall timescale for the cloud ( e.g. dekel & silk 1986 ) . for the system under consideration , the star formation rate is @xmath23 @xmath24 . this rate is almost three orders of magnitude larger than that the maximum rate adopted by maclow & ferrara ( 1998 ) in their study . it is , however , comparable to the star formation rate in the faint blue galaxies as established from their redshifts and b magnitudes ( babul & rees 1992 ) . furthermore , since the star formation is distributed over a region one kiloparsec in radius , the star formation rate per unit area of @xmath25 @xmath26 is comparable to that seen in typical starburst galaxies ( lehnert & heckman 1996 ) . soon after the onset of starburst , the massive stars will go supernova . if each supernova releases @xmath27 ergs , the rate of total energy released by the supernova explosions into the interstellar matter is @xmath28 where @xmath29 is the number of supernovae per @xmath30 of stars formed , and @xmath31 is the star formation rate in units of @xmath24 . in the solar neighbourhood , one has roughly one supernova for every 150 @xmath32 of baryons that form stars ; hence , @xmath33 . if starbursts make only high mass stars ( see , for example , rieke et al . 1993 ) , then @xmath34 would be larger . for present purposes , we assume that @xmath35 . most of the energy released by the supernova will radiated away and only a small fraction , @xmath36 , will go towards heating the ism ( see larson 1974 ; dekel & silk 1986 ; babul & rees 1992 ) . the total rate of energy input to gas is then @xmath37 . we assume that this energy input lasts for @xmath38yr . the lifetimes of supernova type ii progenitors range from few@xmath39 few@xmath40 years , and we assume that the very first generation of sn explosions will disrupt the interstellar medium and quench further star formation . as we will show in the following section , the energy input described above is sufficient to generate an outflows from the dwarf galaxy . this is neither surprising nor the thrust of the present work . many previous works ( eg . larson 1974 ; dekel & silk 1986 ; babul & rees 1992 ) have already made this point . here , we are more interested in studying how different local intergalactic conditions affect the outflow and the eventual fate of the expelled gas . specifically , we focus on the influence of the thermal and ram pressures engendered by the intergalactic medium . at the start of the simulation , the igm gas is uniformly distributed across the entire simulation volume except in the galaxy , the central 1 kpc region of the halo . this distribution is not in equilibrium with the gravitational potential of the halo and in the absence of outflows from the galaxy , would accrete into the halo . in our simulations , this accretion is not important . the associated timescale is comparable to the total duration of a single simulation run and is much longer than all the dynamical timescale established by the outflows . to study the influence of thermal component of the intergalactic pressure , we vary the value of @xmath41 from @xmath42 to @xmath43 k @xmath44 . the lower limit corresponds to an igm of @xmath45 @xmath44 heated to the temperature of @xmath46 k by photoionisation and the upper limit corresponds to pressures thought to exist in central regions of galaxy clusters . as we have already noted , observations indicate that starburst dwarf galaxies are found in all type of environments , especially at intermediate redshifts . to study the effects of ram pressure , we allow the galaxy halo to have velocity , with respect to the igm , ranging from @xmath47 to @xmath48 km s@xmath49 . the parameters for the cases discussed in this paper are summarized in table 1 . we study the outflow and its interaction with the igm using an axisymmetric euler code ( norman & winkler 1985 ; yoshioka & ikeuchi 1990 ; murakami & ikeuchi 1994 ) . following norman & winkler ( 1985 ) , we include artificial viscosity in our simulations in order to treat shocks . we use the code to solve ( in cylindrical coordinates ) the continuity equation : @xmath50 the euler equation : @xmath51 = - \nabla p - \rho \nabla \phi , \label{eq - motion}\ ] ] the poisson equation : @xmath52 and the energy equation : @xmath53 in the above equations , @xmath54 is the specific internal energy , @xmath55 and @xmath56 are the heating and cooling rates respectively , and @xmath57 is the density of dark halo matter . we ignore the self - gravity of the baryonic system . we also ignore the depletion of gas as it forms stars as well as mass input from stellar winds and supernova explosions . the rest of the symbols have their usual meanings . the heating rate , @xmath55 , is zero everywhere except in the galaxy itself , where for a brief time , the supernova explosions inject thermal energy into the ism at a rate @xmath58 , where @xmath59kpc . for the cooling rate , we consider two possibilities : a cooling rate for gas with primordial abundance ( hereafter , referred to as ` primordial cooling ' ) and a cooling rate for gas with cosmic abundance ( @xmath60 ) determined by allen ( 1973 ) ( hereafter , referred to as ` metal cooling ' ) . in order to compute the primordial cooling rate , we take into account radiative recombination , collisional ionisation , thermal bremsstrahlung , line emission and dielectric recombination ( see umemura & ikeuchi 1987 ) for the gas of @xmath61 . for the metal cooling rate , we use a broken power - law fit of the results of raymond , cox & smith ( 1976 ) who adopted the cosmic abundances ( that is , @xmath62 , @xmath63 , and @xmath60 ) tabulated by allen ( 1973 ) . in computing the heating rate due to supernova explosions , we have already taken into account the fact that most of the energy will be lost via cooling radiation . to keep the cooling functions on during this time would mean that we would be double counting the cooling losses . therefore , the cooling function is switched on after the supernova explosions cease . we now discuss the results of our simulations . first , we shall consider cases where the outflows from dwarf systems are impeded only by the thermal pressure of the igm . subsequently , we shall consider cases where the dwarf system is acted upon by thermal as well as ram pressure . energy input from supernova explosions causes the interstellar medium in the galaxy to heat up to @xmath64 k. as a consequence of the resulting pressure differential between the gas in the bubble and the igm , the bubble expands and an outflow is established . at its maximum , the leading edge of the outflow has a velocity of order @xmath65 . the outflow / expansion represents the conversion of some of the sn - injected thermal energy into kinetic energy . as in all outflow - type situations ( e.g. see castor , mccray & weaver 1975 and references therein ) , the interaction between the outflow and the intergalactic medium leads to the formation of a dense shell consisting of the swept - up ism and igm . in figures [ fig-1 ] [ fig-4 ] , we show the evolution of the outflow for models -2 m , 0 m , 3 m and 4 m ( see table 1 ) . the results for gas subject to primordial cooling are qualitatively similar . in cases where the igm temperature is @xmath66 k ( models -2 m and 0m / p ) , the leading edge of the outflow is supersonic and the interface between the outflow and the unperturbed igm is demarcated by a shock . in models where the igm temperature is @xmath67 k , the outflow is always subsonic and , as expected , the gas ahead of the forming shell is also perturbed . in figure [ fig-5 ] , we show the radial pressure , density , temperature , and velocity profiles for models 0p and 0 m . the left and right panels show the profiles for models 0p and 0 m respectively during the expansion phase . in figures [ fig-6 ] and [ fig-7 ] , we show the expanding ( left - hand panels ) and contracting ( right - hand panels ) for models 4 m and 4p , respectively . during the heating phase , the pressure in the bubble exceeds the igm pressure and the heated gas begins to expand outward . at the head of the outflow , the swept - up ism and igm begins to form a shell . once the outflow is established and the heating stops , the gas in the cavity cools as a result of radiative as well as adiabatic cooling . the higher density gas closer to the center of the gas cools much more rapidly than lower density cavity gas behind the shell as a result of efficient radiative cooling . eventually , all of the gas in the cavity cools . once the pressure in the cavity drops below that of the external igm , the shell begins to decelerate . ( the sweeping up of the igm also decelerates the shell but in all but the low pressure cases , [ @xmath68 , the effect is not important . ) with continued expansion , the pressure in the cavity may fall below that in the shell and a contact discontinuity forms at the boundary between the two . once the shell velocity becomes comparable to the sound speed in the external igm , the shell velocity stagnates . if a contact discontinuity has formed between the cavity gas and the inner boundary of the shell , then in the shell frame of reference , the inner shell boundary begins to expand inward into the cavity , led by a shock front which halts and thermalizes the outflowing cavity gas and also smoothes out the sharp pressure gradient . this inward expansion of the inner shell boundary has also been noted by ciardi & ferrara ( 1997 ) . in high pressure environments , the stagnation radius and the halting radius are , for all practical purposes , equivalent as the stagnation point is reached just before the shell is stopped . the inward expansion of the inner shell boundary is , however , evident in the right panels ( collapsing phase ) of figures [ fig-6 ] and [ fig-7 ] . in low pressure environments , the shell decelerates very slowly and there can be a large lag between the time when the shell velocity stagnates and when it comes to a halt . during this time , the outer shell continues to expand outward while the inner boundary begins to expand inward , and the shell thickness grows . for @xmath69 ( figure [ fig-5 ] ) , the stagnation occurs at @xmath70 kpc . as expected , the main difference between the metal and primordial cooling cases is the increased efficiency of radiative cooling in the former case . this is obvious both from the rapid evolution of the temperature profiles , the thinness of the shell and the dramatic drop in the temperature of the gas remaining in the galaxy . in fact once the heating stops , a small cooling flow is established as the gas flows back towards the potential minimum . the maximum radius to which the shell expands depends on the igm pressure . if the igm pressure is low ( @xmath71 ) , the shell can be driven well beyond the virial radius of the halo and into the intergalactic space . if , on the other hand , the igm pressure is as high as in the central regions of clusters , the shell expands by a very small amount before being halted . from a physical point of view , the evolution of the bubble - shell system is schematically similar to that of ` superbubbles ' as sketched out by maclow , mccray & norman ( 1989 ) although , it should be noted that are differences between the configuration that they studied and those considered here . one consequence of this is that the bubble - shell expansion in our simulations is not self - similar . the simulation results do , however , suggest that the radius of maximum expansion is related to the value of the igm pressure as : @xmath72 with @xmath73 for the metal cooling cases tending to be slightly smaller than that for the corresponding primordial cooling cases . in addition , preliminary numerical experiments indicate that @xmath73 also depends , albeit weakly in the case of shallow potential wells of the kind under consideration in this paper , on the depth of the gravitational potential well of the halo in the sense that @xmath73 decreases as @xmath74 increases . during the expansion phase , the amount of intergalactic mass that is swept - up by the shell - bubble system is : @xmath75 in all but cases 0p / m and -2 m , the external pressure brings the shell - bubble system to a halt well before the swept - up mass exceeds the mass originally in the sn - heated bubble . for models -2 m , the shell continues expanding during the entire course of the simulation , expanding to distances greater than @xmath76 kpc . in this case , the shell - bubble system would need to expand out to @xmath77 kpc before the swept - up mass exceeds the initial mass . we can safely assume that the system will indeed expand out to this radius since @xmath73 for this configuration is estimated to be @xmath78 kpc . however , we do not follow the expansion out to such radii . for model 0p / m , the bubble - shell system expands beyond @xmath79 kpc , the radius at which the swept - up mass equals that initial mass in the bubble . thereafter , the leading front of the shell continues to evolve much like an ` oort snowplow ' . @xmath80 kpc is also roughly the stagnation radius and therefore , while the outer shell radius continues to expand , the inner boundary expands inwards as described previously , sweeping through the cavity in @xmath81 years . this inward propagating shock is stable against rayleigh - taylor instabilities because both the density gradient and pressure gradient have the same sign . and in this regard , the inward moving shock is very different from the shell collapse that occurs in high pressure environments and that we discuss below . once the shell is brought to a halt , both gravity and the igm pressure begin to force the system to contract . the timescale for the shell to fall back onto the central galaxy , assuming that all the gas has piled up in the shell and that the shell remains intact during the collapse , can be estimated as @xmath82 where we have used equation ( [ max - shell - radius ] ) . this estimate matches the actual collapse time within a factor of @xmath83 . in computing @xmath84 , we have assumed that in comparison to the pressure force , the gravitational force can be neglected ; the pressure force exceeds the gravitational force by two orders of magnitude or more . the bottom two panels of figures [ fig-3 ] and [ fig-4 ] show the collapse of the shell in cases 3 m and 4 m , while the right panels in figures [ fig-6 ] and [ fig-7 ] show the the radial pressure , density , temperature , and velocity profiles for models 4 m and 4p , respectively , at different times during the collapse phase . the evolution of the sn - heated gas bubble as well as the shell during the collapse phase depends sensitively on the efficiency of cooling . in the metal cooling case , the cooling timescale of the gas in the bubble is generally shorter than the dynamical timescale and gas / shell collapses in a ` simple ' fashion . in the primordial cooling case , however , the cooling timescale is larger than the dynamical timescale . as the gas / shell collapses , the gas is heated and the resulting increase in the pressure causes the shell to bounce . ( see curves for @xmath85yr and @xmath86yr in the right - hand panels of figure [ fig-7 ] . ) the same , though a bit more pronounced , occurs in simulations with no radiative cooling . once the shell starts to collapse , it begins to buckle and deform into tentacle - like structures . this deformation is due to initially small perturbations that are present in the shell being enhanced by raleigh - taylor ( r - t ) instability . as discussed by chevalier ( 1976 ) and others , a pressure - driven flow is r - t unstable if the pressure gradient ( source of acceleration ) and the density gradient have opposite signs . in numerical simulations of supernova explosions ( e.g. nagasawa , nakamura , & miyama 1988 ; arnett , fryxell , & muller 1991 ; hachisu et al . 1991 ) , r - t instability manifests as well - defined mushroom - like features . this shape is a consequence of the fact that the structures develop outward . in present case , the shell is contracting . the features associated with the r - t instability are also mushroom - like ; however , the heads of the mushrooms develop inward where there is less volume and hence , merge with each other , giving rise to tentacle - like structures . whether r - t instability materializes in a numerical simulation and the extent to which it does depends sensitively on the resolution of the numerical simulation @xcite . in figure [ fig-8 ] , we show the evolution of shell - bubble for model 3 m during the collapse phase . the two left panels show the results corresponding to resolution d@xmath87 kpc and the two right panels show the results for exactly the same simulation but with d@xmath88 kpc . in both cases , the shell buckles and tentacle - like features arise as it is forced to contract . in the higher resolution case , however , there are many more tentacle - like features , the shell itself is thinner and has a greater tendency to fragment forming small clouds . this is especially true of the lagging sections of shell . the presence of clouds and extended tentacles ensures that the collapse is not uniform . the lag between when the first parts of the shell reach the central region of the galaxy and when the clouds fall in is @xmath89 yr . once the shell has fragmented and the pressure surrounding the clouds has equilibrated , the clouds are only subject to gravitational forces . as we mentioned in the introduction , galaxies in high - thermal - pressure environments such as clusters of galaxies are also moving , often supersonically , through the intracluster medium . the intracluster medium flowing through a galaxy results in the gas in the galaxy being subjected to ram pressure forces ( in addition to the thermal pressure forces ) . the impact of the ram pressure is to strip away the gas in a galaxy , eventually denuding the galaxy of its gas content . ram pressure stripping is thought to be the dominant mechanism by which galaxies in cluster environment lose their gas ( gunn & gott 1972 ; gisler 1976 ; lea & de young 1976 ) and consequently , a great deal of effort has gone into trying to understand the process , especially through the use of two - dimensional hydrodynamic simulations ( eg . takeda et al . 1984 ; gaetz et al . 1987 ; portnoy et al . 1993 ; balsara et al . 1994 ) . in the case of galaxies moving through the intracluster medium at velocities comparable to or larger than the velocity dispersions of typical clusters ( @xmath90 ) , the ram pressure forces can equal or exceed the thermal pressure forces ( we shall continue to denote thermal pressure as @xmath91 ) : @xmath92 where @xmath93 is the velocity dispersions of clusters . ram pressure effects can have significant impact on the outflow from the galaxies . at the very least , the expanding gas shell will not be spherically symmetric in spite of the fact that the outflow is . upstream , the expanding shell is subject to both thermal and ram pressure ( @xmath94 ) while on the downstream side , the shell is largely unaffected by ram pressure . consequently , the shell should therefore be oval - shaped . the halting distance of the shell should also reflect this asymmetry . for example , the upstream stopping distance of the shell associated with a galaxy moving at @xmath48 km / s through a @xmath95 icm should be comparable to the maximum expansion radius of the shell associated with a galaxy embedded in @xmath96 icm ( see equation [ max - shell - radius ] ) . the snapshots in figure [ fig-9 ] show the effect of ram pressure on the outflow from a dwarf galaxy described in the example above . the galaxy is moving at a velocity of @xmath97 through a medium characterized by @xmath95 ( model 3mw ) . for reasons mentioned above , the expanding shell has an oval shape ( figure [ fig-9]a ) . once the upstream expansion of the shell is halted , ram pressure begins to drag the material in the shell downstream , distorting the shell ( figure [ fig-9]b ) . the downstream side shell is also distorted by the eddying flow towards the center of potential well . the shell buckles and begins to fragment into high density clouds ( figure [ fig-9]c , d ) . in time , the upstream segment of the shell is pushed back into the galaxy while the remains of the rest of the shell is distorted into a hyperboloid - like surface . the galaxy - shell system resembles a comet , with the main galaxy forming the head and the shell material being the tail . the dense clouds are cold and are potential sites of star formation . if this was to occur , one would expect the resulting galaxy to have a relative high surface brightness ` head ' attached to a diffuse tail . while the evolution of the outflow / shell in models 4pw/4mw are qualitatively similar to that in models 3pw/3mw ( described above ) , this is not the case when the ambient thermal pressure is as high as @xmath98 ( models 5mw/5pw ) . in such circumstances , the combined pressure ( ram @xmath99 thermal ) greatly exceeds the thermal pressure of the supernova heated gas . even during the heating phase , heated gas is unable to expand upstream . instead , the shell that forms at the interface between the supernova - heated gas and ambient intergalactic medium is quickly swept downstream and the galaxy loses most its gas in @xmath100 years ( figure [ fig-10 ] ) . perpendicular to the flow , the heated gas manages to expand slight but the resulting larger cross - section presented to the oncoming wind only hastens the sweeping away of the gas . thus far , we have been considering the impact of ram pressure on a dwarf galaxy that is moving at a velocity comparable to what one would expect of a galaxy in a rich cluster . we have also considered cases where the galaxy is moving at somewhat lower velocities , velocities comparable to what one would expect in poor clusters and galaxy groups . specifically , we consider cases where @xmath101 and @xmath102 , respectively . in these simulations , the thermal pressure is not overwhelmed by ram pressure and during the expansion phase , the results are similar to that in the no - wind case except that the shape of the expanding shell is not quite spherical . the main effect of the wind is to gradually drag downstream the more slowly collapsing tentacles and clumps . most of the expelled gas manages to collapse back into the central regions but it too is eventually dragged out of the halo and carried downstream . the clouds that form during the crushing and the fragmentation of the shell tend to have densities @xmath103 and sizes @xmath104@xmath105 kpc . the cloud masses range between @xmath106@xmath107 . once formed , the clouds are accelerated by the wind . for a cloud at rest , the timescale for cloud to be accelerated to the escape velocity of the dwarf galaxy halo by ram pressure is @xmath108 where @xmath109 , @xmath110 , @xmath111 , @xmath112 , and @xmath113 . the clouds remain in the vicinity of the galaxy ( i.e. within the dark halo of the galaxy ) for approximately @xmath114 years before they are dragged away . the clouds are also subject to thermal evaporation , which will enhance the stripping rate @xcite . one of the most important results that we draw from our work is that in cluster environments , the confinement of the supernovae - heated outflows from dwarf galaxies is complicated by the effects of ram pressure . if the ram pressure acting on a dwarf galaxy is much less than the thermal pressure of the local intergalactic medium , then , as described by babul & rees ( 1992 ) , the outflow from the galaxy is indeed halted by the thermal pressure and the confined gas subsequently falls back onto the galaxy , providing fuel for a possible second burst of star formation . otherwise , ram pressure alters the confinement picture in a very significant fashion . only a small fraction of the expelled gas manages to collapse back onto the central galaxy ; most of it is swept away . if this is the case , it is difficult to understand how trends reported , for example , by secker ( 1996 ) would arise unless there exist two different populations of de galaxies , an original population that is centrally concentrated within the cluster and whose members have lower velocities , and a more extended , higher velocity population comprising of galaxies that fell into the cluster at some latter time . the outflows from the former group would be subject to thermal confinement and one would expect such galaxies to be brighter and redder on the average , much like the population of nucleated des . the more extended ` infall ' population of de galaxies , on the other hand , are likely to recover a very small fraction of the outflowing gas : if the outflow occurs while the galaxies are outside the cluster , thermal icm pressure there is too low to effect any confinement . if the outflow occurs after the galaxies fall into the cluster , the ram pressure acting on the galaxies would sweep away the bulk of the gas . in cases where the ram pressure is important , the gas that is swept away tends to be concentrated in small clouds fragments of the shell that formed at the interface between the icm and the outflowing material . ordinarily , transfer of heat from the igm and into the clouds via conduction would cause the clouds to evaporate . the magnetic fields that permeate cluster environments are likely to suppress thermal conduction and therefore , the clouds will maintain their integrity . the clouds are dense enough to support small episodes of star formation and because of their velocities and spatial distribution , we would expect that if the galaxy was observed while the clouds were forming stars , it would resemble a comet , with the actual galaxy forming the head and the distributed star forming regions tracing out the ` comet tail ' . there are quite a few galaxies with ` comet - like ' morphologies in @xmath115 cluster 3c324 ( dickinson , private communications ; dickinson 1996 ) . drawing upon the results of our simulations , we speculate that these objects are galaxies whose interstellar medium is being stripped away and that some of the stripped material is undergoing star formation , giving rise to the diffuse tail - like structures . this would suggest the structures should have blue colors , possibly bluer than the colors of the central galaxy although it is difficult to quantify the expected difference in color between the ` head ' and the diffuse ` tail ' because infalling material collapsing onto the central galaxy may also cause the central galaxy to experience a burst of star formation . furthermore , we would argue that the tail - like structures are transient features that will eventually disappear . as noted above , the ram pressure accelerates the clouds to relatively high velocities and , although once formed the stars are immune to effects of ram pressure , they will eventually disperse because of the velocity imparted at the time formation ( in effect , the cloud velocity at the time ) . in cluster environments , where in fact stripping is most likely to occur , the high velocity stars would give rise to a diffuse population of intraclusters stars . such a population of stars have recently been detected in fornax ( theuns & warren 1997 ) and virgo ( ferguson , tanvir & von hippel 1998 ) . as already noted , the evolution of the gas streaming out of dwarf galaxies in environments where the icm pressure is low , is very different from that of dwarfs in hot , high density regions . the outflow triggered by the first generations of supernova explosions will give rise to a mass shell that , for all practical purposes , expands away from the galaxy and carries away its gas supply . the field dwarf galaxies , therefore , are likely to experience only one short episode of star formation . the dense expanding shells and dense clumps will give rise to ly @xmath0 absorption lines in quasar spectra if lines of sight to the quasars intersect such structures . here we consider the profiles of such absorption lines and the hi column densities associated with the absorption . as seen in the simulations , almost all of the galactic gas , @xmath116 , and the swept - up igm are in the expanding shell . when the shell radius , @xmath117 , is much larger than 1kpc , the shell mass is @xmath118 where @xmath119 is the mass of the hydrogen atom , @xmath120 , @xmath121 , and @xmath122 . for simplicity , we assume that the gas has primordial abundance ( @xmath61 ) and therefore , @xmath123 . the swept - up gas mass dominates the shell mass once the shell expands beyond @xmath124kpc . if the igm density is higher , the swept - up mass can come to dominate sooner . for @xmath125 , the igm density in outer regions of the clusters or the mean igm density of the universe at @xmath126 , the swept - up mass becomes comparable to the galactic gas when @xmath127 kpc . the average hydrogen density of the shell is then @xmath128 where @xmath129 is the width of the shell , and @xmath130 . if the gas is photoionised by a metagalactic uv flux and is in ionisation equilibrium , the number density of neutral hydrogen is given by @xmath131 where @xmath132 is the hydrogen recombination rate , and @xmath133 is the photo - ionisation rate ( see , for example , black 1981 ) . if @xmath134 is the uv flux at ly limit ( @xmath135 ) and the spectrum of the background uv radiation is @xmath136 , then the neutral hydrogen number density is @xmath137 where @xmath138 is the gas temperature . the hi column density at zero impact parameter through the shell is @xmath139 the column density depends on the shell radius . when the igm density is low , @xmath140 decreases rapidly with increasing shell radius . however , once the swept - up mass becomes comparable to the galactic mass , the above equation suggests that @xmath140 becomes proportional to @xmath141 but as we have seen in figure [ fig-5 ] , the shell thickness does not remain constant . at large radii , the shell thickness too increases as mass is swept - up and this increase in the shell width slows the growth of the hi column density . figure [ fig-11 ] shows the hi column density along a line - of - sight that has impact parameter , @xmath142 , as a function of time for model -2 m . the radius of the expanding shell is indicated by the upper horizontal axis . the radius is defined as the outermost boundary of the shell . for the background uv radiation , we use @xmath143 and neglect any self - shielding of the uv flux . the hi column density through the center grows with time because of the cooling inflow of gas . on the other hand , the maximum value of the hi column density for lines with @xmath144kpc decreases as the shell radius grows . the shell motion and structure affects the shape of the absorption line . a typical line - of - sight intersects a shell in two places and therefore , one would expect to see two absorption lines separated in velocity space by @xmath145 because of the expansion of the shell . when the separation between the two lines is small , thermal broadening of the lines can , however , cause the lines to overlap and blur , giving the impression of one broad line . alternatively , the combination of the two lines can give rise to double - horn features . the absorption profiles that take into account the velocity field and thermal broadening can be calculated according to wang ( 1995 ) . figures [ fig-12 ] and [ fig-13 ] show the absorption line profiles for model -2 m at @xmath146yr ( figure [ fig-12 ] ) and for model 0 m at @xmath147yr ( figure [ fig-13 ] ) . the impact parameters are the same as in figure 11 . the double - horn line profiles caused by the two absorption line features are readily observed , particularly at low impact parameters . as the impact parameter is increased and the component of the expansion velocity along the line - of - sight becomes smaller , the separation between the two lines decreases and the overall width of the absorption feature become narrower . the double - horn feature in our plots are similar to those described by wang ( 1995 ) except that wang assumed a steady state outflow with lower gas density and relatively higher gas temperature resulting in shallower absorption lines . the results for model 0 m and -2 m can be taken as examples of what we would expect if galactic winds were responsible for ly @xmath0 clouds at @xmath148 and at @xmath149 , respectively . recent keck observations of ly @xmath0 forests by lu et al.(1998 ) show us interesting absorption lines which have double horn profile : two absorption systems at @xmath150 in the spectrum of qso1107 + 4847 and at @xmath151 in the spectrum of qso1422 + 2309 . these profiles are very much like one shown in figure [ fig-13 ] . they are associated with civ absorption lines and located in the visinity of quasars within @xmath152 . if there are nearby galaxies in a cluster around the quasars , the profiles can be explained as consequence of metal - enriched galactic outflows . ciardi & ferrara ( 1997 ) have proposed that a hot secondary halo that forms when the inner boundary of the shell re - expands back into the cavity would also produce ly @xmath0 absorption lines . the beginnings of this inward re - expansion is seen in our simulation of model 0 m . the outer radius expanded out to the radius at which we stopped all our simulations well before the secondary halo had been established . we were , therefore , not able to explore its effects on quasar spectra . finally , the fact that outflow from low - mass galaxies can extend out to distances of 40 kpc or more indicates that such galaxies may have played an important role in polluting the intergalactic medium with metals at high redshifts . this possibility was first discussed by silk , wyse & shields ( 1987 ) . the ubiquity of the carbon features observed in quasar absorption systems at @xmath153 ( cowie et al . 1995 ; tytler et al . 1995 ; songaila & cowie 1996 ) implies that metals were dispersed over a large region , if not uniformly , fairly early in the history of the universe . the fact that the generally accepted hierarchical clustering models for structure formation suggest that at high redshifts , the universe was dominated by halos with shallow potential wells and , as we have shown , the fact that the winds from these wells can spread out over large distances supports the scenario where early generations of dwarf galaxies are responsible for polluting the intergalactic medium . in this paper , we have sought to study the interaction between supernova - powered gas outflows from low - mass galaxies and the local intergalactic medium ( igm ) . we find that even if the supernova explosions are , in principle , able to expel the gas from a low - mass galaxy , they may not be able to do so if the thermal pressure of the local intergalactic medium is sufficiently high . the thermal pressure of the igm resists the outflow , eventually bringing it to a halt . the confinement radius , the radius to which the gas can expand before being brought to a halt , depends on the igm pressure as @xmath154 . the higher the igm pressure , the smaller the confinement radius . we find that the thermal pressure of the hot intracluster medium in clusters of galaxies , for example , is large enough to prevent the gas from expanding much beyond the galaxy . the interface between the outflow and quiescent igm is demarcated by a dense expanding shell formed by the gas swept - up by the outflow . once halted , the igm pressure pushes the shell back into the galaxy . the collapsing shell is susceptible to rayleigh - taylor instability resulting in non - uniform collapse of the gas . the instability enhances small perturbations in the shell , causes it to deform into tentacle - like structures and eventually fragments into small clouds . in high density , high temperature regions , the dwarf galaxies are also likely to be moving at high velocities and therefore , subject to the effects of ram pressure . when ram pressure is comparable or higher than the ambient thermal pressure , it can distort and fragment the shell into high density clouds that are then dragged away from the galaxy and carried downstream . these high density clouds are potential sites of star formation and the spatial distribution of stars newly born in these clouds will trace out a diffuse tail - like structure . we speculate that comet - like galaxies with diffuse tail seen in z=1.15 cluster 3c324 are such galaxies . the structure exhibited by these galaxies is temporary ; it will dissolve away as the stars stream away and become part of a diffuse population of stars in the intracluster environment . in contrast , the relatively unhindered outflows in low density , low temperature environments can drive the shells of swept - up gas out to distances of 40 kpc or more from the galaxy . such shells , if they intersect a quasar line - of - sight , would give rise to ly @xmath0 absorption lines of the kind seen in quasar spectra . assuming that the universe is permeated by a metagalactic uv flux , the absorption features correspond to hi column densities typically of order @xmath155 @xmath44 . at small impact parameters , the velocity field of the expanding shell gives rise to double - horned absorption profiles . this feature weakens and disappears as the impact parameter increases . finally , the fact that outflow from low - mass galaxies can extend out to distances of 40 kpc or more indicates that such galaxies may have played an important role in polluting the intergalactic medium with metals at early epochs . this research was partially carried out at the canadian institute for theoretical astrophysics . we would like to thank to m. dickinson for his helpful comments and discussions . we also thank the anonymous referee for helpful comments and poignant suggestions . i.m . acknowledges nserc for a fellowship and a.b . acknowledges support from nserc through an operating grant . 5p & p & @xmath162 & @xmath163 & @xmath42 & 0 . & 1.5 + 4p & p & @xmath164 & @xmath163 & @xmath165 & 0 . & 3.8 + 3p & p & @xmath166 & @xmath163 & @xmath167 & 0 . & 8.8 + 2p & p & @xmath168 & @xmath163 & @xmath169 & 0 . & 18.0 + 0p & p & @xmath170 & @xmath171 & @xmath167 & 0 . & @xmath172 + + 5 m & m & @xmath162 & @xmath163 & @xmath42 & 0 . & 1.5 + 4 m & m & @xmath164 & @xmath163 & @xmath165 & 0 . & 3.6 + 3 m & m & @xmath166 & @xmath163 & @xmath167 & 0 . & 8.6 + 2 m & m & @xmath168 & @xmath163 & @xmath169 & 0 . & 18.0 + 0 m & m & @xmath170 & @xmath171 & @xmath167 & 0 . & @xmath172 + -2 m & m & @xmath42 & @xmath171 & @xmath173 & 0 . & @xmath172 + + 5pw & p & @xmath162 & @xmath163 & @xmath42 & 1000 . + 4pw & p & @xmath164 & @xmath163 & @xmath165 & 1000 . + 3pw & p & @xmath166 & @xmath163 & @xmath167 & 1000 . + + 5mw & m & @xmath162 & @xmath163 & @xmath42 & 1000 . + 4mw & m & @xmath164 & @xmath163 & @xmath165 & 1000 . + 3mw & m & @xmath166 & @xmath163 & @xmath167 & 1000 . + 3mw6 & m & @xmath166 & @xmath163 & @xmath167 & 600 . + 2mw4 & m & @xmath168 & @xmath163 & @xmath169 & 400 . + + units are @xmath174 for @xmath157 ; k for @xmath158 ; @xmath175 for @xmath159 ; km s@xmath49 for @xmath160 ; and kpc for @xmath161 . + @xmath156 cooling function : p for primordial cooling and m for metal cooling .

we have carried out 2d hydrodynamical simulations in order to study the interaction between supernova - powered gas outflows from low - mass galaxies and the local intergalactic medium ( igm ) . we are specifically interested in investigating whether a high pressure igm , such as that in clusters of galaxies , can prevent the gas from escaping from the galaxy , as suggested by babul and rees ( 1992 ) . we find that this is indeed the case as long as ram pressure effects are negligible . the interface between the outflow and ambient igm is demarcated by a dense expanding shell formed by the gas swept - up by the outflow . a sufficiently high igm pressure can bring the shell to a halt well before it escapes the galaxy . galaxies in such high pressure environments are however , more likely than not , to be ploughing through the igm at relatively high velocities . hence , they will also be subject to ram pressure , which acts to strip the gas from the galaxy . we have carried out simulations that take into account the combined impact of ram pressure and thermal pressure . we find that ram pressure deforms the shell into a tail - like structure , fragments it into dense clouds and eventually drags the clouds away from the galaxy . the clouds are potential sites of star formation and if viewed during this transient phase , the galaxy will appear to have a low - surface brightness tail much like the galaxies with diffuse comet - like tail seen in z=1.15 cluster 3c324 . the stars in the tail would , in time , stream away from the galaxy and become part of the intracluster environment . in contrast , the relatively unhindered outflows in low density , low temperature environments can drive the shells of swept - up gas out to large distances from the galaxy . such shells , if they intersect a quasar line - of - sight , would give rise to ly @xmath0 absorption lines of the kind seen in quasar spectra . in addition , the fact that outflows from low - mass galaxies can extend out to distances of 40 kpc or more indicates that such galaxies may have played an important role in polluting the intergalactic medium with metals . galaxies : dwarf inter galactic medium

the central theme of this paper is , as indicated by the title , _ dimension theory_. basically , an equivalence relation @xmath4 is given on a structure @xmath3 , and our goal is to elucidate the quotient structure @xmath42 . we shall be interested in cases where @xmath3 is a _ complete lattice _ endowed with a notion of orthogonality subject to a number of axioms . to make it clear that the motivations for this are widespread and by no means confined to lattice theory , we start by discussing what is known about some fundamental examples . in chapter [ ch : clesp ] , we will apply our general theory to these examples , thus showing the improvements that it brings to them . one of the most basic examples of what could be called a `` dimension theory '' arises from measure theory . to pick a favorite , we first consider the lebesgue measure @xmath43 on the real line @xmath44 . it is defined on the boolean algebra @xmath45 of all lebesgue - measurable subsets of @xmath44 . however , it fails total additivity of measure , for every subset of @xmath44 is the union of singletons , which have lebesgue measure zero . to bring back total additivity , the standard way is to say that @xmath43 is defined not on @xmath45 , but on the quotient algebra @xmath46 , where @xmath47 is the ideal of null sets . the boolean algebra @xmath14 and the resulting map from @xmath14 to @xmath48 $ ] , which we still denote by @xmath43 , have the following properties : * @xmath14 is a _ complete _ boolean algebra . * the map @xmath43 is _ unrestrictedly additive _ , that is , the following equality holds : @xmath49 for any _ disjoint _ family @xmath50 of elements of @xmath14 . the notation @xmath51 stands for the _ join _ ( i.e. , supremum ) of the set @xmath52 in @xmath14 . * for all @xmath5 , @xmath15 , if @xmath6 is a translate of @xmath5 ( that is , @xmath53 for some real number @xmath54 ) , then @xmath55 . rule ( b ) above seems somehow puzzling at first glance , because of the apparent possibility of an uncountable index set @xmath56 . however , since the boolean algebra @xmath14 is countably saturated , all infinite joins in @xmath14 are , really , _ countable _ joins , so that , in ( b ) , all the @xmath57-s are majorized by the join of countably many of them . for @xmath5 , @xmath15 , we define the relation @xmath25 to hold , if there are disjoint families @xmath50 and @xmath58 of elements of @xmath14 such that @xmath59 and @xmath60 , and @xmath61 is a translate of @xmath57 , for all @xmath62 . it is not difficult to verify that @xmath4 is an _ equivalence relation _ on @xmath14 . furthermore , by ( b ) and ( c ) above , @xmath25 implies that @xmath55 , for all @xmath5 , @xmath15 . it is harder to verify that the converse of the above fact also holds , namely : @xmath55 implies that @xmath25 , for all @xmath5 , @xmath15 . this fact is due to s. banach and a. tarski , see @xcite , or ( * ? ? ? * theorem 9.17 ) . hence the quotient set @xmath63 is isomorphic , _ via _ the measure @xmath43 , to the interval @xmath48 $ ] . a moment s reflection shows that @xmath63 can be endowed with a _ partial addition _ , defined by the rule @xmath64+[y]=[x\vee y],\quad\text{for all disjoint } x,\,y\in b,\ ] ] that endows it with a structure of _ partial commutative monoid _ ( see definition [ d : partcm ] ) , and that the measure @xmath43 factors through an isomorphism of partial monoids between @xmath63 and @xmath48 $ ] . we see in this particular case that @xmath63 is a _ total _ monoid , that is , the addition of @xmath63 is defined everywhere . now let us consider the _ converse _ of the above paragraph . that is , we are given a boolean algebra @xmath14 , endowed with an equivalence relation @xmath4 , and we wish to find the structure of @xmath63 . while this problem in full generality can lead to almost any structure , we focus the study by making the following assumptions on @xmath14 and @xmath4 , that are satisfied for the example above : * @xmath14 is a complete boolean algebra . * @xmath65 implies that @xmath66 , for all @xmath67 . * ( see axiom ( l6 ) of definition [ d : measchlatt ] ) the relation @xmath4 is _ unrestrictedly refining _ , that is , for every @xmath68 and every disjoint family @xmath69 of elements of @xmath3 , if @xmath70 , then there exists a decomposition @xmath71 , with @xmath72 disjoint , such that @xmath73 for all @xmath62 . * ( see axiom ( l7 ) of definition [ d : measchlatt ] ) the relation @xmath4 is _ unrestrictedly additive _ , that is , for all disjoint families @xmath72 and @xmath69 of elements of @xmath3 , if @xmath73 for all @xmath62 , then @xmath74 . the most basic example of this situation is for @xmath75 , the powerset algebra of an infinite set @xmath76 , where @xmath4 is the relation of _ equipotency _ on subsets of @xmath76 , that is , @xmath77 if and only if there exists a bijection from @xmath78 onto @xmath79 . if @xmath40 is the unique ordinal such that @xmath80 , then @xmath42 is isomorphic to the monoid @xmath81 endowed with the addition that extends the natural addition of the set @xmath82 of nonnegative integers and such that @xmath83 , for all @xmath84 and all ordinals @xmath54 , @xmath85 such that @xmath86 , see page . so , if @xmath87 is the map defined by the rule @xmath88 , for all @xmath89 , then @xmath90 factors through @xmath4 , thus defining an isomorphism from @xmath63 onto @xmath91 . as we shall see in this paper , it is still possible , in the general case , to obtain a `` measure '' @xmath90 on @xmath14 such that @xmath63 is isomorphic to the range of @xmath90 . the range of the measure @xmath90 is not necessarily @xmath48 $ ] and not even some @xmath91 ( as in the example above ) , but rather a certain set of continuous functions from a complete boolean space ( i.e. , extremally disconnected compact hausdorff topological space ) @xmath76 to a monoid of the form @xmath92 ( or a submonoid of this monoid ) . a similar result is achieved by d. maharam in @xcite , in a slightly different context for instance , all sums and joins are countable joins , while @xmath14 satisfies the countable chain condition . this is not the only restriction imposed in maharam s work , as , for example , axiom iii , page 281 in @xcite , that rules out what we will call later the `` type iii '' case . let @xmath17 be a ( von neumann ) regular , right self - injective ring , let @xmath93 be a nonsingular injective right @xmath17-module . we order the set @xmath3 of all direct summands of @xmath93 by inclusion and we endow it with the relation of isomorphism , @xmath94 . the _ dimension theory _ of @xmath93 is the study of the structure of @xmath95 . we say that a family @xmath96 of elements of @xmath3 is _ orthogonal _ , if the sum of the submodules @xmath97 is a direct sum . we recall some fundamental properties of @xmath3 and @xmath94 ( references will be given in section [ s : rsireg ] ) : * @xmath3 is a complete lattice , that is , every subset of @xmath3 has a supremum . + it is known that the infimum of a family @xmath96 of elements of @xmath3 is their _ intersection _ , @xmath98 . * @xmath3 is _ complemented _ , that is , every element @xmath78 of @xmath3 has a complement ( that is , an element @xmath79 of @xmath3 such that @xmath99 ) . * @xmath3 is _ meet - continuous _ , that is , for every @xmath100 and every upward directed family @xmath101 of elements of @xmath3 , the following equality holds : @xmath102 * @xmath3 is _ modular _ , that is , the equality @xmath103 holds , for all @xmath78 , @xmath79 , @xmath104 such that @xmath105 . * ( see axiom ( l6 ) of definition [ d : measchlatt ] ) the relation @xmath94 is _ unrestrictedly refining _ , that is , for every @xmath100 and every orthogonal family @xmath101 of elements of @xmath3 , if @xmath106 , then there exists an orthogonal decomposition @xmath107 such that @xmath108 for all @xmath62 . * ( see axiom ( l7 ) of definition [ d : measchlatt ] ) the relation @xmath94 is _ unrestrictedly additive _ , that is , for all orthogonal families @xmath96 and @xmath101 of elements of @xmath3 , if @xmath108 for all @xmath62 , then @xmath109 . we observe that the supremum in @xmath3 of a family @xmath96 of elements of @xmath3 is not given by the sum of submodules @xmath110 , but by its _ injective hull _ , @xmath111 ( which can be identified with a unique submodule of @xmath93 because @xmath93 is injective and nonsingular ) . as in subsection [ su : ameasth ] , the quotient set @xmath95 can be endowed with a structure of _ partial commutative monoid _ , under the addition given by the rule @xmath112+[y]=[x\oplus y]\quad\text{if } x\cap y={\{0\}},\ ] ] for all @xmath78 , @xmath113 . essentially by using axioms ( 1)(6 ) above , the structure of @xmath95 has been completely elucidated in several particular cases . for example , in case @xmath93 is _ directly finite _ ( i.e. , @xmath93 is not isomorphic to any proper direct summand of itself ) , @xmath95 is isomorphic to a lower subinterval ( with respect to the componentwise ordering ) of a monoid of the form @xmath114 where @xmath37 and @xmath38 are complete boolean spaces ; see chapter 11 in k.r . goodearl and a.k . boyle @xcite . in the general case , there are a monoid @xmath93 of the form given by and a direct power @xmath115 of a monoid of the form @xmath116 ( for a certain ordinal @xmath40 ) such that @xmath95 embeds into @xmath117 , see chapters 12 and 13 in @xcite , and the variations in ( * ? ? ? * chapter 12 ) . further results along these lines were obtained by c. busqu @xcite , who showed , in particular , that the second factor of the embedding above , namely the map @xmath118 , actually sends @xmath95 to @xmath119 for a suitable complete boolean space @xmath76 ( containing @xmath120 ) ( * ? ? ? * proposition 4.7 ) . however , these embeddings do not provide an isomorphism of @xmath95 onto a _ lower subset _ of a monoid of continuous functions . the difficulties are already visible in case @xmath17 is a complete boolean algebra ( viewed as a ring ) , due to an example of k. eda @xcite : there exists a complete boolean algebra @xmath17 such that the injective hull of the free @xmath17-module of rank @xmath121 contains a direct sum of @xmath122 copies of itself ( see the discussion of problem 18 in @xcite ) . here the image of the embedding obtained from @xcite , @xcite , and @xcite contains a function with all values at least @xmath123 , but not the constant function with value @xmath122 . for elements @xmath124 , @xmath125 , and @xmath126 in a lattice @xmath3 with zero , we say that @xmath127 , if @xmath128 and @xmath129 . we say that @xmath3 is _ sectionally complemented _ , if for all @xmath124 , @xmath130 such that @xmath131 , there exists @xmath132 such that @xmath133 . if @xmath3 is _ modular _ , that is , the implication @xmath134 holds , for all @xmath5 , @xmath6 , @xmath7 , then the partial operation @xmath135 gives @xmath3 a structure of _ partial commutative monoid_. completeness and meet - continuity of @xmath3 are defined as in ( 1 ) and ( 3 ) of subsection [ su : rsireg ] . so , in particular , if @xmath93 is a nonsingular injective right module over a right self - injective regular ring @xmath17 , then the lattice of all direct summands of @xmath93 is complete , meet - continuous , sectionally complemented , and modular . the classical von neumann _ continuous geometries _ , see j. von neumann @xcite or f. maeda @xcite , are obtained by adding the conditions that @xmath3 has a unit ( that is , a largest element ) and is join - continuous . at this point , we seem to be stymied because of the following problem . we can not claim outright that our lattice - theoretical context could lead to generalizations of subsection [ su : rsireg ] , for there is no such thing _ a priori _ as `` isomorphism of submodules '' between the elements of @xmath3 . in the case of continuous geometries , it is easy to remedy this by replacing isomorphism by _ perspectivity _ , which turns out to be _ transitive _ ( this is a difficult result , due to j. von neumann @xcite ) . elements @xmath124 and @xmath125 of a lattice @xmath3 are _ perspective _ , in notation @xmath136 , if there exists @xmath132 such that @xmath137 and @xmath138 . for continuous geometries , the structure of @xmath42 is completely understood , see @xcite and , for the general , reducible case , t. iwamura @xcite namely , @xmath42 is isomorphic to a lower segment of the positive cone of a dedekind complete lattice - ordered group . the paper j. harding and m.f . janowitz @xcite shows how a reducible continuous geometry can be represented as the space of continuous sections of a bundle of irreducible continuous geometries , thus shedding more light on the transition from irreducible continuous geometries to reducible ones . however , for a general complete , meet - continuous , sectionally complemented , modular lattice @xmath3 , the relation of perspectivity @xmath4 on @xmath3 is not transitive as a rule see , for example , the obvious case where @xmath3 is the subspace lattice of an infinite - dimensional vector space over a field . hence , we have to find a better candidate than @xmath4 to replace isomorphism of submodules . a natural guess is of course the _ transitive closure _ @xmath139 of @xmath4 ( usually called _ projectivity _ ) , but this relation fails to be additive , as defined in axiom ( l7 ) of definition [ d : measchlatt ] , and as isomorphism of submodules should be . the final answer is , in fact , nontrivial , and it follows from the theory of _ normal equivalences _ introduced by the second author in chapters 1013 of @xcite . namely , there is ( fortunately ! ) _ exactly one _ `` reasonable '' candidate for isomorphism , and it is the binary relation @xmath140 on @xmath3 defined by the rule @xmath141 for all @xmath124 , @xmath130 . the transitivity of @xmath140 is proved in theorem 13.2 of @xcite , while the complete additivity of @xmath140 ( axiom ( l7 ) of definition [ d : measchlatt ] , see also item ( 6 ) of subsection [ su : rsireg ] ) is proved as in proposition 13.9 of @xcite by replacing countable families by arbitrary families . the quotient @xmath142 is then a lower subset of the so - called _ dimension monoid _ @xmath143 of @xmath3 , which , as its name indicates , is a ( commutative ) monoid . the dimension theory of @xmath3 is elucidated here in theorem [ t : dimmonrcmlatt ] . in the context of subsection [ su : rsireg ] , that is , @xmath3 is the lattice of all direct summands of a given nonsingular injective right module over a right self - injective regular ring , it is then the case that @xmath140 is identical to submodule isomorphism on @xmath3 , see lemma 10.2 and theorem 13.2 of @xcite . of crucial importance for all the proofs of these results is a result of i. halperin and j. von neumann @xcite that states that @xmath144 and @xmath16 implies that @xmath25 , for all @xmath5 , @xmath24 . this result is extended in @xcite to countably meet - continuous lattices , where it is used to prove that the quotient @xmath142 is then a so - called _ generalized cardinal algebra _ , see proposition 13.10 of @xcite . however , even in case the lattice - theoretical version of direct finiteness ( see subsection [ su : rsireg ] ) holds in @xmath3 , no analogue of an embedding into monoids of the form had been found before the present work . we recall that an _ aw*-algebra _ is a c*-algebra @xmath18 such that the right annihilator of any subset @xmath78 of @xmath18 has the form @xmath145 , for a projection @xmath26 of @xmath18 ( a _ projection _ of @xmath18 is an element @xmath26 of @xmath18 such that @xmath146 ) . we denote by @xmath3 the set of projections of @xmath18 . let @xmath131 hold , if @xmath147 ( equivalently , @xmath148 ) , for all @xmath124 , @xmath130 . thus @xmath149 is a partial ordering on @xmath3 . orthogonality of any projections @xmath124 and @xmath125 , in notation @xmath150 , is defined by @xmath151 , and ( murray - von neumann ) equivalence is defined by the rule @xmath152 much of the structure of @xmath3 was developed axiomatically by i. kaplansky in his monograph @xcite . some of the axioms and methods we use were inspired by kaplansky s work , as was the structure theory for nonsingular injective modules constructed by goodearl and boyle @xcite . readers familiar with either of those works will recognize the parallels below . again , the quotient @xmath42 can be endowed with a structure of partial commutative monoid , where addition is given by the rule @xmath153+[b]=[a+b],\quad\text{if } ab=0,\text { for all } a,\,b\in l,\ ] ] where @xmath154 $ ] denotes the @xmath4-equivalence class of a projection @xmath26 of @xmath18 . the amount of known general information on the structure of @xmath42 is more fragmentary than for the examples considered in previous sections , due to fewer axioms satisfied . for example , the analogues of properties ( 3 ) ( meet - continuity ) and ( 4 ) ( modularity ) considered in subsection [ su : rsireg ] fail for projections of aw*-algebras as a rule . in f.j . murray and j. von neumann @xcite , a @xmath48$]-valued `` dimension function '' is constructed on the projections of any _ w*-factor _ ( i.e. , indecomposable von neumann algebra ) ; kaplansky showed that the same construction could be carried out for aw*-factors . still in the indecomposable case , it is known that the closed two - sided ideals are well - ordered , see f.b . wright @xcite . most of what was known about @xmath42 in the general case could be obtained from more general , often lattice - theoretical works that we shall discuss now . a common feature of the structures considered in subsections [ su : ameasth][su : projaw * ] is that they all involve a complete , sectionally complemented lattice @xmath3 , a binary relation @xmath2 on @xmath3 , and an equivalence relation @xmath4 on @xmath3 . it has been observed early that even apart from the classical study of continuous geometries , the dimension theory of a given structure could be done by just studying the associated structure @xmath155 . furthermore , these structures will be _ ordered _ structures , so that we shall write @xmath35 instead of @xmath155 : * it is in s. maeda @xcite that the most general axiomatization of the structures @xmath35 is given . it holds for all the examples considered in subsections [ su : ameasth][su : projaw * ] , and this allows to construct `` dimension functions''corresponding to the measures of subsection [ su : ameasth]on @xmath3 that , in the `` finite '' case , separate the elements of @xmath3 . * in l.h . loomis @xcite , another axiomatization is used , that involves an _ orthocomplementation _ on @xmath3 , thus it does not apply to the examples considered in subsections [ su : rsireg ] and [ su : cmsmlatt ] . * in p.a . fillmore @xcite , a further axiomatization of the structures @xmath156 is introduced , that does not assume completeness of @xmath3 but rather _ countable _ completeness , and that assumes an orthocomplementation ( thus , again , it does not encompass subsections [ su : rsireg ] and [ su : cmsmlatt ] ) . one of the main results is that the structure @xmath42 is a generalized cardinal algebra ( as in subsection [ su : cmsmlatt ] ) . furthermore , under some countability assumptions , @xmath3 is complete and @xmath42 is isomorphic to a lower subset of a monoid of the form , see ( * ? ? ? * theorem 3.12 ) . nevertheless , in each class of examples considered in subsections [ su : ameasth][su : projaw * ] , some of the dimension - theoretical properties that one could have expected to hold were still missing from the known results . for example , there has been no general treatment of the reducible type iii case ; it was seemingly not even clear whether or not it had to be treated as a pathology . in view of the various examples presented in section [ s : backgr ] and of what is known about them , the main goals of this paper are the following : * to capture in a convenient set of axioms the various properties of the structures @xmath0 encountered in these examples . this set of axioms should be sufficient to develop a _ complete _ dimension theory of these structures , that is , a complete description of the structures @xmath42 , without additional assumptions such as finiteness or chain conditions . * to develop a set of monoid - theoretical axioms that should be satisfied by the structures @xmath42 . * although the set of axioms obtained in ( 2 ) is quite complicated , our third goal will be to give a simple description of the structures satisfying the axioms of ( 2 ) in terms of continuous functions on complete boolean spaces . we shall now give some details about our road to these goals . the relevant structures @xmath35 will be called _ espaliers _ , see definition [ d : measchlatt ] . the axiom system defining espaliers is stronger than the axiom system @xmath157 , @xmath158 , , @xmath159 , @xmath160 , , @xmath161 considered by s. maeda in @xcite . nevertheless , these axioms are sufficient for our purposes for instance , all the examples considered in section [ s : backgr ] are espaliers . the only drastic generalization that we will introduce is to state that the underlying partial ordering of an espalier @xmath35 defines a _ partial _ , as opposed to total , lattice , so that for elements @xmath124 and @xmath125 of @xmath3 , the meet ( i.e. , infimum ) @xmath162 of @xmath163 always exists , but the join ( i.e. , supremum ) of @xmath163 exists only in case @xmath163 is majorized . this small generalization affects neither the proofs nor even the results the structures @xmath42 are partial structures anyway and it paves the way for further algebraic constructions on espaliers , such as _ amalgamation_. in parallel to this , we shall develop a system of monoid - theoretical axioms , ( m1)(m6 ) ( see definition [ d : dimint ] ) , that captures the structures ( partial commutative monoids ) @xmath42 , for an espalier @xmath3 . this axiom system is rather complicated , but it completely isolates what monoid theory we need to understand the structures @xmath42 . among these axioms is a variant of _ conditional completeness _ ( see axiom ( m2 ) ) , that is , every nonempty subset admits an infimum for the algebraic ( pre)ordering ( see definition [ d : algpr ] ) , but there are other , less natural - looking axioms , such as ( m6 ) . the partial commutative monoids satisfying axioms ( m1)(m6 ) will be called _ continuous dimension scales_. they are unrelated to h. lin s `` continuous scales '' introduced in @xcite . the relation between espaliers and continuous dimension scales is then given by the following ( see theorem [ t : dimesp ] ) . theorem a let @xmath35 be an espalier . then the partial commutative monoid @xmath42 of all @xmath4-equivalence classes of elements of @xmath3 is a continuous dimension scale . at first glance , theorem a may appear as the ultimate goal of this paper . however , it provides only a list of properties of the partial monoids @xmath42 , without giving any representation in terms of known structures . moreover , although the axioms describing the structure of espalier seem to be almost the weakest possible to obtain a complete dimension theory , and thus , in some sense , unavoidable , this might not seem to be the case _ a priori _ for the axioms describing continuous dimension scales . we counter this by proving that there are no `` missing '' axioms for continuous dimension scales relative to espaliers . theorem b a partial commutative monoid @xmath13 is a continuous dimension scale if and only if @xmath164 for some espalier @xmath35 . theorem b follows from the fact that most of our classes of examples of espaliers are _ universal _ in the sense that arbitrary continuous dimension scales can be represented ( isomorphically ) as lower subsets of the continuous dimension scales @xmath42 arising from these examples see , for example , theorems [ t : measduniv ] , [ t : meetcontinduniv ] , [ t : l(r)duniv ] , [ t : l(a)duniv ] . as for a concrete representation of continuous dimension scales , we exhibit them as lower subsets ( for the algebraic preordering ) of product spaces of the form @xmath165 where @xmath37 , @xmath38 , and @xmath39 are complete boolean spaces and , for any ordinal @xmath40 , the monoids @xmath91 , @xmath166 , and @xmath167 are defined as @xmath41 endowed with the natural addition and ordering , together with the interval topology ( see section [ s : notterm ] ) . see page for more details . theorem c let @xmath13 be a partial commutative monoid . then @xmath13 is a continuous dimension scale if and only if it can be embedded as a lower subset into a product monoid of the form @xmath165 where @xmath37 , @xmath38 , and @xmath39 are complete boolean spaces . a more precise version of theorem c is formulated in theorem [ t : embdimint ] . the concrete version of theorem b is that any lower subset of a monoid of the form @xmath165 can be represented as @xmath42 , for a suitable espalier @xmath3 . more precisely , we show that @xmath3 may arise from each of the above contexts abstract measure theory , nonsingular injective modules over self - injective regular rings , meet - continuous complemented modular lattices , and projection lattices of aw*-algebras , see sections [ s : ameasth][s : projaw * ] . for projection lattices of w*-algebras , there is an additional restriction on the spaces @xmath37 , @xmath38 , @xmath39namely , they are _ hyperstonian _ , see corollary [ c : w*hyperduniv ] . in addition , it is worth noticing that although the embedding in theorem c is not unique as a rule , it is determined by the condition that it `` commutes with projections '' and its value at the elements of a _ finitary unit _ of @xmath13 ( definition [ d : finun ] ) , see theorem [ t : uneps ] . finally , all this extends to `` continuous dimension scales '' that are no longer _ sets _ , but rather _ proper classes_. the corresponding common extensions of the abovementioned `` existence '' and `` uniqueness '' statements hold , and they are presented in theorem [ t : genembdi ] . in order to make the results and methods of this paper accessible to the widest audience , we have avoided the use of forcing and boolean - valued models for most proofs . exceptions to this rule are the proofs of d - universality for the classes of boolean espaliers ( theorem [ t : measduniv ] ) and espaliers of projections of aw*-algebras ( theorem [ t : bigaw*drng ] ) , as reasonable `` forcing - free '' proofs do not seem to be available . disjoint unions of sets will be denoted by @xmath168 , @xmath169 , so that , for example , @xmath170 means that @xmath171 and that @xmath172 , for all distinct @xmath173 , @xmath174 . following the usual set - theoretical terminology , we denote by @xmath175 the set of all natural numbers . we identify any natural number @xmath176 with @xmath177 . any ordinal @xmath54 is identified with the set of all ordinals less than @xmath54 . a cardinal is an initial ordinal . following well - established set - theoretical practice , for an ordinal @xmath54 , the notations @xmath178 and @xmath179 both denote the @xmath54-th infinite cardinal , except that the first one is viewed as an ordinal while the second one is viewed as a cardinal . if @xmath180 is a partially preordered set , a subset @xmath78 of @xmath180 is a _ lower subset _ ( resp . , _ upper subset _ ) of @xmath180 if @xmath181 and @xmath182 ( resp . , @xmath183 ) implies that @xmath183 ( resp . , @xmath182 ) , for all @xmath5 , @xmath184 . for an element @xmath124 of @xmath180 , we denote by @xmath185 $ ] ( resp . , @xmath186 ) the lower subset ( resp . , upper subset ) of @xmath180 generated by @xmath124 . a subset @xmath78 of @xmath180 is _ coinitial _ , if @xmath187 is equal to @xmath180 . if @xmath180 has a least element @xmath188 , a subset @xmath78 of @xmath180 is _ dense _ in @xmath180 , if @xmath189 is coinitial in @xmath190 . we say that @xmath78 is an _ antichain _ of @xmath180 , if @xmath191 and @xmath185\cap(b]={\{0\}}$ ] for any distinct @xmath124 , @xmath192 . if @xmath78 and @xmath79 are subsets of @xmath180 , we abbreviate the statement @xmath193 by @xmath194 . if @xmath195 ( resp . , @xmath196 ) , we write @xmath197 ( resp . , @xmath198 ) . the _ interval topology _ on @xmath180 is the least topology of @xmath180 for which all the intervals of the form @xmath185 $ ] or @xmath186 , for @xmath199 , are closed . we shall consider the interval topology only in the totally ordered , complete case . the relevant result is then the following , see , for example , @xcite . [ p : frink ] let @xmath200 be a totally ordered set . we suppose that @xmath201 is _ complete _ , that is , every subset of @xmath201 has an infimum in @xmath201 . then the interval topology of @xmath201 is compact hausdorff . if @xmath202 is a partially ordered group , @xmath203 denotes the positive cone of @xmath202 . we put @xmath204 . we say that @xmath202 is _ directed _ , if is upward directed as a partially ordered set ; we say that @xmath202 satisfies the _ interpolation property _ , if for all @xmath205 , @xmath206 , @xmath207 , @xmath208 such that @xmath209 , there exists @xmath210 such that @xmath211 . we say that @xmath202 is _ dedekind complete _ , if it is directed and every nonempty majorized subset of @xmath202 has a supremum . it is well - known that every dedekind complete partially ordered group is abelian , see , for example , ( * ? ? ? * theorem 28 ) . we shall write such groups using additive notation . for any point @xmath5 in a topological space @xmath76 , we denote by @xmath212 ( or @xmath213 if @xmath76 is understood ) the set of all open neighborhoods of @xmath5 in @xmath76 . for a subset @xmath78 of @xmath76 , we denote by @xmath214 the _ interior _ of @xmath78 and by @xmath215 the _ closure _ of @xmath78 in @xmath76 . if @xmath216 is a totally ordered set , endowed with its interval topology , a map @xmath217 is _ lower semicontinuous _ , _ upper semicontinuous _ ) , if the set @xmath218 ( resp . , @xmath219 ) is closed , for every @xmath220 . a topological space @xmath76 is _ extremally disconnected _ , if the closure of every open subset of @xmath76 is open . we use the terminology _ complete boolean space _ as a synonym for _ extremally disconnected compact hausdorff topological space_. see section [ s : cmpbooleanspc ] for more detail on these concepts . complete boolean spaces are also called _ stone spaces _ ( or _ stonian spaces _ ) in the literature . many monoid - theoretical objects we shall deal with through this paper are not monoids , but just _ partial _ monoids . the following fundamental example provides us with a large supply of partial monoids . [ ex : fundpartmon ] let @xmath221 be a commutative monoid . for a subset @xmath13 of @xmath93 satisfying the two following properties 1 . @xmath222 ; 2 . @xmath223 implies that @xmath5 , @xmath224 , for all @xmath5 , @xmath225 , we endow @xmath13 with the partial addition @xmath226 defined by @xmath227 for any @xmath124 , @xmath228 . we call @xmath13 a _ partial submonoid _ of @xmath93 . observe that we do not merely consider _ all _ subsets of @xmath93 , but only those that satisfy the conditions ( i ) and ( ii ) above they are exactly the nonempty lower subsets of @xmath93 for the _ algebraic preordering _ of @xmath93 , see definition [ d : algpr ] . it turns out that the properties of partial submonoids of commutative monoids are captured by the following definition . [ d : partcm ] a _ partial commutative monoid _ is a structure @xmath229 , where @xmath230 is a partial binary operation on @xmath13 which satisfies the following properties : * @xmath230 is _ associative _ , that is , for all @xmath124 , @xmath125 , @xmath231 , @xmath232 is defined if and only if @xmath233 is defined , and then , both have the same value . * @xmath230 is _ commutative _ , that is , for all @xmath124 , @xmath228 , @xmath234 is defined if and only if @xmath235 is defined , and then , both have the same value . * there exists an element , denoted by @xmath188 ( _ necessarily unique _ ) , of @xmath13 such that @xmath236 , for all @xmath237 . we generalize to this context the classical definition of the algebraic preordering on a commutative monoid . [ d : algpr ] let @xmath229 be a partial commutative monoid . the _ algebraic preordering _ on @xmath13 is the ( reflexive , transitive ) binary relation @xmath149 defined on @xmath13 by the rule @xmath238 an element @xmath239 is an _ order - unit _ , if every element of @xmath13 lies below @xmath240 ( defined ) , for some @xmath84 . the following definition is of course a direct generalization of example [ ex : fundpartmon ] . [ d : partsubm ] a _ partial submonoid _ of a partial commutative monoid @xmath13 is a lower subset @xmath241 of @xmath13 ( for the algebraic preordering of @xmath13 ) containing @xmath188 as an element , endowed with the partial addition defined by @xmath242 we omit the trivial proof of the following result . every partial submonoid of a partial commutative monoid is a partial commutative monoid . the following class of embeddings will be of special interest . [ d : lowemb ] let @xmath18 and @xmath14 be partial commutative monoids , and let@xmath243 . we say that @xmath244 is a _ lower embedding _ , if the following conditions hold : 1 . @xmath244 is a homomorphism of partial monoids . @xmath244 is one - to - one , and @xmath245 implies that @xmath181 , for all @xmath5 , @xmath246 . 3 . the range of @xmath244 is a lower subset of @xmath14 , with respect to the algebraic preordering of @xmath14 . hence , a lower embedding from @xmath18 into @xmath14 identifies @xmath18 with a lower subset ( with respect to the algebraic preordering ) of @xmath14 , endowed with the structure of partial submonoid as in definition [ d : partsubm ] . the following result shows that all partial commutative monoids can be obtained from example [ ex : fundpartmon ] . [ p : pcmcm ] every partial commutative monoid admits a lower embedding into a commutative monoid . let @xmath229 be a partial commutative monoid . let @xmath247 be any object such that @xmath248 , and put @xmath249 . we define on @xmath250 the binary operation @xmath251 defined by the rule @xmath252 it is easy to verify that @xmath253 is a commutative monoid and that the inclusion map from @xmath13 into @xmath250 is a lower embedding . a noticeable effect of proposition [ p : pcmcm ] is to make computations in partial commutative monoids much more convenient . for example , suppose that we have to prove that an equality of the form @xmath254 holds in a given partial commutative monoid @xmath13 , _ via _ a sequence of equalities @xmath255 , where @xmath18 , @xmath14 , and the @xmath256 are finite sums of elements of @xmath13 . we assume in addition that the sum defining @xmath18 is defined in @xmath13 . instead of having to verify that all the terms @xmath256 are defined in @xmath13 and pairwise equal , it is sufficient to argue in @xmath250 that @xmath255 , without having to worry about undefined terms . this applies , in particular , to the following lemmas [ l : leqstilldef ] , [ l : invpermplus ] , and [ l : setassoc ] . [ l : leqstilldef ] let @xmath229 be a partial monoid , with algebraic preordering @xmath149 . let @xmath124 , @xmath125 , @xmath257 , @xmath258 . if @xmath234 is defined and @xmath259 and @xmath260 , then @xmath261 is defined , and @xmath262 . in any given partial commutative monoid @xmath13 , we define inductively the statement @xmath263 to hold , for @xmath264 , @xmath124 , @xmath205 , , @xmath265 , as follows : 1 . @xmath266 if and only if @xmath267 . 2 . @xmath268 if and only if @xmath269 . if the operation of @xmath13 is denoted by @xmath135 , then we shall write @xmath270 instead of @xmath271 . [ l : invpermplus ] let @xmath229 be a partial commutative monoid . for all @xmath264 , all @xmath124 , @xmath205 , , @xmath265 , and every permutation @xmath272 of @xmath176 , @xmath273 by lemma [ l : invpermplus ] , for a finite set @xmath56 and elements @xmath124 , @xmath274 ( for @xmath62 ) of @xmath13 , we can define unambiguously the statement @xmath275 to hold , if @xmath276 , where @xmath176 is the cardinality of @xmath56 and @xmath272 is any bijection from @xmath176 onto @xmath56 . [ l : setassoc ] let @xmath229 be a partial commutative monoid . let @xmath56 and @xmath277 be finite sets , let @xmath278 be a surjective map , let @xmath72 be a family of elements of @xmath13 , and let @xmath237 . then the following are equivalent : 1 . 2 . for all @xmath279 , the term @xmath280 is defined , and , if we denote its value by @xmath281 , then @xmath282 . [ d : refppty ] we say that a partial commutative monoid @xmath229 has the _ refinement property _ , or is a _ partial refinement monoid _ , if for all @xmath205 , @xmath206 , @xmath207 , @xmath283 such that @xmath284 , there are elements @xmath285 in @xmath13 , for @xmath173 , @xmath286 , such that the equalities @xmath287 and @xmath288 hold , for all @xmath289 . the information contained in the equalities @xmath287 and @xmath288 for all @xmath289 will often be condensed in the format of a _ refinement matrix _ as follows : @xmath290 these notations can be easily generalized to refinement matrices of arbitrary , finite or even infinite , dimensions . these notations are also very widely used in @xcite . refinement monoid _ as a commutative monoid satisfying the refinement property . in a spirit similar to proposition [ p : pcmcm ] , we shall now prove ( see proposition [ p : prmrm ] ) that every partial refinement monoid can be obtained as a lower subset of a refinement monoid . the proof of proposition [ p : pcmcm ] does not apply for this result , because @xmath250 fails in general to satisfy refinement even if @xmath13 has refinement . we shall use instead a procedure adapted to refinement monoids . [ p : prmrm ] every partial refinement monoid @xmath13 admits a lower embedding into a refinement monoid @xmath291 . in addition , one can take @xmath291 to be generated by @xmath13 as a monoid , and such that the canonical embedding from @xmath13 into @xmath291 is universal among the homomorphisms of partial monoids from @xmath13 to commutative monoids . the following construction is a particular case of the construction presented in chapter 4 of @xcite with the notations used there , @xmath292 . however , in this context , a direct verification is easy , so we give an outline here . let @xmath13 be a partial refinement monoid . we endow the set @xmath293 of all finite , nonempty sequences of elements of @xmath13 with the binary relation @xmath140 defined by the rule @xmath294 by using the refinement property in @xmath13 , it is not difficult to verify that @xmath140 is an equivalence relation on @xmath293 . for any @xmath295 , we denote by @xmath296 $ ] the equivalence class of @xmath297 modulo @xmath140 . we endow the quotient @xmath298 with the binary addition @xmath230 defined by @xmath299+[t]=[s{\mathbin{{}^{\frown}}}t],\text { for all } s,\,t\in\mathbb{s},\ ] ] where @xmath300 denotes the _ concatenation _ of @xmath297 and @xmath301 . it is straightforward to verify that @xmath291 , endowed with @xmath230 , is a refinement monoid . for any @xmath237 , we denote by @xmath302 the equivalence class modulo @xmath140 of the finite sequence @xmath303 of length one . then @xmath304 is the zero element of @xmath291 , and @xmath305 is a lower embedding from @xmath13 into @xmath291 . for the remainder of the proof , we shall identify @xmath13 with its image in @xmath291 under the embedding @xmath305 . thus the elements of @xmath291 are exactly the finite sums @xmath306 , where @xmath307 and @xmath205 , , @xmath308 , and the equality @xmath309 holds if and only if there exists a refinement matrix of the form @xmath310 for some elements @xmath285 ( for @xmath311 and @xmath312 ) of @xmath13 . obviously @xmath291 is generated by @xmath13 as a monoid . now we verify the second assertion of proposition [ p : prmrm ] . let @xmath93 be a commutative monoid and let @xmath313 be a homomorphism of partial monoids . let @xmath314 be the map defined by the rule @xmath315 then @xmath316 , @xmath317 , and @xmath318 implies that @xmath319 , for all @xmath297 , @xmath320 . hence @xmath321 can be factored through @xmath140 , thus yielding a homomorphism @xmath322 that extends @xmath323 . since @xmath13 generates @xmath291 as a monoid , @xmath324 is the only homomorphism with this property . for any partial refinement monoid @xmath13 , we take @xmath291 to be the refinement monoid having all the properties described in proposition [ p : prmrm ] . by the given universal property , @xmath291 is unique up to isomorphism . we will also identify @xmath13 with its canonical image in @xmath291 . our next lemma collects some basic information about @xmath291 . for @xmath325 , we put @xmath326 [ d : conical ] a partial refinement monoid @xmath13 is _ conical _ , if @xmath327 implies that @xmath328 , for all @xmath5 , @xmath224 . in other words , @xmath329 implies that @xmath66 , for all @xmath330 . [ l : basicrefs ] let @xmath13 be a partial refinement monoid . then the following assertions hold : 1 . @xmath331 is a lower subset of @xmath291 , for all @xmath325 . if @xmath13 is cancellative , then @xmath291 is cancellative . 3 . if @xmath13 is conical , then @xmath291 is conical . \(i ) is an easy consequence of refinement in @xmath291 . \(ii ) folklore . a proof can be found in lemma 3.6 in @xcite . \(iii ) let @xmath5 , @xmath332 such that @xmath327 . write @xmath333 and @xmath334 for some @xmath335 , @xmath325 and @xmath336 , , @xmath337 , @xmath338 , , @xmath339 . then , for all @xmath311 , @xmath340 in @xmath291 , thus , since @xmath13 is a lower subset of @xmath291 , @xmath341 in @xmath13 . hence , as @xmath13 is conical , @xmath342 , so @xmath66 . hence @xmath343 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in this section , we shall fix a conical partial refinement monoid @xmath13 . we shall denote by @xmath149 the algebraic preordering of @xmath13 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ d : ideal ] an _ ideal _ of @xmath13 is a nonempty subset @xmath56 of @xmath13 such that @xmath344 if and only if @xmath345 and @xmath346 , for all @xmath124 , @xmath228 such that @xmath234 is defined . we define elements @xmath124 and @xmath125 of @xmath13 to be _ orthogonal _ , in notation , @xmath150 , if @xmath347 implies that @xmath66 , for all @xmath330 . if @xmath78 and @xmath79 are subsets of @xmath13 , then we define @xmath348 to hold if @xmath9 for all @xmath349 . we shall put @xmath350 if @xmath351 , a singleton , then we write @xmath352 instead of @xmath353 . in particular , @xmath348 if and only if @xmath354 , if and only if@xmath355 . [ l : xbotid ] 1 . @xmath356 and @xmath357 implies that @xmath358 , for all @xmath124 , @xmath125 , @xmath231 such that @xmath234 is defined . the set @xmath359 is an ideal of @xmath13 , for all @xmath360 . @xmath150 and @xmath124 , @xmath361 implies that @xmath234 is defined and @xmath362 . \(i ) let @xmath363 . by refinement , there are @xmath257 , @xmath258 such that @xmath259 , @xmath260 , and @xmath364 . so @xmath365 , whence @xmath366 . similarly , @xmath367 , so @xmath66 , thus proving @xmath358 . \(ii ) is an obvious consequence of ( i ) . \(iii ) by the definition of @xmath149 , there are @xmath257 , @xmath258 such that @xmath368 . by applying refinement to the equality @xmath369 and by using the assumption that @xmath150 , we obtain @xmath370 such that @xmath371 and @xmath372 . since @xmath373 is defined , @xmath234 is defined , and @xmath374 . [ not : sumsubsets ] for @xmath325 and @xmath375 , , @xmath376 , we put @xmath377 we shall also write @xmath378 instead of @xmath379 . if @xmath380 for all @xmath173 , then we shall abbreviate this further by @xmath381 . if @xmath382 for all @xmath383 , then we shall write @xmath384 , or @xmath385 , instead of @xmath378 , and we shall say that the sum of the @xmath97 is _ orthogonal_. [ l : oplusid ] let @xmath325 and let @xmath386 , for @xmath387 , be nonempty subsets of @xmath13 such that @xmath388 . then the following hold : 1 . @xmath389 , for all @xmath387 . in particular , @xmath386 is an ideal of @xmath13 . 2 . for all @xmath330 , there exists a unique decomposition @xmath390 such that @xmath391 for all @xmath387 . \(i ) the sum of all the @xmath392 is orthogonal , thus so is the sum of all @xmath392 , for @xmath393 . furthermore , @xmath394 , for all @xmath393 , so @xmath395 . hence , by using lemma [ l : xbotid](ii ) , @xmath396 . conversely , let @xmath397 . by assumption , there exists a decomposition @xmath398 , where @xmath399 , for all @xmath312 . but @xmath391 , thus @xmath400 ; whence @xmath342 , so @xmath401 . hence @xmath389 . by lemma [ l : xbotid](ii ) , it follows that @xmath386 is an ideal of @xmath13 . \(ii ) suppose @xmath402 , with elements @xmath57 , @xmath403 , for all @xmath387 . since @xmath13 satisfies refinement , there exists a refinement matrix of the form @xmath404 with elements @xmath405 , for all @xmath173 , @xmath312 . but if @xmath383 , then @xmath394 , whence @xmath406 . hence , @xmath407 , for all @xmath387 . [ r : embdssi ] the direct product @xmath408 can be naturally endowed with a structure of partial monoid , by defining the addition componentwise . in the context of lemma [ l : oplusid ] , we obtain a map @xmath409 this map is a one - to - one homomorphism of partial monoids . however , it is _ not _ , in general , surjective : for arbitrary @xmath391 , for @xmath387 , the sum @xmath410 may not be defined . but of course , if @xmath13 is a ( total ) monoid , then @xmath244 is an isomorphism . [ p : embdssi ] in the context of remark , @xmath244 is a lower embedding from @xmath13 into @xmath408 . only part ( iii ) of the definition of a lower embedding is not completely trivial . let @xmath330 and @xmath411 such that @xmath412 . put @xmath413 , so @xmath414 , for all @xmath387 . by the definition of @xmath244 , @xmath410 is defined , and equal to @xmath5 . by lemma [ l : leqstilldef ] , @xmath415 is also defined . denote its value by @xmath6 . by the definition of @xmath244 , @xmath416 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ standing hypothesis : @xmath13 is a conical partial refinement monoid . we denote again by @xmath149 the algebraic preordering of @xmath13 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ d : proj ] a _ projection _ of @xmath13 is an endomorphism @xmath26 of @xmath229 such that @xmath417 in particular , if @xmath26 is a projection of @xmath13 , then @xmath418 , for all @xmath330 . thus @xmath419 . furthermore , @xmath26 preserves the algebraic preordering of @xmath13 , see definition [ d : algpr ] . [ p : charproj ] let @xmath26 be an endomorphism of @xmath13 . then the following are equivalent : 1 . @xmath26 is a projection of @xmath13 . 2 . there are ideals @xmath420 and @xmath421 of @xmath13 such that 1 . 2 . @xmath423 , for all @xmath424 such that @xmath425 is defined . ( ii)@xmath426(i ) is easy . ( i)@xmath426(ii ) assume ( i ) . we put @xmath427 and @xmath428 . by the definition of a projection , @xmath429 . since @xmath430 , it follows that @xmath422 . in particular , @xmath420 and @xmath421 are ideals of @xmath13 ( see lemma [ l : oplusid](i ) ) . for @xmath330 , let @xmath431 such that @xmath432 . if @xmath433 in @xmath13 such that @xmath391 for all @xmath289 , then , by lemma [ l : oplusid](ii ) , @xmath434 and @xmath435 . every projection of @xmath13 is idempotent . in the context of proposition [ p : charproj](ii ) , we observe that @xmath427 while @xmath436 . in particular , @xmath26 is determined by @xmath420 alone , so we shall call @xmath26 the _ projection of @xmath13 onto @xmath420_. a _ direct summand _ of @xmath13 is a subset @xmath78 of @xmath13 such that @xmath437 , for some @xmath438 . of course , by lemma [ l : oplusid ] , @xmath78 is then an ideal of @xmath13 , and @xmath439 , so @xmath440 . it follows that the direct summands of @xmath13 are exactly the ranges of the projections of @xmath13 . furthermore , by exchanging the roles of @xmath420 and @xmath421 , we obtain another projection , which we shall denote by @xmath441 . formally , @xmath441 is the unique projection of @xmath13 such that @xmath442 and @xmath443 . we observe that @xmath444 . let @xmath445 denote the set of projections of @xmath13 . we shall also often use the notation @xmath446 . we shall now study the structure of @xmath445 , towards proposition [ p : projba ] . [ l : meetcomppr ] let @xmath26 , @xmath447 . the the following holds : 1 . 2 . let @xmath449 denote the projection from @xmath13 onto @xmath450 . then @xmath451 . \(i ) it is obvious that all ideals @xmath450 , @xmath452 , @xmath453 , and @xmath454 are pairwise orthogonal . let @xmath330 . since @xmath455 , there exists a decomposition @xmath433 , where @xmath456 and @xmath457 . for @xmath289 , @xmath458 , thus @xmath459 , for some @xmath460 and @xmath461 . since @xmath462 and @xmath463 are ideals of @xmath13 , @xmath464 , @xmath465 , @xmath466 , and @xmath467 . observe that @xmath468 . \(ii ) since both @xmath26 and @xmath31 act as the identity on @xmath450 , so does @xmath469 . furthermore , @xmath470 for all @xmath471 and @xmath472 for all @xmath473 , so , @xmath474 . by symmetry , @xmath475 . we shall put @xmath476 , for all @xmath26 , @xmath447 . the structure @xmath477 is a semilattice . we endow @xmath445 with the partial ordering @xmath149 defined by @xmath478 for this partial ordering , @xmath479 is , of course , the infimum of @xmath480 . the least element of @xmath445 is @xmath188 ( the zero map ) , while the greatest element of @xmath445 is @xmath481 ( the identity on @xmath13 ) . [ l : pleqortq ] let @xmath26 , @xmath447 . then the following holds : 1 . @xmath482 if and only if @xmath483 if and only if @xmath484 holds , for all @xmath330 . 2 . @xmath485 if and only if @xmath486 . \(i ) if @xmath482 , then @xmath487 . suppose now that @xmath483 . let @xmath330 . the inequality @xmath484 holds for all @xmath488 ( because then @xmath489 ) and for all @xmath490 ( because then @xmath491 ) , so it holds for all @xmath330 since @xmath492 . if @xmath484 for all @xmath330 , then @xmath493 since @xmath494 is an ideal of @xmath13 , so @xmath495 . hence @xmath482 . \(ii ) by lemma [ l : meetcomppr ] , @xmath485 if and only if @xmath496 . hence , @xmath485 if and only if @xmath497 , if and only if@xmath486 by ( i ) above . [ c : botaa ] the map @xmath498 is an involutive anti - automorphism of @xmath499 . we already know that @xmath444 , for all @xmath500 . furthermore , by lemma [ l : pleqortq ] , @xmath482 implies that @xmath501 , for all @xmath26 , @xmath447 . since @xmath499 is a meet - semilattice , we thus obtain the following . [ c : bslatt ] @xmath499 is a lattice . so we denote by @xmath502 the supremum of @xmath480 , for all @xmath26 , @xmath447 . we can strengthen corollary [ c : bslatt ] right away . [ p : projba ] @xmath499 is a boolean algebra . by corollary [ c : bslatt ] , @xmath499 is a lattice . furthermore , @xmath441 is a complement of @xmath26 , for all @xmath500 . hence , to conclude the proof , it suffices to prove distributivity . the argument below is classical , and it can be traced back to glivenko s work , see , for example , @xcite . so , let @xmath26 , @xmath31 , @xmath503 . we put @xmath504 then @xmath505 , which implies , by lemma [ l : pleqortq](ii ) , that @xmath506 and @xmath507 , thus , meeting both inequalities , @xmath508 . by corollary [ c : botaa ] , @xmath509 , so it follows that @xmath510 , that is , by lemma [ l : pleqortq](ii ) , @xmath511 ; whence @xmath512 . but the converse inequality @xmath513 is obvious , thus @xmath514 . for @xmath26 , @xmath31 , @xmath503 , let @xmath515 hold just in case @xmath516 and @xmath485 . [ l : joindisj ] let @xmath26 , @xmath447 such that @xmath485 . then @xmath517 let @xmath330 , and put @xmath516 . we apply the definition of a projection to @xmath26 and to @xmath31 . so there are @xmath518 and @xmath519 such that @xmath520 . we observe that @xmath521 and @xmath522 . by applying the refinement property to the equality @xmath523 and by observing that @xmath524 , we obtain @xmath370 such that @xmath525 and @xmath526 . on the one hand , @xmath527 , thus @xmath528 , see lemma [ l : meetcomppr ] . on the other hand , @xmath529 . hence , @xmath530 , so @xmath531 . [ not : weve ] for @xmath5 , @xmath6 , @xmath532 , @xmath533 is the statement @xmath534 we define , dually , the statement @xmath535 . note that @xmath10 is uniquely defined by either statement only in case @xmath149 is antisymmetric . similarly , one can define the notations @xmath536 and @xmath71 . [ p : pvwq(x ) ] let @xmath26 , @xmath447 , let @xmath330 . then the following statements are satisfied : @xmath537 we put @xmath538 and @xmath539 . by lemma [ l : pleqortq ] , @xmath540 let @xmath224 such that @xmath541 . since @xmath462 and @xmath494 are ideals of @xmath13 ( see lemma [ l : oplusid](i ) ) , @xmath542 , so @xmath543 . thus @xmath544 . hence @xmath545 . by lemma [ l : pleqortq ] , @xmath546 . let @xmath224 such that @xmath547 . thus , _ a fortiori _ , @xmath548 . since @xmath445 is a boolean algebra , @xmath549 . it follows , by lemma [ l : joindisj ] , that @xmath550 , thus @xmath551 by lemma [ l : xbotid](iii ) . hence @xmath552 . if @xmath13 is a _ total _ ( as opposed to partial ) monoid , then the projections of @xmath13 correspond to direct decompositions of @xmath13 , thus , they preserve arbitrary suprema and infima . for our partial structures , the corresponding result still holds . [ l : projcont ] let @xmath26 be a projection of @xmath13 . for every family @xmath72 of elements of @xmath13 and every @xmath237 , 1 . if @xmath553 , then @xmath536 implies that @xmath554 . 2 . suppose that any two elements of @xmath13 have a meet . then @xmath71 implies that @xmath555 . the natural settings of lemma [ l : projcont ] are in situations where @xmath13 is antisymmetric as well . however , that condition is not , strictly speaking , necessary , if we use the interpretation of the symbols @xmath556 and @xmath557 given in notation [ not : weve ] . \(i ) of course , @xmath558 , for all @xmath173 . let @xmath125 be a minorant of @xmath559 . since @xmath56 is nonempty , @xmath125 belongs to @xmath462 , so @xmath560 . since @xmath125 is also a minorant of @xmath561 , @xmath562 . hence , @xmath563 . \(ii ) of course , @xmath564 , for all @xmath173 . let @xmath125 be a majorant of @xmath559 . since @xmath565 is also a majorant of @xmath559 and by assumption , @xmath566 exists and it is a majorant of @xmath559 . from @xmath567 it follows that @xmath568 is defined . furthermore , @xmath569 , for all @xmath62 , thus @xmath570 . therefore , @xmath571 . [ d : lsdiff ] suppose that @xmath13 is antisymmetric . for @xmath124 , @xmath125 , @xmath231 , let @xmath572 mean that @xmath126 is the least @xmath330 such that @xmath573 . we say that @xmath126 is the _ least difference _ of @xmath125 and @xmath124 . [ l : lstdiff ] suppose that @xmath13 is antisymmetric , let @xmath131 in @xmath13 . if @xmath574 exists , then @xmath575 . put @xmath572 . since @xmath131 , there exists @xmath576 such that @xmath577 , thus , by the definition of the least difference , @xmath578 , whence @xmath579 . therefore , since @xmath13 is antisymmetric , @xmath580 . [ l : sdproj ] suppose that @xmath13 is antisymmetric and that @xmath574 exists for all @xmath124 , @xmath228 such that @xmath131 . then @xmath581 , for all @xmath124 , @xmath228 such that @xmath131 and all @xmath500 . from @xmath582 it follows that @xmath583 , thus @xmath584 . conversely , put @xmath585 . then @xmath586 by definition , while we also have @xmath587 , thus , adding the two inequalities together ( and observing that , since @xmath588 , @xmath589 is defined ) , we obtain the inequality @xmath590 . it follows that @xmath591 , whence , by applying @xmath26 , we obtain that @xmath592 . therefore , @xmath593 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ standing hypothesis : @xmath13 is a conical partial refinement monoid . we denote again by @xmath149 the algebraic preordering of @xmath13 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ d : gencompmon ] we say that @xmath13 has _ general comparability _ , if for all @xmath5 , @xmath224 , there exists @xmath500 such that @xmath594 and @xmath595 . we give a sufficient condition that implies general comparability . [ l : gencompax ] suppose that @xmath13 satisfies the following axioms : 1 . @xmath596 , @xmath597 such that @xmath598 , @xmath599 , and @xmath9 . 2 . @xmath600 , for all @xmath237 . then @xmath13 satisfies general comparability . let @xmath124 , @xmath228 . consider @xmath126 , @xmath5 , @xmath6 as in ( i ) . by ( ii ) , there exists @xmath500 such that @xmath601 and @xmath602 . so @xmath603 and @xmath604 , whence @xmath605 and @xmath606 . [ l : meetjoins ] suppose that @xmath13 has general comparability and that the algebraic preordering of @xmath13 is antisymmetric . let @xmath124 , @xmath228 . the following assertions hold : 1 . the pair @xmath163 has an infimum . if the pair @xmath163 is majorized , then it has a supremum . a partially ordered set satisfying ( i ) and ( ii ) above is sometimes called a _ chopped lattice_. by general comparability , there exists @xmath500 such that @xmath605 and @xmath607 . so @xmath608 is defined , and @xmath609 , @xmath125 . furthermore , it is easy to verify that @xmath610 . similarly , if the pair @xmath163 is majorized by an element @xmath611 , then @xmath612 is defined , and @xmath613 . it is easy to verify that @xmath614 . for @xmath124 , @xmath228 , let @xmath615 hold , if @xmath616 . we also say that @xmath125 _ absorbs _ @xmath124 . we recall the following axiom , see @xcite : @xmath617 if @xmath618 , @xmath619 , @xmath620 , and @xmath621 , @xmath622 , then @xmath623 is defined ( because @xmath624 and @xmath625 is defined ) and @xmath626 . hence we obtain the following weaker version of the pseudo - cancellation property : @xmath627 [ l : gcimppc ] suppose that @xmath13 has general comparability . then @xmath13 satisfies the pseudo - cancellation property . suppose @xmath618 in @xmath13 . by using refinement , we find a refinement matrix as follows : @xmath628 by general comparability , there exists @xmath500 such that @xmath629 and @xmath630 . let @xmath621 , @xmath631 such that @xmath632 and @xmath633 . then @xmath634 it follows that @xmath635 . put @xmath636 . by using lemma [ l : joindisj ] , we obtain the equalities @xmath637 [ c : gcsep ] suppose that @xmath13 has general comparability . then @xmath13 is _ separative _ , that is , it satisfies the statement @xmath638 let @xmath124 , @xmath125 , @xmath231 such that @xmath618 and @xmath609 , @xmath125 . by lemma [ l : gcimppc ] , there are @xmath639 , @xmath257 , @xmath258 such that @xmath640 , @xmath641 , and @xmath257 , @xmath642 . in particular , @xmath257 , @xmath643 , thus , since @xmath644 and @xmath625 are defined , @xmath645 and @xmath646 are defined , see lemma [ l : leqstilldef ] . note that @xmath647 . however , @xmath609 , @xmath125 , thus , since @xmath648 , we obtain that @xmath649 , so @xmath650 . similarly , @xmath651 . therefore , @xmath652 . [ d : dirfinmon ] an element @xmath126 of @xmath13 is * _ directly finite|ii _ , if @xmath653 implies that @xmath66 , for all @xmath330 , * _ cancellable _ , if @xmath654 implies that @xmath655 , for all @xmath5 , @xmath224 . we say that @xmath13 is _ stably finite _ , if every element of @xmath13 is directly finite . we denote by @xmath656 the subset of @xmath13 consisting of all directly finite elements . it is obvious that every cancellable element is directly finite . by lemma [ l : gcimppc ] , we obtain immediately the following converse . [ l : dfimcanc ] suppose that @xmath13 has general comparability . then every directly finite element of @xmath13 is cancellable . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ standing hypothesis : @xmath13 is a conical partial refinement monoid . we denote again by @xmath149 the algebraic preordering of @xmath13 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ for elements @xmath124 and @xmath125 of @xmath13 , it follows from proposition [ p : pvwq(x ) ] that the set of all projections @xmath26 of @xmath13 such that @xmath605 is closed under finite join . we shall now consider a stronger statement . [ d : boolval ] for @xmath124 , @xmath228 , we shall denote by @xmath657 the largest projection @xmath26 of @xmath13 such that @xmath605 if it exists . hence , @xmath658 . we say that @xmath13 is _ boolean - valued _ , if the boolean value @xmath657 is defined , for all @xmath124 , @xmath228 . [ not : bva = b ] for @xmath124 , @xmath228 , if both @xmath657 and @xmath659 are defined , we put @xmath660 . [ l : cc(a ) ] assume that @xmath13 has general comparability . let @xmath237 , and suppose that @xmath661 is defined . then the following assertions hold : 1 . @xmath662 . 2 . @xmath663 . 3 . @xmath664 . \(i ) put @xmath665 . for @xmath488 such that @xmath666 , we have @xmath667 . hence @xmath668 . conversely , let @xmath669 . by general comparability , there exists @xmath447 such that @xmath670 and @xmath671 . since @xmath672 , the equalities @xmath673 hold . it follows from the definition of @xmath26 that @xmath674 . therefore , @xmath675 , so @xmath488 . \(ii ) follows immediately from ( i ) , while ( iii ) follows immediately from ( i ) , ( ii ) , and the fact that @xmath661 is a projection of @xmath13 . [ l : bva=0 ] assume that @xmath13 has general comparability . . then @xmath661 exists if and only if @xmath664 . if @xmath661 exists , then @xmath664 by lemma [ l : cc(a)](iii ) . conversely , suppose that @xmath664 . so there exists a unique projection @xmath26 of @xmath13 such that @xmath676 . from @xmath677 and @xmath678 it follows that @xmath679 . let @xmath447 such that @xmath680 . we claim that @xmath681 , for any @xmath330 . indeed , let @xmath224 such that @xmath682 . from @xmath683 it follows that @xmath684 , thus @xmath685 , so @xmath343 , thus establishing our claim . so @xmath686 , whence @xmath674 by lemma [ l : pleqortq](i ) . therefore , @xmath665 . [ d : remov ] let @xmath124 , @xmath228 . we say that @xmath124 is _ removable _ from @xmath125 , and we write @xmath687 , if the following conditions hold : 1 . @xmath573 implies that @xmath688 , for all @xmath330 . in particular , we observe that @xmath687 implies that @xmath689 ( the converse does not hold as a rule ) . in particular , in case @xmath13 is antisymmetric , @xmath687 implies that @xmath615 . [ l : trleqtr ] let @xmath124 , @xmath125 , @xmath231 such that either @xmath690 or @xmath691 . then @xmath692 . in both cases , it is trivial that @xmath693 . suppose that @xmath690 . let @xmath330 such that @xmath694 . so @xmath573 , thus , since @xmath687 , @xmath688 , that is , @xmath695 for some @xmath6 . hence @xmath696 . but @xmath687 , thus @xmath689 , so @xmath697 . so @xmath692 . suppose now that @xmath691 . let @xmath330 such that @xmath694 . so @xmath698 , thus ( since @xmath699 ) @xmath700 . so , again , @xmath692 . [ l : gcivptr ] suppose that @xmath13 is antisymmetric and that @xmath13 has pseudo - cancellation . for all @xmath124 , @xmath125 , @xmath231 , the following assertions hold : 1 . @xmath701 implies that there exists @xmath702 such that @xmath703 . 2 . if @xmath131 in @xmath13 , then @xmath687 if and only if @xmath703 implies that @xmath704 , for all @xmath330 . \(i ) since @xmath131 , there exists @xmath224 such that @xmath705 . hence @xmath706 , thus , by pseudo - cancellation , there exists @xmath707 such that @xmath708 . by refinement , there are @xmath709 and @xmath702 such that @xmath710 . since @xmath13 is antisymmetric , @xmath711 . hence , @xmath712 , with @xmath702 . \(ii ) we prove the nontrivial direction . so , suppose that @xmath703 implies @xmath704 , for all @xmath330 . now let @xmath330 such that @xmath573 . by ( i ) above , there exists @xmath713 such that @xmath705 . by assumption , @xmath714 ; whence @xmath688 . [ l : projtr ] suppose that @xmath13 is antisymmetric and satisfies general comparability . let @xmath124 , @xmath228 . 1 . if @xmath687 , then @xmath715 , for all @xmath500 . 2 . let @xmath716 be a family of projections of @xmath13 . we assume that both @xmath717 and @xmath718 are defined . if @xmath719 for all @xmath62 , then @xmath720 . 3 . let @xmath264 , let @xmath721 be a finite sequence of projections of @xmath13 , and let @xmath722 . if @xmath719 for all @xmath387 , then @xmath723 . \(i ) it is clear that @xmath605 . now let @xmath330 such that @xmath724 . so , @xmath725 since @xmath687 , it follows that @xmath726 ; whence @xmath727 . by lemma [ l : gcivptr](ii ) , @xmath715 . \(ii ) observe first that @xmath728 . let @xmath330 such that @xmath729 . observe that @xmath730 for all @xmath62 ; hence @xmath731 . similarly , @xmath732 . therefore , @xmath733 , for all @xmath173 , hence , since @xmath719 , @xmath734 . this holds for all @xmath173 , whence @xmath735 , so @xmath736 . the conclusion follows from lemma [ l : gcivptr](ii ) . \(iii ) by proposition [ p : pvwq(x ) ] , @xmath737 and @xmath738 . the conclusion follows then from ( ii ) . [ c:2 - 5 . ? ] suppose that @xmath13 is antisymmetric and satisfies general comparability . for all @xmath124 , @xmath125 , @xmath231 , if @xmath739 , then @xmath740 . by general comparability , there exists @xmath500 such that @xmath741 and @xmath742 . then , as in the proof of lemma [ l : meetjoins ] , @xmath743 . by lemma [ l : projtr](i ) , @xmath744 and @xmath745 . hence , lemma [ l : projtr](iii ) implies that @xmath740 . [ d : purinf ] an element @xmath124 of @xmath13 is _ purely infinite _ , if @xmath746 . we denote by @xmath747 the set of all purely infinite elements of @xmath13 . we observe that the only element of @xmath13 which is both directly finite and purely infinite is @xmath188 . [ l:2 - 5 . ? ] suppose that @xmath13 is antisymmetric and satisfies general comparability . then @xmath747 is closed under finite infima and suprema . this is clear from the descriptions of pairwise infima and suprema given in the proof of lemma [ l : meetjoins ] . [ l : a+b = b(pi ) ] suppose that @xmath13 is antisymmetric . let @xmath124 , @xmath228 such that @xmath131 . if either @xmath124 or @xmath125 is purely infinite , then @xmath615 . [ l : bvtr ] suppose that @xmath13 is antisymmetric , boolean - valued , and that it has general comparability . let @xmath748 and @xmath228 such that @xmath131 . put @xmath749 . then @xmath750 . by the definition of @xmath26 , @xmath751 . since @xmath13 is antisymmetric , @xmath752 . furthermore , @xmath753 ( because @xmath131 ) . let @xmath330 such that @xmath754 by general comparability , there exists @xmath447 such that @xmath755 by applying @xmath31 to , we obtain that @xmath756 however , @xmath757 , thus @xmath758 , so @xmath759 . hence , by lemma [ l : a+b = b(pi ) ] , @xmath760 , so , by , @xmath761 . by the definition of @xmath26 , @xmath762 , thus , since @xmath763 , @xmath764 , that is , @xmath674 . hence @xmath765 , thus , by , @xmath766 . hence , by and by lemma [ l : a+b = b(pi ) ] , @xmath767 . we conclude the proof by lemma [ l : gcivptr](ii ) . we now introduce a useful definition . [ d : cc(a ) ] for @xmath237 , the _ central cover _ of @xmath124 , denoted by @xmath768 , is defined as @xmath769 . note that @xmath770 , by lemma [ l : cc(a)](i ) . [ c : remequiv ] suppose that @xmath13 is antisymmetric , boolean - valued , and that it has general comparability . let @xmath748 and @xmath228 such that @xmath131 . then @xmath687 if and only if @xmath771 for all nonzero projections @xmath772 . assume first that @xmath687 , and let @xmath773 be a projection such that @xmath774 . then @xmath775 , and it follows from the assumption @xmath687 that @xmath776 . thus @xmath777 , so @xmath778 , and hence @xmath779 . conversely , assume that @xmath771 for all nonzero projections @xmath772 , set @xmath749 , and observe that @xmath780 . by lemma [ l : bvtr ] , @xmath781 , and so @xmath782 . therefore @xmath687 . [ l : basiccc ] assume that @xmath13 is antisymmetric , boolean - valued , and satisfies general comparability . let @xmath237 and let @xmath500 . 1 . @xmath783 if and only if @xmath784 . 2 . @xmath785 . 3 . suppose that @xmath71 , for a family @xmath72 of elements of @xmath13 . then @xmath786 . 4 . @xmath787 , for all @xmath124 , @xmath228 . \(i ) @xmath783 if and only if @xmath788 , if and only if @xmath789 , if and only if @xmath790 , if and only if @xmath784 . \(ii ) for all @xmath447 , @xmath791 if and only if@xmath792 , if and only if@xmath793 , if and only if @xmath794 . hence @xmath795 therefore , @xmath796 . \(iii ) by ( i ) , for any @xmath500 , @xmath783 if and only if @xmath784 , if and only if @xmath797 for all @xmath173 ( by lemma [ l : projcont](ii ) ) , if and only if @xmath798 for all @xmath173 , if and only if@xmath799 . the conclusion of ( iii ) follows . \(iv ) it suffices to prove the inequality @xmath800 since @xmath801 , @xmath125 , the inequality @xmath802 is obvious . conversely , put @xmath803 . by general comparability , there are @xmath31 , @xmath503 such that @xmath804 , @xmath805 , and @xmath806 . it follows that @xmath807 hence @xmath808 . similarly , @xmath809 , so @xmath810 . this completes the proof of . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ standing hypothesis : @xmath13 is a partial refinement monoid satisfying the following additional properties : _ * @xmath13 is antisymmetric . * @xmath13 has general comparability . * @xmath13 is boolean - valued . * every element of @xmath13 is the sum of a directly finite element and a purely infinite element . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ l : perpdfpi ] for all @xmath237 , there exists @xmath500 such that @xmath565 is directly finite and @xmath811 is purely infinite . by assumption on @xmath13 , there are elements @xmath5 and @xmath6 in @xmath13 such that @xmath812 , @xmath5 is purely infinite , and @xmath6 is directly finite . by general comparability , there exists @xmath500 such that @xmath594 and @xmath813 . since @xmath6 is directly finite and @xmath814 , @xmath815 is directly finite . but @xmath815 is purely infinite , thus @xmath603 , and so @xmath816 is directly finite . since @xmath595 with @xmath817 purely infinite , @xmath818 by lemma [ l : a+b = b(pi ) ] . therefore , @xmath811 is purely infinite . [ c : a / infty ] for any @xmath237 , the following assertions hold : 1 . there exists a largest purely infinite element @xmath621 of @xmath13 such that @xmath819 . the element @xmath621 is also the largest @xmath820 such that @xmath821 . 3 . there exists a unique @xmath631 such that @xmath822 and @xmath823 . the element @xmath824 is directly finite . by lemma [ l : perpdfpi ] , there exists @xmath500 such that @xmath565 is directly finite and @xmath811 is purely infinite . set @xmath825 and @xmath826 . observe that @xmath819 and @xmath621 is purely infinite , so @xmath707 . \(ii ) for any @xmath820 , @xmath821 implies that @xmath827 . since @xmath565 is directly finite , @xmath828 , and thus @xmath829 . \(i ) for any purely infinite @xmath370 such that @xmath830 , it follows from lemma [ l : a+b = b(pi ) ] that @xmath831 , whence @xmath832 by part ( ii ) . ( iii ) , ( iv ) we already have @xmath822 with @xmath823 and @xmath824 directly finite . for any @xmath833 , if @xmath834 with @xmath835 , then @xmath836 , thus , by refinement ( and since @xmath837 ) , @xmath838 . for any @xmath237 , we shall denote by @xmath839 the largest purely infinite element @xmath621 of @xmath13 such that @xmath819 . [ l : a+b / infty ] let @xmath124 , @xmath228 such that @xmath234 is defined . then @xmath840 first , @xmath841 is purely infinite and below @xmath234 , thus @xmath842 . conversely , put @xmath843 . then @xmath844 , thus , by canceling the directly finite parts of @xmath124 and @xmath125 ( use lemma [ l : dfimcanc ] ) , @xmath845 , thus , again , @xmath846 . in particular , @xmath847 . our next result involves the _ least difference _ function introduced in definition [ d : lsdiff ] . [ p : bsminusa ] let @xmath131 in @xmath13 ; then @xmath574 exists . by lemma [ l : perpdfpi ] , there exists @xmath447 such that @xmath848 is directly finite and @xmath849 is purely infinite . let @xmath850 such that @xmath851 put @xmath852 . since @xmath853 , the inequality @xmath674 holds . observe also the following equality : @xmath854 since @xmath131 and @xmath849 is purely infinite , it follows from lemma [ l : bvtr ] that @xmath855 , that is , since @xmath765 , @xmath856 in particular , we obtain the relation @xmath857 since @xmath858 , @xmath859 is defined . so we obtain that @xmath860 furthermore , let @xmath330 such that @xmath573 . so @xmath861 , that is , @xmath862 . thus , since @xmath848 is directly finite and by lemma [ l : dfimcanc ] , we obtain @xmath863 furthermore , @xmath864 , thus , by , we obtain that @xmath865 by adding and together , we thus obtain that @xmath866 . so we have verified that @xmath572 . a similar result holds for the existence of the `` largest difference '' . [ p : blminusa ] let @xmath131 in @xmath13 . then there exists a largest element @xmath126 of @xmath13 such that @xmath867 , and then @xmath580 . the element @xmath126 of the statement above will be denoted by @xmath868 , the _ largest difference _ of @xmath125 and @xmath124 . by the definition of the algebraic preordering , there exists @xmath576 such that @xmath869 . so @xmath870 is defined ( because @xmath871 ) and @xmath872 . from @xmath873 it follows that @xmath874 . if @xmath330 is such that @xmath875 , then , by pseudo - cancellation ( see lemma [ l : gcimppc ] ) , @xmath876 for some @xmath877 , so @xmath878 . [ c:+mj ] let @xmath124 , @xmath228 , let @xmath78 be a nonempty subset of @xmath13 . we assume that @xmath879 is defined for all @xmath183 . 1 . if @xmath880 , then @xmath881 . if @xmath882 and @xmath883 is majorized , then @xmath884 . \(i ) pick @xmath183 . since @xmath879 is defined and @xmath688 , @xmath234 is defined . furthermore , @xmath885 . conversely , let @xmath886 . by lemma [ l : meetjoins ] , @xmath887 is defined , and @xmath888 . by proposition [ p : bsminusa ] , @xmath889 , so @xmath890 . therefore , by adding @xmath124 on both sides of this inequality , we obtain that @xmath891 . \(ii ) pick a majorant @xmath126 of @xmath883 . in particular , @xmath892 . by proposition [ p : blminusa ] , @xmath893 , so @xmath894 . since @xmath895 , @xmath234 is defined and @xmath362 . this holds for any majorant @xmath126 of @xmath883 . since @xmath234 is itself a majorant of @xmath883 , it is the supremum of @xmath883 . the fundamental definition underlying this chapter is the following . [ d : dimint ] a _ continuous dimension scale _ is a partial commutative monoid @xmath13 which satisfies the following axioms . * @xmath13 has refinement ( see definition [ d : refppty ] ) , and the algebraic preordering on @xmath13 is antisymmetric . * every nonempty subset of @xmath13 admits an infimum . equivalently , every majorized subset of @xmath13 admits a supremum . * @xmath13 has general comparability ( see definition [ d : gencompmon ] ) . * @xmath13 is boolean - valued ( see definition [ d : boolval ] ) . * every element @xmath124 of @xmath13 can be written @xmath812 , where @xmath5 is directly finite ( definition [ d : dirfinmon ] ) and @xmath6 is purely infinite ( definition [ d : purinf ] ) . * let @xmath124 , @xmath125 be purely infinite elements of @xmath13 . if @xmath687 ( see definition [ d : remov ] ) , then the set of all purely infinite elements @xmath5 of @xmath13 such that @xmath896 and @xmath897 ( see , page ) has a least element . a continuous dimension scale @xmath13 is _ bounded _ , if it has a largest element . all axioms ( m1)(m5 ) have been considered in chapter [ ch : partmon ] . axiom ( m6 ) is a newcomer , whose importance will appear in section [ s : alphap ] . we shall first give an alternative axiomatization of continuous dimension scales . in order to prepare for this , we first prove the following result , which extends the result of lemma [ l : gencompax ] . [ p : altax ] let @xmath13 be a partial refinement monoid satisfying the following properties : * @xmath13 is antisymmetric . * any two elements of @xmath13 have a meet . * @xmath13 satisfies axiom . then the following assertions are equivalent : 1 . @xmath13 satisfies the following axioms : * @xmath596 , @xmath597 such that @xmath598 , @xmath599 , and @xmath9 . * @xmath600 , for all @xmath237 . * @xmath574 exists , for all @xmath124 , @xmath228 such that @xmath131 . @xmath13 is boolean - valued and it satisfies general comparability . ( i)@xmath426(ii ) let @xmath13 satisfy ( n1 ) , ( n2 ) , and ( n3 ) . the fact that @xmath13 satisfies general comparability follows from lemma [ l : gencompax ] . now we prove that @xmath13 is boolean - valued . so let @xmath124 , @xmath228 . by ( n3 ) , @xmath898 exists . for any projection @xmath26 of @xmath13 , @xmath899 by ( n2 ) , @xmath900 . by lemma [ l : xbotid](ii ) , @xmath901 and @xmath902 are ideals of @xmath13 , thus , since @xmath903 , @xmath904 , hence , by lemma [ l : bva=0 ] , @xmath905 exists . therefore , by , @xmath657 exists , and @xmath906 . ( ii)@xmath426(i ) suppose that @xmath13 is boolean - valued and satisfies general comparability . we verify that @xmath13 satisfies ( n1)(n3 ) . ( n1 ) by general comparability , there exists @xmath500 such that @xmath605 and @xmath606 . let @xmath5 , @xmath224 such that @xmath907 and @xmath908 . furthermore , since @xmath605 and @xmath909 , the element @xmath608 is defined . from @xmath910 and @xmath911 it follows that @xmath9 . finally , @xmath912 hence we have obtained ( n1 ) . ( n2 ) follows immediately from lemma [ l : cc(a ) ] . ( n3 ) follows immediately from proposition [ p : bsminusa ] . [ c : altax ] let @xmath13 be a partial commutative monoid . then @xmath13 is a continuous dimension scale if and only if it satisfies the axioms , , , , , , and . [ rk : al1order ] it follows from corollary [ c : altax ] that for a partial commutative monoid @xmath13 , to be a continuous dimension scale is equivalent to the conjunction of the _ second - order _ axiom ( m2 ) and a finite list of _ first - order _ axioms . as a corollary of this alternative description of continuous dimension scales , we observe the following . [ d : dirprod ] the _ direct product _ of a family @xmath913 of partial commutative monoids is obtained by endowing the ordinary cartesian product @xmath914 with the partial addition defined by @xmath915 for all @xmath72 , @xmath916 . of course , definition [ d : dirprod ] is an obvious generalization of the definition of the product of finitely many partial commutative monoids introduced in remark [ r : embdssi ] . it is trivial that the direct product of any family of partial commutative monoids is a partial commutative monoid . far less trivial is the following preservation result . [ l : findpdi ] any direct product of a family of continuous dimension scales is a continuous dimension scale . we use the characterization of continuous dimension scales obtained in corollary [ c : altax ] . the proof is relatively long but very easy , so we will not give the details of it but rather the basic idea . a key point is to verify that the operations @xmath917 , @xmath918 , @xmath919 , @xmath920 ( for @xmath181 ) , and the relations @xmath181 , @xmath921 , @xmath9 , and @xmath922 can be `` read componentwise '' , that is , for example , if @xmath923 and @xmath924 , then @xmath181 if and only if @xmath925 for all @xmath173 , @xmath926 , and so on . once these simple facts are established , the verification of the axioms ( m1 ) , ( m2 ) , ( m5 ) , ( m6 ) , ( n1 ) , ( n2 ) , and ( n3 ) is routine . for a continuous dimension scale @xmath13 and elements @xmath124 and @xmath125 in a lower subset @xmath241 of @xmath13 , the orthogonality of @xmath124 and @xmath125 means the same in @xmath13 and in @xmath241 . we capture this pattern in a definition . [ d : abs ] 1 . a _ statement _ @xmath927 in the language of partial commutative monoids is _ absolute _ , if for any continuous dimension scale @xmath13 , every lower subset @xmath241 of @xmath13 , and all elements @xmath206 , , @xmath928 , @xmath13 satisfies @xmath929 if and only if @xmath241 satisfies @xmath929 . a _ definable function _ @xmath930 in the language of partial commutative monoids is _ absolute _ , if for any continuous dimension scale @xmath13 , every lower subset @xmath241 of @xmath13 , and all elements @xmath206 , , @xmath928 , @xmath929 is defined in @xmath13 if and only if it is defined in @xmath241 , and then both values are equal . [ l : abs ] the following statements 1 . @xmath181 ; 2 . @xmath9 ; 3 . @xmath931 ; 4 . @xmath932 ; 5 . @xmath921 and the following function * @xmath933 are absolute . most items are trivial , except perhaps ( v ) and ( vi ) . let @xmath241 be a lower subset of a continuous dimension scale @xmath13 , let @xmath131 in @xmath241 . since @xmath13 is a continuous dimension scale , @xmath572 is defined in @xmath13 . by lemma [ l : lstdiff ] , @xmath580 , thus , since @xmath241 is a lower subset of @xmath13 , @xmath934 . it readily follows that @xmath572 in @xmath241 , which concludes the proof of ( vi ) ( because @xmath126 always exists ) . as @xmath687 if and only if @xmath131 and @xmath935 , item ( v ) follows immediately . as an easy consequence of corollary [ c : altax ] and lemma [ l : abs ] , we obtain the following . [ l : segdi ] let @xmath13 be a continuous dimension scale , let @xmath241 be a lower subset of @xmath13 , viewed as a partial submonoid of @xmath13 . then @xmath241 is a continuous dimension scale . we refer to lemma [ l : segdimint ] for more information on lower subsets of continuous dimension scales . we observe that trying to use the original axioms ( m1)(m6 ) for the proof of lemma [ l : findpdi ] would have been much more difficult , since we would have needed to understand the projections of the product @xmath936 . by using corollary [ c : altax ] , the proof is still somewhat tedious , but essentially trivial . we present another way to produce continuous dimension scales . [ l : dirun ] let @xmath56 be an upwards directed partially ordered set , let @xmath937 be a family of continuous dimension scales such that @xmath386 is a lower subset of @xmath392 , for all @xmath938 in @xmath56 . then @xmath939 is a continuous dimension scale . the set @xmath13 is , of course , endowed with the union of all the partial commutative monoid operations on all the @xmath386-s . by using lemma [ l : abs ] , it is easy to verify that @xmath13 satisfies ( m1 ) , ( m5 ) , ( n1 ) , ( n2 ) , and ( n3 ) . let @xmath78 be a nonempty subset of @xmath13 ; so there exists @xmath62 such that @xmath940 . denote by @xmath941 the meet of @xmath942 , for all @xmath943 in @xmath56 . then @xmath944 implies that @xmath945 , in particular , all the @xmath941-s belong to @xmath386 , and the meet of all the @xmath941-s in @xmath386 is also the meet of @xmath78 in @xmath13 . hence @xmath13 satisfies ( m2 ) . let @xmath687 in @xmath747 . there exists @xmath62 such that @xmath124 , @xmath946 . it follows from lemma [ l : abs ] that the statement @xmath687 holds in all @xmath392 with @xmath943 , thus , since @xmath392 is a continuous dimension scale , the set of all elements @xmath947 such that @xmath896 and @xmath897 ( we use lemma [ l : abs ] ) has a least element , say , @xmath948 . it follows again from lemma [ l : abs ] that @xmath949 , for all @xmath943 ; denote by @xmath126 this element , then @xmath950 and @xmath126 is minimum in the set of all elements @xmath951 such that @xmath896 and @xmath897 . hence @xmath13 satisfies ( m6 ) . by corollary [ c : altax ] , @xmath13 is a continuous dimension scale . we now provide fundamental examples of continuous dimension scales . for an ordinal @xmath40 , the monoids @xmath91 , @xmath166 , and @xmath167 are defined in the introduction:[pg : zr2gam ] @xmath952 we call the elements of @xmath82 , @xmath953 , and @xmath954 the _ finite elements _ of @xmath91 , @xmath166 , and @xmath167 , respectively . we endow each of the sets @xmath91 , @xmath166 , and @xmath167 with the addition that extends the natural addition on the finite elements and on the alephs ( so @xmath955 if @xmath956 ) , and such that every finite element is absorbed by every aleph . hence @xmath91 , @xmath166 , and @xmath167 are monoids ( and not just partial ones ) . we also observe that the finite elements of @xmath91 , @xmath166 , and @xmath167 are precisely the directly finite ones . therefore , the algebraic ordering on each of the structures @xmath91 , @xmath166 , and @xmath167 is the natural _ total _ ordering . [ p : zrtwothdimint ] for every ordinal @xmath40 , the monoids @xmath91 , @xmath166 , and @xmath167 are totally ordered continuous dimension scales . all axioms ( m1)(m6 ) are trivially satisfied , except perhaps refinement . let @xmath13 be one of the structures @xmath91 , @xmath166 , or @xmath167 . since every element of @xmath13 is either cancellable or purely infinite , @xmath13 satisfies the following weak form of the pseudo - cancellation property : @xmath957 by proposition 1.23 of @xcite , since @xmath13 is totally ordered , it has refinement . of course , one could also have verified refinement directly , but at the expense of a few more calculations . for example , start with the observation that @xmath82 , @xmath953 , and @xmath954 have refinement ; note the easy fact that adjoining a new infinity element to any refinement monoid produces another refinement monoid ; use ordinal induction . we shall also see in proposition [ p : lgrpdi ] that the positive cone of any dedekind complete lattice ordered group is a continuous dimension scale . furthermore , proposition [ p : zrtwothdimint ] will be considerably extended in theorem [ t : c(o , k)dimint ] . we recall that every dedekind complete partially ordered group is abelian , see ( * ? ? ? * theorem 28 ) . much more is true . [ p : lgrpdi ] let @xmath202 be a dedekind complete lattice - ordered group . then @xmath203 is a continuous dimension scale . it is trivial that the algebraic preordering on @xmath203 is antisymmetric . it is well - known that @xmath203 ( or , more generally , the positive cone of any lattice - ordered group ) satisfies refinement , see , for example , ( * ? ? ? * thorme 1.2.16 ) . hence @xmath203 satisfies ( m1 ) . axiom ( m2 ) is just a reformulation of the fact that @xmath202 is dedekind complete . axioms ( m5 ) and ( m6 ) are trivially satisfied , because every element of @xmath203 is directly finite . to verify that @xmath203 satisfies axiom ( m3 ) , it suffices to verify that it satisfies the assumptions of lemma [ l : gencompax ] . for @xmath124 , @xmath958 , if we put @xmath610 , @xmath959 , and @xmath960 , then @xmath598 , @xmath599 , and @xmath16 . this takes care of assumption ( i ) . for @xmath5 , @xmath961 , we define @xmath9 to hold , if @xmath962 , and we put @xmath963 the _ polar _ of @xmath78 . since @xmath202 is dedekind complete , the direct factors of @xmath202 are exactly the polar subsets of @xmath202 , see ( * ? ? ? * thorme 11.2.4 ) . in particular , @xmath964 , for all @xmath965 . assumption ( ii ) follows . finally , we verify that @xmath203 satisfies axiom ( m4 ) . let @xmath124 , @xmath958 . we put @xmath966 . for a polar subset @xmath967 of @xmath202 , @xmath202 is the orthogonal sum of @xmath967 and @xmath968 ( see , for example , ( * ? ? ? * theorem 27 ) ) . in particular , the projection @xmath969 of @xmath202 onto @xmath967 ( relatively to @xmath968 ) is an idempotent homomorphism of lattice - ordered groups . by proposition [ p : charproj ] , the projections of @xmath203 are exactly the restrictions to @xmath203 of the maps of the form @xmath969 , for a polar subset @xmath967 of @xmath202 . for any such @xmath967 , @xmath970 if and only if@xmath971 , that is , @xmath972 . hence , the restriction of @xmath973 to @xmath203 is the largest projection @xmath26 of @xmath203 such that @xmath605 . [ l : refsm1 ] let @xmath13 be a conical partial refinement monoid . if @xmath13 is cancellative and satisfies axiom , then @xmath291 is the positive cone of a dedekind complete lattice - ordered group . by proposition [ p : prmrm ] , @xmath291 is a refinement monoid , and it is generated by @xmath13 as a monoid . by lemma [ l : basicrefs](ii ) , @xmath291 is a cancellative commutative monoid , thus it is the positive cone @xmath203 of a directed , partially preordered abelian group @xmath202 . since @xmath13 is conical , so is @xmath291 ( see lemma [ l : basicrefs](iii ) ) , and hence @xmath202 is , in fact , partially ordered . since @xmath974 has refinement , @xmath202 is an interpolation group , see proposition 2.1 in @xcite . [ cl : vwsg ] let @xmath124 , @xmath125 , @xmath231 such that @xmath126 is the infimum of @xmath163 in @xmath13 . then @xmath126 is also the infimum of @xmath163 in @xmath202 . let @xmath210 be a minorant of @xmath163 . since @xmath202 has interpolation , there exists @xmath961 such that @xmath188 , @xmath975 , @xmath125 . so @xmath224 , hence @xmath976 by the assumption on @xmath126 . [ cl : latord ] @xmath202 is lattice - ordered . we put @xmath977 . it follows from claim [ cl : vwsg ] that @xmath78 contains @xmath13 . let @xmath124 , @xmath192 , we prove that @xmath978 . let @xmath820 , put @xmath979 . for any @xmath210 , @xmath980 so @xmath981 . so @xmath78 is closed under addition , whence @xmath982 . then , replacing @xmath13 by @xmath203 in the definition of @xmath78 yields , by a similar argument , that @xmath162 is defined , for all @xmath124 , @xmath983 . [ cl : vwsg ] let @xmath78 be a nonempty subset of @xmath13 and @xmath237 . if @xmath124 is the supremum of @xmath78 in @xmath13 , then it is also the supremum of @xmath78 in @xmath202 . let @xmath983 be a majorant of @xmath78 . then , using claim [ cl : latord ] , @xmath162 is also a majorant of @xmath78 , but @xmath984 , thus @xmath985 . since @xmath124 is the supremum of @xmath78 in @xmath13 , @xmath986 . @xmath987 satisfies axiom . let @xmath988 be a nonempty subset of @xmath987 , majorized by some element of @xmath987 , say , @xmath234 , where @xmath124 , @xmath228 . we prove that @xmath78 admits a supremum in @xmath202 . by ( i ) , @xmath987 satisfies refinement , thus , for all @xmath62 , there are @xmath989 and @xmath990 in @xmath13 such that @xmath991 . since @xmath561 is a nonempty subset of @xmath13 , majorized by @xmath124 , it has , by assumption , a supremum in @xmath13 , say , @xmath621 . observe that @xmath992 ( in @xmath13 ) for all @xmath173 , and put @xmath993 . then @xmath994 is a nonempty subset of @xmath13 , majorized by @xmath125 , thus it has a supremum , say , @xmath824 , in @xmath13 . for all @xmath62 , @xmath995 , thus , by adding @xmath274 to this inequality , we obtain that @xmath996 . so , to conclude the proof , it suffices to prove that @xmath997 is the least common majorant of @xmath78 . so let @xmath5 be a majorant of @xmath78 . it follows from claim [ cl : vwsg ] that @xmath621 is the supremum of @xmath561 in @xmath202 . then @xmath998 , for all @xmath62 , thus @xmath999 . put @xmath1000 ; observe that @xmath1001 . for @xmath62 , @xmath1002 so @xmath1003 . since @xmath1004 and @xmath202 is lattice - ordered , @xmath1005 , so @xmath1006 . this holds for all @xmath173 , thus , by claim [ cl : vwsg ] , @xmath1007 , so @xmath1008 . the conclusion of the proof is then easy : @xmath1009 is a lower subset of @xmath1010 , for each @xmath264 , and all the sets @xmath1009 satisfy ( m2 ) , thus their union , namely , @xmath203 , also satisfies ( m2 ) . therefore , @xmath202 is dedekind complete . we shall first present a very general result about continuous functions from extremally disconnected topological spaces to totally ordered sets with their interval topology . some particular cases of this result are well - known . for example , if @xmath76 is a complete boolean space , then @xmath1011 is a dedekind complete lattice ordered group , see stze 1 and 3 in @xcite and theorem 14 in @xcite . we also refer to ( * ? ? ? * lemma 9.1 ) for a version of proposition [ p : extlsc ] for @xmath76 basically disconnected and @xmath216 an arbitrary closed interval of @xmath44 . we recall a basic result of general topology , see ( * ? ? ? * theorem 10 ) . [ p : extrdisc]a topological space is the ultrafilter space of a complete boolean algebra if and only if it is a complete boolean space . [ p : extlsc ] let @xmath76 be an extremally disconnected topological space , let @xmath216 be a complete totally ordered set . we endow @xmath216 with its interval topology . let @xmath217 be a lower semicontinuous map . then the map @xmath1012 defined by the rule @xmath1013,\quad\text{for all } x\in\omega,\ ] ] is continuous , and it is the least continuous map @xmath1014 such that @xmath1015 . obviously , @xmath1016 . for any @xmath220 , we define subsets of @xmath76 by @xmath1017 we record a few basic facts about the sets @xmath1018 , @xmath1019 , @xmath1020 . [ cl : basicfg ] for all @xmath220 , the following assertions hold : 1 . @xmath1018 is closed . 2 . @xmath1021 . 3 . @xmath1022 . \(i ) follows from the lower semicontinuity of @xmath323 . \(ii ) follows from the fact that @xmath1016 . \(iii ) by ( ii ) , @xmath1023 contains @xmath1024 . put @xmath1025 . for any @xmath1026 , we have @xmath1027 and @xmath1028\leq\alpha$ ] , hence @xmath1029 , that is , @xmath1030 . hence @xmath1031 is contained in @xmath1019 , thus , since @xmath1031 is open , in @xmath1024 . to prove that @xmath1032 is continuous , it is sufficient to prove that @xmath1033 is open and that @xmath1019 is closed , for any @xmath220 . we start with @xmath1033 . for any @xmath1034 , there exists @xmath1027 such that @xmath1028<\alpha$ ] , that is , there exists @xmath1035 such that @xmath1036 . since @xmath1031 is open , it follows from claim [ cl : basicfg](iii ) that @xmath1037 , whence @xmath1038 . let now @xmath1039 , we prove that @xmath1030 . suppose first that @xmath54 is right isolated , that is , @xmath1040={\{\alpha,\gamma\}}$ ] for some @xmath1041 . then @xmath1042 , and so @xmath1019 is open . since @xmath76 is extremally disconnected , @xmath1043 is clopen . on the other hand , it follows from claim [ cl : basicfg](i , ii ) that @xmath1044 . therefore , by claim [ cl : basicfg](iii ) , @xmath1045 , so @xmath1030 . suppose now that @xmath54 is not right isolated , and put @xmath1046 . let @xmath1027 . by assumption , @xmath1047 , so there exists @xmath1048 such that @xmath1049 . by the definition of @xmath1032 , there exists @xmath1050 such that @xmath1051\leq\beta$ ] , that is , @xmath1052 . since @xmath1053 is open , @xmath1054 as well , so @xmath1055 since @xmath1056 . therefore , every neighborhood of @xmath5 meets @xmath1057 , that is , @xmath1058 . however , since @xmath76 is extremally disconnected and @xmath1057 is open , @xmath1059 is open as well , thus @xmath1060 . furthermore , @xmath1061 , thus @xmath1062 , thus , by claim [ cl : basicfg](iii ) , @xmath1063 . in particular , @xmath1064 . this holds for all @xmath1065 , whence @xmath1029 , that is , @xmath1030 . therefore , in both cases , @xmath1019 is closed . so we have proved the continuity of @xmath1032 . [ cl : fcontf * ] if @xmath323 is continuous , then @xmath1066 . we prove in fact that for any @xmath1067 , if @xmath323 is continuous at @xmath5 , then @xmath1068 . we have already observed that @xmath1069 . to prove the converse inequality , suppose first that @xmath1070 is right isolated , and let @xmath1071 such that @xmath1072={\{f(x),\gamma\}}$ ] . since @xmath1073 and since @xmath323 is continuous , there exists @xmath1027 such that for all @xmath1048 , @xmath1074 , that is , @xmath1075 . thus @xmath1076 , hence @xmath1077 . suppose now that @xmath1070 is not right isolated . since @xmath323 is continuous at @xmath5 , for any @xmath1071 , there exists @xmath1027 such that @xmath1074 holds for all @xmath1048 . hence @xmath1078\leq\gamma$ ] . this holds for all @xmath1071 and @xmath1070 is not right isolated , thus , again , @xmath1076 , hence @xmath1077 . if @xmath1079 is a continuous map from @xmath76 to @xmath216 , then , by claim [ cl : fcontf * ] , @xmath1080 ; the minimality statement follows . we introduce a few convenient notations . [ not : kappa+ ] we denote by @xmath1081 the proper class of all ordinals , and we extend the definitions of @xmath91 , @xmath166 , and @xmath167 ( see section [ s : basicrz2 ] ) , defining further proper classes @xmath1082 , @xmath1083 , and @xmath1084 as follows : @xmath1085 for @xmath1086 , we define @xmath1087 as the successor cardinal of @xmath1088 if @xmath1088 is an infinite cardinal , and we put @xmath1089 . that is , @xmath1087 is the immediate successor of @xmath1088 in @xmath1084 . [ not : fresu ] let @xmath76 be a set , let @xmath1090 be a subset of @xmath76 , let @xmath216 be a set with a distinguished zero element @xmath188 , let @xmath217 . we denote by @xmath1091 the map from @xmath76 to @xmath216 defined by the rule @xmath1092 [ not : omiki ] let @xmath76 be a topological space , written as a disjoint union @xmath1093 , where @xmath325 and @xmath1094 , , @xmath1095 are clopen subsets of @xmath76 . let @xmath1096 , , @xmath1097 be topological spaces . we define the set @xmath1098 as the set of all maps @xmath1099 such that @xmath1100 , for all @xmath387 . of course , @xmath1101 is naturally isomorphic to the direct product @xmath1102 . however , we find the present notation more convenient for such statements as proposition [ p : gendelta ] . [ t : c(o , k)dimint ] let @xmath76 be an extremally disconnected topological space , written as a disjoint union @xmath1103 , for clopen subsets @xmath37 , @xmath38 , @xmath39 of @xmath76 . let @xmath40 be an ordinal . then the space @xmath1104 is a continuous dimension scale . let @xmath216 be one of the totally ordered sets @xmath91 , @xmath166 , or @xmath167 , endowed with its interval topology and its natural monoid structure . we prove that @xmath1105 is a continuous dimension scale . the proof of the general case of theorem [ t : c(o , k)dimint ] follows since the direct product of any family of continuous dimension scales is again a continuous dimension scale , see lemma [ l : findpdi ] . we first observe that @xmath13 is a _ total _ ( as opposed to partial ) commutative monoid . it is obvious that the algebraic preordering of @xmath13 is antisymmetric and that @xmath13 has a largest element . furthermore , let @xmath1106 be a nonempty family of elements of @xmath13 . define @xmath217 as the componentwise join of all the @xmath1107 , for @xmath62 . then @xmath323 is lower semicontinuous . by proposition [ p : extlsc ] , there exists a least continuous map @xmath1012 such that @xmath1016 . then @xmath1032 is the supremum of @xmath1108 in @xmath13 . hence @xmath13 satisfies axiom ( m2 ) . we now observe that @xmath1091 ( see notation [ not : fresu ] ) belongs to @xmath13 , for any @xmath1109 and any clopen subset @xmath1090 of @xmath76 . if @xmath323 is the constant function with value @xmath54 , for @xmath220 , then we shall write @xmath1110 instead of @xmath1091 , and we shall of course write @xmath54 instead of @xmath1111 . we shall use the following notation . for @xmath323 , @xmath1112 , we put @xmath1113\!]}}&={{\{x\in\omega\midf(x)\leq g(x)\}}},\\ { { [ \![}f = g{]\!]}}&={{\{x\in\omega\midf(x)=g(x)\}}},\\ { { [ \![}f < g{]\!]}}&={{\{x\in\omega\midf(x)<g(x)\}}}. \end{aligned}\ ] ] we observe that @xmath1114\!]}}$ ] and @xmath1115\!]}}$ ] are closed , while @xmath1116\!]}}$ ] is open . for any @xmath1109 , put @xmath1117\!]}}$ ] . then @xmath1118 is open , thus , since @xmath76 is extremally disconnected , @xmath1119 is clopen . we put @xmath1120 . observe that @xmath1121 has only infinite values ; in particular , it is purely infinite . [ cl : cancdf][cl : candft ] let @xmath1109 . then the following are equivalent : 1 . @xmath323 is cancellable in @xmath13 . @xmath323 is directly finite in @xmath13 . 3 . @xmath1122 . ( i)@xmath426(ii ) is trivial . ( ii)@xmath426(iii ) suppose that @xmath1123 . then the clopen set @xmath1124 is nonempty , and @xmath1125 , so @xmath323 is not directly finite . ( iii)@xmath426(i ) if @xmath1122 , then @xmath1118 is dense in @xmath76 . now let @xmath324 , @xmath1126 such that @xmath1127 . then @xmath1128 for all @xmath1129 ( because the values of @xmath323 on @xmath1118 are finite ) , thus , since @xmath1118 is dense in @xmath76 , @xmath1130 . hence @xmath323 is cancellable . we thus obtain axiom ( m5 ) . [ cl : pi+df ] every element of @xmath13 can be written under the form @xmath1131 , where @xmath324 is cancellable and @xmath1132 is purely infinite . let @xmath1109 . by claim [ cl : cancdf ] , @xmath1133 is cancellable . we put @xmath1134 . since @xmath1135 and @xmath1132 is purely infinite , the conclusion of the claim holds . now a weak form of pseudo - cancellation . @xmath13 satisfies the following statement : @xmath1136 this follows immediately from claims [ cl : candft ] and [ cl : pi+df ] , by putting @xmath1137 . as a consequence , the equality @xmath1138 holds ( see corollary [ c : a / infty ] ) , for any @xmath1109 , that is , @xmath1121 is , indeed , the largest @xmath1112 such that @xmath1139 . furthermore , we observe that for @xmath323 , @xmath1112 , the componentwise meet @xmath1140 of @xmath1141 belongs to @xmath13 , and it is the meet of @xmath1141 in @xmath13 . obviously , @xmath1142 , for all @xmath323 , @xmath324 , @xmath1126 . by using proposition 1.23 of @xcite , it follows that @xmath13 satisfies the refinement property . so , @xmath13 satisfies axiom ( m1 ) . now we characterize the projections of @xmath13 . for any clopen set @xmath1090 of @xmath76 , the map @xmath1143 defines obviously a projection of @xmath13 . [ cl : sgencomp ] @xmath13 has general comparability . let @xmath323 , @xmath1112 . put @xmath1144\!]}}}}$ ] . since @xmath76 is extremally disconnected , @xmath1031 is clopen . it is obvious that @xmath1145 and that @xmath1146 . [ cl : projpu ] the projections of @xmath13 are exactly the @xmath1147 , where @xmath1090 is a clopen subset of @xmath76 . let @xmath26 be a projection of @xmath13 . put @xmath1148 and @xmath1149 . then @xmath1150 and @xmath1151 , thus there exists a clopen subset @xmath1090 of @xmath76 such that @xmath1152 . so @xmath1153 , for all @xmath1109 . conversely , @xmath1154 , thus @xmath1155 . so @xmath1156 , for all @xmath1109 . [ cl : compbvfleg ] @xmath13 satisfies axiom . let @xmath323 , @xmath1112 , we prove that there exists a largest projection @xmath26 of @xmath13 such that @xmath1157 . for @xmath1090 a clopen subset of @xmath76 , @xmath1158 if and only if @xmath1159 , where we put @xmath1160\!]}}$ ] . since @xmath76 is extremally disconnected , @xmath1161 is clopen , hence , by claim [ cl : projpu ] , @xmath1162 is the largest projection @xmath26 of @xmath13 such that @xmath1157 . by claims [ cl : projpu ] and [ cl : compbvfleg ] , it follows that for any @xmath1109 , @xmath1163 , where we put @xmath1164\!]}}}}$ ] . it remains to verify ( m6 ) . [ cl : charrem ] for all @xmath323 , @xmath1165 , @xmath1166 if and only if @xmath1015 and@xmath1167\!]}}\cap{{[\![}f = g{]\!]}}$ ] is nowhere dense . we use the alternate form of @xmath1168 given in lemma [ l : gcivptr](ii ) , justified by lemma [ l : gcimppc ] and claim [ cl : sgencomp ] . we put @xmath1169\!]}}\cap{{[\![}f = g{]\!]}}$ ] . observe that @xmath1167\!]}}={{[\![}\aleph_0\leq g{]\!]}}$ ] ( because @xmath1165 ) , hence it is clopen . hence @xmath18 is closed . suppose first that @xmath1166 . of course , it follows that @xmath1015 . towards a contradiction , suppose that @xmath18 is not nowhere dense . since @xmath76 is extremally disconnected , @xmath1170 is nonempty and clopen . put @xmath1171 . then @xmath1172 , thus , since @xmath1166 , @xmath1130 , so @xmath1173 , a contradiction since @xmath1090 is nonempty and contained in @xmath1167\!]}}$ ] . conversely , suppose that @xmath1015 and that @xmath18 is nowhere dense . let @xmath1126 such that @xmath1172 . let @xmath1174\!]}}\setminus a$ ] . since @xmath1175 with @xmath1176 and @xmath1177 , @xmath1178 . so @xmath324 and @xmath1132 agree on @xmath1167\!]}}\setminus a$ ] , with @xmath18 nowhere dense and @xmath1167\!]}}$ ] clopen , hence @xmath324 and @xmath1132 agree on @xmath1167\!]}}$ ] . it is obvious that both @xmath324 and @xmath1132 are zero on @xmath1179\!]}}$ ] , so , finally , @xmath1130 . this proves that @xmath1166 . towards a proof of ( m6 ) , let @xmath323 , @xmath1165 with @xmath1166 , and set @xmath1180\!]}}\cap{{[\![}f\nobreak = \nobreak g{]\!]}}$ ] . we define a map @xmath1181 by the rule @xmath1182\!]}})\\ f(x)&(\text{if } x\in a)\\ f(x)^+&(\text{if } x\in{{[\![}0<g{]\!]}}\setminus a ) , \end{cases } \qquad\text{for any } x\in\omega.\ ] ] in the display above , @xmath1070 is an element of @xmath1084 , and @xmath1183 denotes the successor of @xmath1070 in @xmath1084 , see notation [ not : kappa+ ] . at this point , we observe the obvious inequality @xmath1184 . [ cl : olfusc ] the map @xmath1185 is upper semicontinuous . let @xmath1067 , and put @xmath1186 . since @xmath1187\!]}}}=0 $ ] and @xmath1179\!]}}$ ] is clopen , @xmath1185 is continuous at every point of @xmath1179\!]}}$ ] . now suppose that @xmath1188 . suppose first that @xmath32 . so @xmath1189 , thus @xmath1190\!]}}\cap{{[\![}g=\alpha{]\!]}}$ ] is an open neighborhood of @xmath5 . let @xmath1048 . if @xmath246 , then @xmath1191 . if @xmath1192 , then @xmath1193 , thus @xmath1194 . therefore , @xmath1195 . since @xmath1196 , it follows that @xmath1185 is continuous at @xmath5 . now suppose that @xmath1197 . then @xmath1198\!]}}$ ] is an open neighborhood of @xmath5 , and @xmath1199 and @xmath1200 for all @xmath1048 . so , if @xmath1201 such that @xmath1202 , that is , @xmath1203 , then @xmath1204 for any @xmath1048 . hence @xmath1185 is upper semicontinuous at @xmath5 . by claim [ cl : olfusc ] and proposition [ p : extlsc ] ( used for the dual partially ordered set of @xmath216 ) , for any upper semicontinuous map @xmath1205 , there exists a largest element @xmath1206 of @xmath13 such that @xmath1207 , and @xmath1206 is given by the formula @xmath1208,\quad\text{for all } x\in\omega.\ ] ] by claim [ cl : olfusc ] , we can apply this to @xmath1209 , thus obtaining an element @xmath1210 of @xmath13 . since the range of @xmath1185 is contained in @xmath167 , so is the range of @xmath1132 , whence @xmath1211 . furthermore , @xmath1184 and , since @xmath323 is continuous , @xmath1212 ( see proposition [ p : extlsc ] ) , thus @xmath1213 it follows from the definition of @xmath1185 that @xmath1214 if and only if@xmath1188 , for any @xmath1067 . since @xmath1167\!]}}$ ] is clopen , it follows that @xmath1167\!]}}={{[\![}0<h{]\!]}}$ ] , whence @xmath1215 . the relation @xmath1216 holds . we have seen that @xmath1217 . put @xmath1218\!]}}\cap{{[\![}f = h{]\!]}}$ ] . towards a contradiction , assume that @xmath1219 . since @xmath18 is closed and nowhere dense , @xmath1220 is a nonempty open subset of @xmath1167\!]}}$ ] . pick @xmath1221 such that @xmath1186 is minimum . then @xmath1222\!]}}$ ] is an open neighborhood of @xmath5 . for any @xmath1048 , @xmath1199 and @xmath1223 , thus , by minimality of @xmath54 , @xmath1224 . this holds for any @xmath1048 , and @xmath1225\!]}}\setminus a$ ] , thus @xmath1226 for any @xmath1048 . since @xmath1031 is an open neighborhood of @xmath5 , @xmath1227 , which contradicts by claim [ cl : charrem ] , @xmath1216 . to conclude the proof of theorem [ t : c(o , k)dimint ] , it suffices to prove that if @xmath1228 is any element of @xmath747 such that @xmath1229 and @xmath1230 , then @xmath1231 . observe first that from the assumption @xmath1230 , lemma [ l : cc(a)](i ) implies that @xmath1232 , and so @xmath1167\!]}}={{[\![}0<k{]\!]}}$ ] . since @xmath1229 , @xmath1233 and @xmath1234\!]}}\cap{{[\![}f = k{]\!]}}$ ] is nowhere dense . for any @xmath1235\!]}}\setminus b$ ] , the inequality @xmath1236 holds , thus @xmath1237 . since @xmath14 is nowhere dense and @xmath1238\!]}}$ ] is open , @xmath1239\!]}}}\leq k|_{{{[\![}0<f{]\!]}}}$ ] . now let @xmath1240\!]}}$ ] . if @xmath1241 , then @xmath1242 . if @xmath1188 , then , since @xmath1167\!]}}={{[\![}0<k{]\!]}}$ ] , @xmath1243 , so @xmath1244 . therefore , we have proved that @xmath1231 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ standing hypothesis : @xmath13 is a continuous dimension scale . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we first prove the following lemma . [ l : orthcpl ] let @xmath124 , @xmath228 and @xmath360 such that @xmath882 . if @xmath1245 , then @xmath150 . the statement @xmath1245 means that @xmath1246 , that is , by lemma [ l : cc(a ) ] , @xmath1247 . but by lemma [ l : projcont](ii ) , the range of any projection of @xmath13 is closed under suprema . in particular , @xmath1248 . we prove here an important structural result about @xmath445 . [ p : b(s)cba ] the boolean algebra @xmath445 is complete . if @xmath716 is any family of projections of @xmath13 , then , for all @xmath330 , 1 . if @xmath553 , then @xmath1249 . 2 . @xmath1250 . furthermore , if @xmath1251 , then @xmath1252 . the cases of ( i ) and ( ii ) where @xmath56 is finite follow from proposition [ p : pvwq(x ) ] . therefore , by replacing in ( i ) ( resp . , in ( ii ) ) the family @xmath716 by the family of all nonempty finite meets ( resp . , finite joins ) of the @xmath1253-s , we can assume without loss of generality that @xmath56 is an upward directed partially ordered set and that @xmath938 in @xmath56 implies @xmath1254 ( resp . , @xmath1255 ) . let us suppose that @xmath1256 we consider only the nontrivial case where @xmath553 . the supremum @xmath1257 is defined and @xmath418 , for all @xmath330 . since @xmath553 , @xmath1258 is also defined . the maps @xmath26 and @xmath31 are endomorphisms of @xmath13 . it is obvious that @xmath419 . let @xmath5 , @xmath6 , @xmath532 such that @xmath1259 . we compute : @xmath1260 a similar argument , based on corollary [ c:+mj](i ) , proves that @xmath31 is an endomorphism of @xmath13 . @xmath1261 . let @xmath5 , @xmath224 . for all @xmath62 , @xmath1262 , thus , since @xmath1263 , @xmath1264 . this holds for all @xmath62 , thus , by lemma [ l : orthcpl ] , @xmath1265 . [ cl : pperpq ] @xmath26 and @xmath31 are projections of @xmath13 , and @xmath1266 . by the definition of a projection ( definition [ d : proj ] ) and by claim [ cl : pperpq ] , it remains to prove that @xmath1267 for all @xmath330 . for all @xmath62 , @xmath1268 . since @xmath1269 , @xmath1270 is defined and @xmath1271 . this holds for all @xmath173 , thus , by corollary [ c:+mj](ii ) , @xmath1272 is defined and @xmath1273 . for all @xmath62 , @xmath1274 , thus @xmath1275 . by taking the infimum over all @xmath173 , we obtain that @xmath1276 ; whence @xmath1277 . therefore , @xmath1267 . it follows easily ( use lemma [ l : pleqortq ] ) that @xmath1278 and that @xmath1279 . this proves simultaneously ( i ) and ( ii ) . finally , suppose that @xmath1251 . we prove that @xmath1252 . for @xmath1280 , this is trivial ( @xmath1281 ) , so suppose that @xmath553 . we have seen above that @xmath1282 , for all @xmath330 . hence , @xmath488 if and only if @xmath1283 , if and only if @xmath1284 for all @xmath173 ( because @xmath1285 for all @xmath173 ) , if and only if @xmath1286 . [ p : xbotbot ] let @xmath78 be a subset of @xmath13 . then @xmath1287 . by using proposition [ p : b(s)cba ] , we define a projection @xmath26 of @xmath13 by the formula @xmath1288 then @xmath1289 , thus , by proposition [ p : b(s)cba ] and lemma [ l : cc(a)](i ) , @xmath1290 it follows from this that @xmath1291 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ standing hypothesis : @xmath13 is a continuous dimension scale . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we shall now define a certain doubly indexed family of elements of @xmath747 . these elements represent in some sense the `` layers '' of @xmath747 , and a process of `` measuring '' @xmath747 against these elements will allow us to pin down the dimension theory of @xmath747 . for any @xmath500 , we define inductively a transfinite sequence of elements @xmath1293 of @xmath747 , for certain elements @xmath1088 of @xmath1084 , as follows . 1 . @xmath1294 . 2 . let @xmath1086 , and suppose that @xmath1293 is defined , and that it is purely infinite . we put @xmath1295 if @xmath78 is nonempty , then , by axiom ( m6 ) , it has a least element . we denote this element by @xmath1296 . if @xmath1297 , then we say that @xmath1296 is undefined . 3 . let @xmath1298 be a limit cardinal . suppose that @xmath1292 has been defined for all @xmath1299 in @xmath1084 , and that these elements form an increasing , majorized sequence of elements of @xmath747 . then we put @xmath1300 otherwise , we say that @xmath1301 is undefined . for any @xmath500 , we define @xmath1302 as the class of all @xmath1303 such that @xmath1292 is defined . observe that @xmath1304 for all @xmath1088 . in particular , it follows that @xmath1305 . the following lemma summarizes some elementary properties of the elements @xmath1292 . [ l : basicekp ] 1 . @xmath1302 is a proper initial segment of @xmath1084 , for all @xmath1306 . @xmath1307 for all @xmath1306 and all @xmath1308 in @xmath1302 . @xmath1309 for all @xmath500 and all @xmath1308 in @xmath1302 . 4 . @xmath1294 , and @xmath1310 for all @xmath500 and all @xmath1311 . let @xmath26 , @xmath447 such that @xmath482 . then @xmath1312 and @xmath1313 , for all @xmath1314 . let @xmath500 and let @xmath54 be an infinite cardinal number such that @xmath1292 is defined . then @xmath1315 is defined if and only if there exists @xmath951 such that @xmath1316 , and then , @xmath1315 is the least such @xmath5 . \(i ) and ( ii ) are obvious . \(iii ) is an easy consequence of lemma [ l : trleqtr ] . \(iv ) by induction on @xmath1311 . the assertion @xmath1294 holds by definition . if @xmath1317 for some @xmath1318 , then , by the definition of @xmath1292 , @xmath1310 . the limit step is an easy consequence of lemma [ l : basiccc ] . \(v ) we prove the statement by induction on @xmath1303 . it is trivial for @xmath1319 . the limit step is an easy consequence of lemma [ l : projcont](ii ) . now suppose that @xmath1317 , for @xmath1318 . since @xmath1320 is defined , @xmath1321 is defined , thus , by the induction hypothesis , @xmath1322 we put @xmath1323 . since @xmath1324 , it follows from lemma [ l : projtr](i ) that @xmath1325 , that is , @xmath1326 . furthermore , by ( iv ) above and by lemma [ l : basiccc](ii ) , @xmath1327 . hence , @xmath1328 so , @xmath1329 , thus , since @xmath1330 , the element @xmath1331 is defined , and @xmath1332 . note , further , that @xmath1333 is purely infinite furthermore , by lemma [ l : projtr](i ) , the following relations hold : @xmath1334 hence , by lemma [ l : projtr](ii ) , we obtain that @xmath1335 from @xmath1336 , it follows that @xmath1337 . by using , we obtain that can be written as @xmath1338 by lemma [ l : basiccc ] and by ( iv ) above , @xmath1339 , hence , implies that @xmath1340 . hence , by taking the image under @xmath26 of each side , we obtain that @xmath1341 . therefore , by , @xmath1342 . \(vi ) we define subsets @xmath78 and @xmath79 of @xmath747 by @xmath1343 so , @xmath1315 is defined if and only if @xmath78 is nonempty , which implies that @xmath79 is nonempty . conversely , suppose that @xmath79 is nonempty . for all @xmath1344 , @xmath1345 , thus , since @xmath1346 and by ( iv ) , @xmath1347 . thus , by lemma [ l : basiccc](ii ) , @xmath1348 . furthermore , @xmath1349 , thus , by lemma [ l : projtr](i ) , @xmath1350 , that is , @xmath1351 . therefore , @xmath1352 . in particular , @xmath1353 , so @xmath1315 is defined . for all @xmath1344 , @xmath1354 , thus @xmath1355 ; whence @xmath1356 . therefore , @xmath1315 is also the least element of @xmath79 . it follows immediately from the definition of the @xmath1357 operation that @xmath1358 provided the left hand side is defined . our next lemma shows that a similar `` linearity '' with respect to the variable @xmath26 holds , thus showing a `` bilinearity '' property of the operation @xmath1357 . [ l:*bilin ] let @xmath1303 , let @xmath716 be a family of elements of @xmath445 . put @xmath1278 . we make the following assumptions : 1 . @xmath1359 is defined for all @xmath62 . @xmath1360 is majorized . then @xmath1292 is defined , and @xmath1361 . we argue by induction on @xmath54 . the supremum @xmath1362 is , by assumption ( ii ) , defined . furthermore , the result of lemma [ l:*bilin ] is obvious for @xmath1319 . now suppose that @xmath1346 . suppose first that @xmath54 is a limit cardinal . by lemma [ l : basicekp](ii ) , @xmath1363 for any cardinal number @xmath1035 . therefore , by the induction hypothesis , @xmath1364 is defined and @xmath1365 . this holds for all @xmath1035 , thus , by definition , @xmath1292 is defined and @xmath1345 , thus , since the converse inequality is obvious , @xmath1366 . now we assume that @xmath1317 , for some @xmath1318 . put @xmath1367 . for any @xmath62 , since @xmath1359 is purely infinite , it follows from corollary [ c : a / infty](i ) that @xmath1368 . furthermore , it follows from lemma [ l : basicekp](v ) that @xmath1369 , thus @xmath1370 . hence , by lemma [ l : basicekp](iv ) , for all @xmath62 , @xmath1371 . since this holds for all @xmath1372 and since @xmath1373 , we obtain the equality @xmath1374 for all @xmath62 , since the relations @xmath1375 and @xmath1376 hold , we deduce from lemma [ l : trleqtr ] that the following relation holds : @xmath1377 in particular , @xmath1378 , thus , by the induction hypothesis , @xmath1364 is defined and @xmath1379 . furthermore , it follows from and from lemma [ l : projtr](i ) that the relation @xmath1380 holds for all @xmath62 . by lemma [ l : basicekp](v ) , @xmath1381 , hence @xmath1382 . this holds for all @xmath62 , thus , by lemma [ l : projtr](ii ) , @xmath1383 . again by lemma [ l : basicekp](v ) , @xmath1384 , so that @xmath1385 . therefore , by , @xmath1292 is defined and @xmath1386 . since the inequality @xmath1387 is obvious , @xmath1388 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ standing hypothesis : @xmath13 is a continuous dimension scale . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ for every @xmath330 , we put @xmath1389 the following result is the main fact about the dimension theory of @xmath747 . [ l : uxdense ] the set @xmath1390 is a coinitial lower subset of @xmath1391 , for all @xmath330 . by replacing @xmath5 by @xmath1392 , we may assume without loss of generality that @xmath5 is purely infinite . the fact that @xmath1390 is a lower subset of @xmath1391 is an obvious consequence of lemma [ l : basicekp ] . now let us prove that @xmath1390 is coinitial in @xmath1391 . so , let @xmath1306 . we find @xmath1393 $ ] such that @xmath1394 . first , if @xmath1395 , then there exists @xmath1393 $ ] such that @xmath470 , thus , obviously , @xmath1394 . so we consider now the case where @xmath1347 . since the sequence of all @xmath1292 , for @xmath1396 , is strictly increasing ( see lemma [ l : basicekp](ii ) ) and continuous at limits , there exists a largest element @xmath54 of @xmath1084 such that @xmath1292 is defined and @xmath1345 . since @xmath1397 and @xmath1348 ( use lemma [ l : basiccc ] ) , the element @xmath1398 is defined , and @xmath1399 . hence , @xmath54 is an infinite cardinal number . now we put @xmath1400 . by lemma [ l : bvtr ] , @xmath1401 if @xmath1402 , then we obtain that @xmath1403 , whence , by , @xmath1316 . by lemma [ l : basicekp](vi ) , @xmath1315 is defined and @xmath1356 , which contradicts the definition of @xmath54 . hence , @xmath1404 , that is , @xmath538 is nonzero . furthermore , @xmath1405 since , on the other hand , @xmath1345 , we obtain that @xmath1406 . therefore , @xmath1407 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ for the remainder of section , we denote by @xmath76 the ultrafilter space of @xmath445 . by propositions and , @xmath76 is a complete boolean space . the clopen sets of @xmath76 are exactly the sets of the form @xmath1408 for all @xmath500 . moreover , we shall fix an ordinal @xmath40 such that @xmath1409 defined implies that @xmath1410 , for all @xmath1306 . the existence of such a @xmath40 is ensured by lemma . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we now define , for any @xmath330 and any @xmath1411 , @xmath1412 the rather involved construction of the elements @xmath1292 will give us more control over the function @xmath1413 just defined than one has over ( analogues of ) the infinite dimension functions on nonsingular injective modules constructed in ( * ? ? ? * chapter xiii ) and ( * ? ? ? * chapter 12 ) . [ l : murelev ] the function @xmath1413 is a continuous map from @xmath76 to @xmath167 , for any @xmath330 . we recall here that @xmath167 is endowed with its interval topology . for any @xmath1414 , we define subsets @xmath1415 and @xmath1416 of @xmath76 by the formulas @xmath1417 @xmath1416 is open , for any @xmath1414 . let @xmath1418 . by the definition of @xmath1413 , there exist @xmath1419 in @xmath167 and @xmath1420 such that @xmath1292 is defined and @xmath1345 . thus , for any @xmath1421 , @xmath1422 , that is , @xmath1423 . to conclude the proof of lemma [ l : murelev ] , it suffices to prove that @xmath1415 is closed , for any @xmath1414 . this is trivial for @xmath1424 . if @xmath1088 is a limit cardinal , then the equality @xmath1425 holds , hence it is sufficient to prove that @xmath1426 is closed , for any @xmath1427 . towards this goal , we first observe that @xmath1428 , thus , by the claim above , @xmath1426 is open . since @xmath76 is extremally disconnected , the closure @xmath1429 of @xmath1426 is clopen , thus it has the form @xmath1430 , for some @xmath500 . if @xmath1431 then @xmath1432 and we are done , so suppose that @xmath1433 . for any @xmath1393 $ ] , @xmath1434 meets @xmath1426 , thus there exists @xmath1435 such that @xmath1436 . hence there exists @xmath1437\cap{\mathfrak{a}}$ ] such that @xmath1438 is defined and @xmath1439 . therefore , the set of all @xmath1440 $ ] such that @xmath1438 is defined and @xmath1439 is coinitial in @xmath1441 $ ] , which proves , by lemma [ l:*bilin ] , that @xmath1315 is defined and @xmath1356 . this means that @xmath1442 . therefore , @xmath1443 is clopen . for all @xmath330 , we put @xmath1444 where @xmath1390 has been defined in . it follows from lemma [ l : uxdense ] that _ @xmath1445 is a dense , open subset of @xmath76_. [ l : mucst ] let @xmath330 . for any @xmath1446 , @xmath1447 is the unique element @xmath54 of @xmath167 such that @xmath1448 let @xmath1449 such that @xmath1450 . by the definition of @xmath1390 , there exists @xmath1396 such that @xmath1451 . in particular , @xmath1452 . let @xmath1453 and @xmath1454 such that @xmath1455 . then @xmath538 belongs to @xmath1456 and @xmath1457 , so @xmath1458 , thus @xmath1459 , from which it follows that @xmath1460 . hence @xmath1461 , so @xmath1462 . hence @xmath1463 , so , finally , @xmath1464 . [ p : pidim ] 1 . @xmath1465 , for all @xmath5 , @xmath224 such that @xmath1466 is defined . 2 . @xmath1467 if and only if @xmath181 , for all @xmath5 , @xmath1468 . the set @xmath1469 $ ] is a lower subset of @xmath1470 . in particular , the restriction of @xmath90 to @xmath747 is a lower embedding from @xmath747 into @xmath1470 . \(i ) put @xmath1471 . let @xmath1472 , and put @xmath1473 and @xmath1474 . by lemma [ l : mucst ] , there exists @xmath1420 such that @xmath1475 hence , @xmath1476 hence , @xmath1477 , and @xmath1478 . therefore , @xmath1479 and @xmath1480 agree on an open dense subset of @xmath76 , so , since they are continuous , they are equal . \(ii ) by ( i ) , @xmath181 implies that @xmath1467 . conversely , for any @xmath1481 , there exist @xmath54 , @xmath1482 such that @xmath1292 and @xmath1364 are defined and equal to @xmath815 and @xmath1483 , respectively . since @xmath1484 , there exists @xmath1411 such that @xmath1420 . then , by lemma [ l : mucst ] , @xmath1464 and @xmath1485 , so @xmath1467 implies , by lemma [ l : basicekp](ii ) , that @xmath956 . hence , @xmath594 , that is , @xmath1486 . this holds for all @xmath26 in the coinitial subset @xmath1487 of @xmath1391 ( see lemma [ l : uxdense ] ) , so @xmath181 . \(iii ) let @xmath951 , and let @xmath1488 such that @xmath1489 . we find @xmath713 in @xmath747 such that @xmath1490 . we put @xmath1491 . [ cl : udense ] @xmath1090 is a coinitial lower subset of @xmath1391 . it is obvious that @xmath1090 is a lower subset of @xmath1391 . let @xmath1306 , we find @xmath1393\cap u$ ] . let @xmath54 be the minimum value of @xmath323 on @xmath1430 . then @xmath1492 is , by continuity of @xmath323 , an open subset of @xmath76 , and @xmath1493 . let @xmath1393 $ ] such that @xmath1494 . then @xmath1495 is constant with value @xmath54 . now let @xmath1496 be a maximal antichain of @xmath1497 . by claim [ cl : udense ] above , @xmath1496 is also a maximal antichain of @xmath1391 . let @xmath1498 denote the constant value of @xmath323 on @xmath1499 , for all @xmath62 . if @xmath1500 , then @xmath1501 . since @xmath1502 , the equality @xmath1503 holds . hence , @xmath1504 is defined and @xmath1505 . it follows that @xmath1506 is a majorized ( by @xmath5 ) subset of @xmath747 , thus it has a supremum , say , @xmath6 . note that @xmath713 . furthermore , @xmath1507 , for all @xmath62 and all @xmath1500 . hence , @xmath1508 and @xmath323 agree on a dense open subset of @xmath76 , so , since they are continuous , @xmath1490 . the following trivial property of @xmath90 will later prove very useful . [ p : mucommproj ] the equality @xmath1509 holds , for all @xmath330 and all @xmath500 . we start with the following easy but fundamental result . [ l : segdimint ] let @xmath13 be a continuous dimension scale , let @xmath241 be a lower subset of @xmath13 , viewed as a partial submonoid of @xmath13 . then the following assertions hold : 1 . @xmath241 is closed under all projections of @xmath13 , and @xmath1510 for all @xmath500 . every projection of @xmath241 extends to a projection of @xmath13 . if @xmath241 is dense in @xmath13 , then every projection of @xmath241 extends to a unique projection of @xmath13 . we recall that by lemma [ l : segdi ] , @xmath241 is a continuous dimension scale . furthermore , @xmath418 for any projection @xmath26 of @xmath13 and any @xmath330 . since @xmath241 is a lower subset of @xmath13 and @xmath418 for any projection @xmath26 of @xmath13 and any @xmath330 , ( i ) holds . now we prove ( ii ) . so let @xmath1511 . by the definition of a projection , @xmath1512 . moreover , it follows from proposition [ p : xbotbot ] that @xmath1513 , where orthogonals are computed in @xmath13 . therefore , there exists a projection @xmath31 of @xmath13 such that @xmath1514 and @xmath1515 . from @xmath1516 it follows that @xmath1517 since @xmath1518 and @xmath1519 are orthogonal in @xmath241 and @xmath241 is a lower subset of @xmath13 , @xmath1518 and @xmath1519 are orthogonal in @xmath13 . hence @xmath1520 , which implies that @xmath1521 from and it follows that @xmath1522 . now we prove ( iii ) . it suffices to prove that if @xmath26 , @xmath447 and @xmath484 for all @xmath1523 , then @xmath482 . suppose otherwise . then @xmath1524 , thus there exists a nonzero element @xmath124 in @xmath1525 . by the assumption of ( iii ) , we can suppose without loss of generality that @xmath1526 . so @xmath1527 while @xmath680 with @xmath1528 , a contradiction . the following series of results allows to relate the structure of the lower subset of directly finite elements of a continuous dimension scale to the dimension function @xmath90 introduced in section [ s : dimfctmu ] . we first introduce a definition . let @xmath13 be a partial commutative monoid . an element @xmath124 of @xmath13 is _ multiple - free _ ( or , in some references , _ abelian _ ) , if @xmath1529 implies that @xmath66 , for all @xmath330 . we denote by @xmath1530 the subset of @xmath13 consisting of all multiple - free elements . it is not hard to verify that in any partial refinement monoid , multiple - free elements are cancellable ( see definition [ d : dirfinmon ] ) . in continuous dimension scales , this is also an immediate consequence of axiom ( m5 ) and lemma [ l : dfimcanc ] . multiple - free elements of a lattice - ordered group are called _ singular _ in @xcite . we recall that @xmath1531 and @xmath1532 , see section [ s : basicrz2 ] . [ not : cfin ] let @xmath76 be a topological space , and let @xmath216 be either @xmath1533 or @xmath1534 , endowed with the interval topology . we denote by @xmath1535 the set of all continuous maps @xmath217 such that @xmath1536 is nowhere dense . we extend notation [ not : omiki ] to @xmath1537 , thus defining , for topological spaces @xmath37 and @xmath38 , @xmath1538 [ p : embcancdimf ] let @xmath13 be a stably finite continuous dimension scale . we suppose that @xmath1530 is dense in @xmath13 . let @xmath76 be the the ultrafilter space of @xmath445 . then there exists a map @xmath1539 that satisfies the following conditions : 1 . @xmath1540 is a lower embedding . 2 . @xmath1541 , for all @xmath330 and all @xmath500 . observe that @xmath13 is cancellative ( see lemma [ l : dfimcanc ] ) . by lemma [ l : refsm1 ] , @xmath291 is the positive cone of a dedekind complete lattice - ordered group , say , @xmath202 . by proposition [ p : lgrpdi ] , @xmath203 is a continuous dimension scale , and by lemma [ l : segdimint](i , iii ) , the restriction map @xmath1542 , @xmath1543 is an isomorphism . hence it suffices to prove that the conclusion of proposition [ p : embcancdimf ] holds in case @xmath1544 , that is , @xmath13 is a _ total _ monoid . by thorme 13.5.6 of @xcite , there exist a complete boolean space @xmath1545 and a lower embedding @xmath1540 of @xmath203 into @xmath1546 . this map @xmath1540 is defined as an `` evaluation map '' on the stone space @xmath1545 , which implies that the condition ( ii ) above is satisfied ( see p. 272 in @xcite for the definition of @xmath1540 ) . furthermore , @xmath1547 is the ultrafilter space of the complete boolean algebra of polar subsets of @xmath202 , thus , @xmath1548 canonically , so we may replace @xmath1545 by @xmath76 . in the case where there are no nontrivial multiple - free elements , we get the following . [ p : embcancdidiv ] let @xmath13 be a stably finite continuous dimension scale with no nontrivial multiple - free element . then there exists a map @xmath1549 that satisfies the following conditions : 1 . @xmath1540 is a lower embedding . 2 . @xmath1541 , for all @xmath330 and all @xmath500 . the proof of proposition [ p : embcancdidiv ] follows the lines of the proof of proposition [ p : embcancdimf ] , with the following changes . the dedekind complete lattice - ordered group @xmath202 has no nontrivial multiple - free element , thus , by theorem 11.2.13 of @xcite , it is divisible . the rest of the proof is the same as for proposition [ p : embcancdimf ] , by using corollaire 13.4.2 of @xcite instead of thorme 13.5.6 of @xcite . as experience proves , it is often useful to state explicitly the definition of the embedding @xmath1540 of propositions [ p : embcancdimf ] and [ p : embcancdidiv ] . the definition that we present here is equivalent to the one given by s.j . bernau s embedding theorem for archimedean lattice - ordered groups , see @xcite , or ( * ? ? ? * theorem 2.4 ) . it is convenient to first define the concept of a _ finitary unit _ in a continuous dimension scale . [ d : finun ] let @xmath13 be a continuous dimension scale . a _ finitary unit _ of @xmath13 is a dense antichain @xmath201 of @xmath656 such that for any @xmath1550 , either @xmath611 is multiple - free or there is no nonzero multiple - free element below @xmath611 . [ l : basis ] every continuous dimension scale has a finitary unit . let @xmath1090 denote the set of all elements @xmath1551 that are either multiple - free or without nonzero multiple - free element below . let @xmath1552 . if there is no nonzero multiple - free element below @xmath124 , then @xmath1553 . if there exists a nonzero multiple - free element @xmath1554 , observe that @xmath1555 . so @xmath1090 is dense in @xmath656 , and the finitary units of @xmath13 are exactly the maximal antichains of @xmath1090 . now let @xmath202 be a dedekind complete lattice - ordered group . every polar subset of @xmath202 is an orthogonal direct summand of @xmath202 , thus @xmath1556 defines an isomorphism from @xmath1557 onto the boolean algebra of polar subsets of @xmath202 . we denote again by @xmath76 the ultrafilter space of @xmath1558 . let @xmath201 be a finitary unit of the continuous dimension scale @xmath203 ( see definition [ d : finun ] ) . we put @xmath1559 for all @xmath1560 and @xmath1411 . then @xmath1561 is a continuous map from @xmath76 to @xmath1534 , for all @xmath1067 , and @xmath1540 is the desired embedding . unlike the map @xmath90 given in , the map @xmath1540 is not intrinsic , for it depends on the choice of a finitary unit of @xmath203 . furthermore , it is not apparent through that under the assumptions of proposition [ p : embcancdimf ] , the map @xmath1540 takes its values in @xmath1562 ( rather than just in @xmath1563 ) . however , it is possible to prove that under those assumptions , since @xmath201 is a finitary unit of @xmath203 , the following equality holds , @xmath1564 for all @xmath1560 and all @xmath1411 . hence the map @xmath1561 is integer - valued . similarly , the proof that the range of @xmath1540 is , in the context of proposition [ p : embcancdimf ] , a lower embedding , uses the assumption that @xmath201 is a finitary unit of @xmath203 . [ d : si , ii , iii ] for a general continuous dimension scale @xmath13 , we define ideals @xmath1565 , @xmath1566 , and @xmath1567 of @xmath13 as follows : @xmath1568 we say that @xmath13 is _ type @xmath1569 _ ( resp . , _ type @xmath1570 _ , _ type @xmath1571 _ ) , if @xmath1572 ( resp . , @xmath1573 , @xmath1574 ) . it follows from proposition [ p : xbotbot ] that the equality @xmath1575 holds ( see notation [ not : sumsubsets ] ) . observe , in particular , that @xmath1576 . we denote by @xmath1577 ( resp . , @xmath1578 , @xmath1579 ) the projection of @xmath13 on @xmath1565 ( resp . , @xmath1566 , @xmath1567 ) . so @xmath1580 in @xmath445 , hence @xmath1103 , where we put @xmath1581 observe that @xmath37 , @xmath38 , and @xmath39 are clopen subsets of @xmath76 . by using propositions [ p : embcancdimf ] and [ p : embcancdidiv ] , we obtain lower embeddings @xmath1582 such that @xmath1583 , for all @xmath1584 , all @xmath1585 , and all @xmath1586 . now we identify @xmath1587 with @xmath1588 , _ via _ lemma [ l : segdimint ] . hence , by combining @xmath1589 and @xmath1590 , we obtain the following result . [ p : gendelta ] let @xmath13 be a continuous dimension scale . then there exists a map @xmath1591 that satisfies the following conditions : 1 . @xmath1540 is a lower embedding . 2 . the values of @xmath1561 are finite on an open dense subset of @xmath76 , for every @xmath1592 . 3 . @xmath1541 , for all @xmath1592 and all @xmath500 . the map @xmath1540 depends on the choice of a finitary unit @xmath201 of @xmath656 , and it is then given by the formula , for all @xmath1592 and all @xmath1593 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ standing hypotheses : @xmath13 is a continuous dimension scale , @xmath1565 , @xmath1566 , @xmath1567 , @xmath37 , @xmath38 , @xmath39 , @xmath1577 , @xmath1578 , @xmath1579 are as in section . _ let @xmath40 be the ordinal and @xmath1594 the dimension function defined in section . we put @xmath1595 . we pick a lower embedding @xmath1596 as in proposition . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ l : locdf ] let @xmath72 be a majorized family of pairwise orthogonal directly finite elements of @xmath13 . then @xmath1597 is directly finite . put @xmath1598 , for all @xmath62 . observe that @xmath1599 , for all @xmath173 . let @xmath330 such that @xmath1600 . it follows that for any @xmath62 , @xmath1601 , thus , since @xmath274 is directly finite , @xmath1602 , that is , @xmath1603 . put @xmath1278 . by lemma [ l : basiccc ] , @xmath1604 , thus , by using lemma [ l : cc(a ) ] and proposition [ p : b(s)cba ] , @xmath1605 but @xmath666 , therefore , @xmath66 . [ l : valmuinf ] let @xmath951 and @xmath1411 . if @xmath1606 , then @xmath1607 . for all @xmath1608 $ ] , there exist , by lemma [ l : uxdense ] , an infinite cardinal number @xmath54 and @xmath1393 $ ] such that @xmath1609 . so , for any @xmath1435 , @xmath1610 . therefore , the set of all @xmath1611 such that @xmath1607 is dense in @xmath1612 . since @xmath1413 is continuous , the conclusion of lemma [ l : valmuinf ] follows . [ l : dflepi ] let @xmath1613 and @xmath1614 . if @xmath1615 , then @xmath131 and @xmath1616 . put @xmath749 and @xmath1617 . then @xmath751 , with @xmath565 directly finite and @xmath1618 purely infinite , thus @xmath1619 , that is , @xmath1620 . by assumption , @xmath1621 , that is , @xmath679 , thus @xmath482 . by general comparability , @xmath1622 , so , in fact , @xmath1623 . therefore , @xmath131 . now put @xmath1624 . for any @xmath1625 , it follows from lemma [ l : valmuinf ] that @xmath1626 . for any @xmath1627 , @xmath1628 because @xmath1629 . therefore , @xmath1630 , for any @xmath1411 . [ l : insegf ] let @xmath1631 be directly finite , let @xmath1614 such that @xmath1632 . then there exists a directly finite @xmath131 in @xmath13 such that @xmath1633 . we put @xmath1634 , and we define a subset @xmath1090 of @xmath1391 by @xmath1635\mid\exists x\in s_{\mathrm{fin}},\ \operatorname{cc}(x)=q\text { and } f\rfloor_{\omega_q}\leq\delta(x)\}}}.\ ] ] [ cl : weirdu ] @xmath1090 is coinitial in @xmath1441 $ ] . by lemma [ l : dflepi ] , every directly finite element of @xmath462 lies below @xmath125 , thus , since @xmath125 is purely infinite and by lemma [ l : leqstilldef ] , @xmath1636 is a total monoid . let @xmath1393 $ ] , we prove that @xmath1637 $ ] is nonempty . since @xmath1638 and @xmath1639 , @xmath494 has a directly finite , nonzero element @xmath6 . observe that @xmath1640 is a nonzero element of @xmath1563 , thus , since @xmath1640 is continuous , there exists @xmath1641 $ ] such that @xmath1642 for all @xmath1643 . without loss of generality , we may assume that @xmath1644 . suppose that @xmath1645 for all @xmath84 . then @xmath1646 , for any @xmath1643 , so @xmath1647 , which contradicts the assumption that @xmath323 is directly finite . hence , there exists a largest nonnegative integer @xmath176 such that @xmath1645 . since @xmath1640 vanishes outside @xmath1648 , there exists @xmath1649 $ ] such that @xmath1650 . therefore , @xmath449 belongs to @xmath1090 ( with witness @xmath1651 ) . by claim [ cl : weirdu ] , there exists a maximal antichain @xmath1053 of @xmath1652 $ ] such that @xmath1653 . for each @xmath1654 , pick a directly finite @xmath1655 such that @xmath1656 and @xmath1657 . observe that @xmath1658 for all @xmath31 , thus @xmath1659 is defined and @xmath1660 . furthermore , by lemma [ l : locdf ] , @xmath5 is directly finite . by lemma [ l : projcont](ii ) , @xmath1661 for all @xmath1654 . for any @xmath1654 and any @xmath1435 , @xmath1662 since @xmath1663 is dense in @xmath1430 and both @xmath323 and @xmath1561 are continuous and vanish outside @xmath1430 , it follows that @xmath1664 . since @xmath1540 is a lower embedding , there exists @xmath1665 in @xmath656 such that @xmath1633 . we now wish to define a homomorphism of partial monoids @xmath1666 by the rule @xmath1667 since @xmath1508 is purely infinite , @xmath1668 is the maximum of @xmath1561 and @xmath1508 . it follows that the existence of @xmath1669 is ensured by the following lemmas [ l : abcinf ] and [ l : acinfb ] . [ l : abcinf ] let @xmath124 , @xmath1670 and @xmath950 . if @xmath1671 , then @xmath1672 . by the refinement property , @xmath1673 for some @xmath260 and @xmath1674 in particular , @xmath1675 and @xmath1676 are directly finite , and , of course , @xmath1677 . by lemma [ l : dflepi ] , @xmath1678 . it follows that @xmath1679 . [ l : acinfb ] let @xmath124 , @xmath950 and @xmath1670 . if @xmath1671 , then @xmath1680 . since @xmath125 is directly finite and @xmath126 is purely infinite , @xmath1681 and @xmath1682 . by lemma [ l : a+b / infty ] , @xmath1683 therefore , @xmath1684 . this shows the existence of a unique homomorphism of partial monoids @xmath1666 satisfying the condition . observe that the following additional condition is satisfied by @xmath1669 ( because it is satisfied by @xmath90 and by @xmath1540 , see propositions [ p : mucommproj ] and [ p : gendelta ] ) : @xmath1685 the purpose of the following lemmas [ l : epsemb ] and [ l : epsid ] is to prove that @xmath1669 is a lower embedding . [ l : epsemb ] the map @xmath1669 is an order - embedding . let @xmath124 , @xmath228 such that @xmath1686 , we prove that @xmath131 . by corollary [ c : a / infty](iii , iv ) , there exists a directly finite @xmath258 such that @xmath1687 . suppose now that @xmath124 is directly finite . we put @xmath1688 . then @xmath1689 , with @xmath565 directly finite and @xmath1690 purely infinite , thus , by lemma [ l : dflepi ] , we obtain that @xmath1691 furthermore , @xmath1692 , hence , by using and the definition of @xmath1669 , @xmath1693 thus , since @xmath1540 is an embedding , we obtain that @xmath1694 it follows from and that @xmath131 . in the general case , there exists , by corollary [ c : a / infty](iii , iv ) , a directly finite @xmath1695 such that @xmath1696 . it follows from the previous paragraph that the inequality @xmath1697 holds . furthermore , dividing by @xmath247 the inequality @xmath1698 and using proposition [ p : pidim ] yields , since both @xmath1699 and @xmath1700 are finite - valued on a dense subset of @xmath76 and both @xmath1701 and @xmath1702 are purely infinite , that @xmath1703 by and , @xmath1704 . [ l : epsid ] the range of @xmath1669 is a lower subset of @xmath1705 . let @xmath228 and let @xmath1631 such that @xmath1706 . we find @xmath131 in @xmath13 such that @xmath1707 . by corollary [ c : a / infty](iii , iv ) , there exists a directly finite @xmath258 such that @xmath1687 . we start with the case where @xmath323 is directly finite . since @xmath1708 and since @xmath1705 satisfies the refinement property , there are @xmath324 , @xmath1709 such that @xmath1135 , @xmath1710 , and @xmath1711 . since @xmath1540 is a lower embedding , there exists @xmath1712 in @xmath656 such that @xmath1713 . since @xmath323 is directly finite and @xmath1714 , @xmath1132 is directly finite , thus , by lemma [ l : insegf ] , there exists a directly finite @xmath1715 such that @xmath1716 . since @xmath1712 , @xmath1715 , and @xmath1717 , @xmath1466 is defined , @xmath1718 , and @xmath1719 . now the general case . since @xmath1705 is a continuous dimension scale , there exists a directly finite @xmath1720 such that @xmath1721 . by the previous paragraph , @xmath1722 for some directly finite @xmath1723 . furthermore , by dividing by @xmath247 the inequality @xmath1724 , we obtain that @xmath1725 , thus , by proposition [ p : pidim ] , there exists @xmath1726 in @xmath747 such that @xmath1727 . since @xmath1723 , @xmath1726 , and @xmath1728 , @xmath1729 is defined , @xmath1730 , and @xmath1731 . we finally arrive at the following more precise version of theorem c. [ t : embdimint ] let @xmath13 be a continuous dimension scale , let @xmath445 be the complete boolean algebra of projections of @xmath13 , and let @xmath76 be the ultrafilter space of @xmath445 , with the decomposition @xmath1103 as given in section . then there exist an ordinal @xmath40 and a lower embedding @xmath1732 such that @xmath1733 , for all @xmath330 and all @xmath500 . conversely , for every ordinal @xmath40 , every complete boolean space @xmath76 , decomposed as @xmath1103 with @xmath37 , @xmath38 , and @xmath39 clopen , every lower subset of the space @xmath1734 endowed with its canonical structure of partial commutative monoid , is a continuous dimension scale . of course , as observed earlier , the following isomorphism @xmath1735 holds , so we could have formulated part of theorem [ t : embdimint ] by using the right hand side of instead of its left hand side . however , the formulation of the relation @xmath1733 would have then been more cumbersome . the first statement ( existence of @xmath1669 ) follows from the construction of @xmath1669 discussed in all previous results of section [ s : embdi ] . conversely , by theorem [ t : c(o , k)dimint ] , every monoid of the form @xmath1104 is a continuous dimension scale , and , by lemma [ l : segdi ] , every lower subset of a continuous dimension scale , viewed as a partial submonoid , is a continuous dimension scale . theorem [ t : embdimint ] follows . any continuous dimension scale is a _ partial _ monoid , which sometimes makes computations cumbersome . however , the following corollary makes it possible to reduce most problems about continuous dimension scales to _ total _ monoids . [ c : embdimint ] let @xmath13 be a continuous dimension scale . then the universal monoid @xmath291 of @xmath13 is a continuous dimension scale , and @xmath1736 defines an isomorphism from @xmath1737 onto @xmath445 . by universality , the map @xmath1669 extends to a unique monoid homomorphism @xmath1738 from @xmath291 to @xmath1705 . since @xmath13 is a lower subset of @xmath291 and @xmath1669 is one - to - one , it follows from ( * ? ? ? * lemma 3.9 ) that @xmath1738 is one - to - one . since @xmath1739 $ ] is a lower subset of the refinement monoid @xmath1705 , the monoid @xmath1740 $ ] , which is equal to the submonoid of @xmath1705 generated by @xmath1739 $ ] , is also a lower subset of @xmath1705 . in particular , @xmath291 is isomorphic to a lower subset of @xmath1705 . the conclusion follows from theorem [ t : c(o , k)dimint ] and lemma [ l : segdimint ] . for the reader s convenience , we restate explicitly the construction of the map @xmath1741 of theorem [ t : embdimint ] , with @xmath1742 , for some large enough ordinal @xmath40 . we first pick a finitary unit @xmath201 of @xmath13 ( see definition [ d : finun ] ) . then we embed @xmath13 into the universal monoid @xmath291 of @xmath13 , see proposition [ p : prmrm ] . we observe that @xmath13 is a lower subset of @xmath291 ( proposition [ p : prmrm ] ) , and that @xmath1543 defines an isomorphism from @xmath1737 onto @xmath1743 ( lemma [ l : segdimint ] ) . the definition of the map @xmath1744 has the parameter @xmath201 , and it is given by the formula , that is , @xmath1745 for all @xmath1746 and all @xmath1747 . the definition of @xmath90 is intrinsic ( it does not depend on the finitary unit @xmath201 ) , but it requires the ordinal @xmath40 to be chosen large enough . it is given by the formula , that is , @xmath1748 for all @xmath330 and all @xmath1411 . finally , @xmath1749 , for all @xmath1750 . we will call this map @xmath1669 the _ canonical embedding from @xmath13 into @xmath1705 , relatively to the finitary unit @xmath201_. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ standing hypotheses : @xmath13 is a continuous dimension scale , @xmath1565 , @xmath1566 , @xmath1567 , @xmath37 , @xmath38 , @xmath39 , @xmath1577 , @xmath1578 , @xmath1579 are as in section . _ let @xmath40 be an ordinal , large enough for the dimension function @xmath1594 introduced in section to be defined . we put @xmath1595 . we fix a finitary unit @xmath201 of @xmath13 , and we let @xmath1596 and @xmath1741 be the canonical maps defined from @xmath201 in sections and . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ throughout the present section until theorem [ t : embdimint ] , we let @xmath1751 be a lower embedding satisfying the conditions @xmath1752 we shall prove that @xmath1753 . we first embed @xmath13 into its universal monoid @xmath291 . by corollary [ c : embdimint ] , @xmath291 is a continuous dimension scale . furthermore , the argument of the proof of corollary [ c : embdimint ] shows that the unique extension of @xmath1754 to a map from @xmath291 to @xmath1705 is a lower embedding . since @xmath1543 defines an isomorphism from @xmath1737 onto @xmath1743 ( lemma [ l : segdimint ] ) , @xmath1755 satisfies the condition . of course , it obviously satisfies . _ therefore , we may assume , without loss of generality , that @xmath13 is a _ total _ monoid , that is , @xmath1756_. next , it follows from theorem [ t : c(o , k)dimint ] that @xmath1705 is a continuous dimension scale . furthermore , by claim [ cl : projpu ] of the proof of theorem [ t : c(o , k)dimint ] , the projections of @xmath1705 are exactly the maps @xmath1757 , for @xmath1758 , in particular , @xmath1759 . thus we shall identify every projection @xmath26 of @xmath13 with the associated projection of @xmath1705 . modulo this identification , the central cover @xmath1760 of any @xmath1631 is exactly the topological closure of the set @xmath1238\!]}}={{\{{\mathfrak{a}}\in\omega\midf({\mathfrak{a}})>0\}}}$ ] . [ l : ccinols ] the equality @xmath1761 holds , for all @xmath330 . put @xmath1762 . from @xmath1763 and it follows that @xmath1764 , whence @xmath1765 . put @xmath1766 . then @xmath1767 , thus , since @xmath1754 is an embedding , @xmath470 . however , @xmath1768 , whence @xmath779 , that is , @xmath1769 . we compute the values of @xmath1669 on the elements of @xmath201 . let @xmath1770 denote the characteristic function of @xmath1430 , for any @xmath500 . [ l : epsone ] the equality @xmath1771 holds , for all @xmath1550 . since @xmath611 is directly finite , @xmath1772 , and it is given by . put @xmath1773 . from @xmath1774 and it follows that @xmath1775 vanishes outside @xmath1430 . now let @xmath1450 . for @xmath1776 and @xmath1777 , the relation @xmath1778 always holds for @xmath1779 ( because then , @xmath1780 , thus @xmath1781 ) , while for @xmath1782 , it is equivalent to the existence of @xmath1453 such that @xmath1783 . however , for any @xmath1453 , @xmath1784 is nonzero , thus @xmath1785 is nonzero , but it is directly finite , thus @xmath1783 if and only if@xmath1786 . the conclusion follows immediately . [ l : epsoni ] the equality @xmath1787 holds , for all @xmath1550 . the conclusion holds by assumption for @xmath1788 . now let @xmath1789 , so @xmath611 is multiple - free and @xmath1790 . moreover , it follows from lemma [ l : ccinols ] that @xmath1791 , therefore , since @xmath1792 and @xmath1790 , we obtain the inequality @xmath1793 let @xmath1794 $ ] such that @xmath1795 . since @xmath1754 is a lower embedding , there exists @xmath1796 such that @xmath1797 , thus ( we recall that @xmath13 is a total monoid ) @xmath1798 , thus , since @xmath1754 is an embedding , @xmath1799 , thus , since @xmath611 is multiple - free , @xmath66 , whence @xmath1431 . this holds for all @xmath1794 $ ] such that @xmath1795 , thus , since @xmath1775 vanishes outside @xmath1800 , we get @xmath1801 . therefore , by and lemma [ l : epsone ] , @xmath1802 . [ l : epsdelta ] the equality @xmath1803 holds , for all @xmath1592 . if the result has been established for @xmath656 , then it obviously holds for @xmath13 . hence we may assume that @xmath1804 , that is , since @xmath13 is total , @xmath13 is the positive cone of some dedekind complete lattice - ordered group ( lemma [ l : refsm1 ] ) . now put @xmath1805 , an open subset of @xmath76 . since every element of @xmath13 meets some element of @xmath201 , it follows from proposition [ p : xbotbot ] that @xmath1545 is dense in @xmath76 . now let @xmath330 . since @xmath1561 belongs to @xmath1806 ( see notation [ not : cfin ] ) , there exists an open dense subset @xmath1807 of @xmath1545 such that @xmath1808 is finite , for all @xmath1809 . since both maps @xmath1810 and @xmath1561 are continuous and @xmath1807 is dense , in order to conclude the proof , it suffices to establish the equality @xmath1811 , for all @xmath1809 . since @xmath1472 , there exists a unique @xmath1550 such that @xmath1812 , hence @xmath1813 let @xmath1777 such that @xmath1814 belongs to @xmath1456 . applying @xmath1754 to the inequality @xmath1815 and using , we obtain the inequalities @xmath1816 thus , by lemmas [ l : epsone ] and [ l : epsoni ] , @xmath1817 . evaluate at @xmath1456 . since @xmath1818 belongs to @xmath1456 , we obtain that @xmath1819 . this holds for all @xmath1820 such that @xmath1821 , whence @xmath1822 now the converse . from @xmath1809 it follows that there exists @xmath1823 such that @xmath1824 . let @xmath325 . there exists a largest @xmath1825 such that @xmath1821 , in fact @xmath1826 . suppose that @xmath1827 . there exists @xmath1828 such that @xmath1829 and @xmath1830 . on the other hand , from @xmath1831 it follows that @xmath1832 whence @xmath1833 , a contradiction ; so we have proved that @xmath1834 . from @xmath1835 and general comparability it follows that the projection @xmath1836 belongs to @xmath1456 . thus , applying @xmath1754 to the inequality @xmath1837 and using together with lemmas [ l : epsone ] and [ l : epsoni ] , we obtain the inequalities @xmath1838 whence , evaluating at @xmath1456 , @xmath1839 , so @xmath1840 this holds for all @xmath325 , thus @xmath1841 . by , the conclusion follows . we need to prove that @xmath1842 , for all @xmath951 . we recall that we have identified the projections of @xmath13 and those of @xmath1705 . for @xmath1303 and @xmath1758 , we shall denote by @xmath1843 ( resp . , @xmath1844 ) the value of @xmath1292 in @xmath13 ( resp . , in @xmath1705 ) , if defined . by @xmath1845 , we denote the function defined on @xmath76 sending any element of @xmath1430 to @xmath54 and any element of @xmath1846 to @xmath188 . [ l : scalols ] the value @xmath1844 is defined and equal to @xmath1845 , for all @xmath1427 and @xmath1758 . by induction on @xmath54 . the case @xmath1319 and the limit step are obvious . suppose that @xmath1317 , for some @xmath1482 . it is easy to verify that @xmath1847 and @xmath1848 ; thus , by the induction hypothesis , @xmath1844 is defined and lies above @xmath1845 . now suppose that @xmath1849 . there exists @xmath1393 $ ] such that @xmath1850 in particular , from @xmath1851 it follows that @xmath1852 . however , by applying @xmath31 to the relation @xmath1853 and using lemma [ l : projtr](i ) , we obtain that @xmath1854 . hence , by and lemma [ l : trleqtr ] , @xmath1855 , whence , since @xmath1852 , we obtain that @xmath779 , a contradiction . since @xmath1754 is a lower embedding , the following lemma is obvious . [ l : lowembrem ] @xmath687 if and only if @xmath1856 , for all @xmath124 , @xmath228 . [ l : epsonpalph ] the equality @xmath1857 holds , for all @xmath1758 and all @xmath1303 such that @xmath1843 is defined . by induction on @xmath54 . for @xmath1319 it is trivial . suppose that @xmath1346 and @xmath1843 is defined . it follows from the induction hypothesis that @xmath1858 , for all @xmath1035 in @xmath1084 ; whence @xmath1859 . since @xmath1754 is a lower embedding , there exists @xmath1860 in @xmath747 such that @xmath1861 . the relation @xmath1862 holds , for all @xmath1035 , thus , by lemma [ l : lowembrem ] , @xmath1863 . furthermore , by lemma [ l : ccinols ] , @xmath1864 hence , by the definition of @xmath1292 , we get that @xmath1865 , so , finally , @xmath1866 and @xmath1857 . [ l : eps=mudi ] the equality @xmath1842 holds , for all @xmath951 . we let @xmath1445 denote the open dense subset of @xmath76 defined in . it suffices to prove that the equality @xmath1867 holds , for all @xmath1446 . since @xmath5 is purely infinite and @xmath1446 , the value @xmath1473 is the unique element of @xmath1084 such that @xmath1843 is defined and equal to @xmath815 , for some @xmath1420 . therefore , we can compute : @xmath1868 by putting together lemmas [ l : epsdelta ] and [ l : eps=mudi ] , we obtain the following . [ c : uneps ] the equality @xmath1869 holds , for all @xmath330 . in order to formulate concisely the corresponding uniqueness result , it is convenient to extend the usual definition of a continuous dimension scale , as follows . we endow each of the proper classes @xmath1082 , @xmath1083 , and @xmath1084 introduced in notation [ not : kappa+ ] with its interval topology . the latter consists , for example , of all open sub__sets _ _ of the corresponding class , the essential fact being that for a topological space @xmath76 ( we emphasize that @xmath76 is a _ set _ ) , the spaces of continuous functions @xmath1870 , @xmath1871 , and @xmath1872 are well - understood ( anyway , any map from a set to @xmath1083 is majorized by some @xmath179 ) . then we naturally extend notation [ not : omiki ] to the case where the @xmath1873-s may also be @xmath1082 , @xmath1083 , or @xmath1084 . by putting this together with lemma [ l : epsone ] and theorem [ t : embdimint ] , we have obtained the following structure theorem for continuous dimension scales . [ t : uneps ] let @xmath13 be a continuous dimension scale , let @xmath445 be the complete boolean algebra of projections of @xmath13 , and let @xmath76 be the ultrafilter space of @xmath445 , with the decomposition @xmath1103 as given in section . let @xmath201 be a finitary unit of @xmath13 . then there exists a unique lower embedding @xmath1874 such that @xmath1733 , for all @xmath330 and all @xmath500 , and @xmath1875 takes its values in @xmath1876 , for all @xmath1877 . furthermore , this embedding satisfies that @xmath1875 takes its values in @xmath1876 , for all @xmath1550 . all the forthcoming section can easily be formulated in such a standard class theory as the bernays - gdel system with choice , bgc . an alternative formulation consists of working in classical set theory zfc and identifying any statement ( with parameters ) with one free variable , say , @xmath1878 , with the `` class '' that it represents , namely , @xmath1879 ; this way , the mention to classes becomes a mere expendable commodity . we shall encounter in chapter [ ch : clesp ] situations where it may appear as artificial to restrict continuous dimension scales to be _ sets _ , as opposed to _ proper classes_. for example , with every right self - injective regular ring @xmath17 , we associate the category @xmath1880 of all nonsingular injective right @xmath17-modules . with the class @xmath1880 is associated a class that meets all attributes of a continuous dimension scale , except that it is not a set ( see section [ s : rsireg ] ) . we shall call such objects _ continuous dimension scales _ ( with capitals ) , and we shall define them shortly . we first do this for monoids . [ d : monoid ] a _ monoid _ is a class @xmath93 , endowed with an associative binary operation @xmath230 and a zero element @xmath188 . partial commutative monoid _ is a class @xmath13 , endowed with a commutative , associative ( in the sense of definition [ d : partcm ] ) partial binary operation @xmath230 , with a zero element @xmath188 . hence the definition of a monoid ( resp . , partial commutative monoid ) extends the one of a monoid ( resp . , partial commutative monoid ) , by allowing proper classes . the problem in defining continuous dimension scales is not that easy to solve . indeed , we wish our `` continuous dimension scales '' to satisfy a version of the main embedding theorem , theorem [ t : embdimint ] . more precisely , we wish every `` continuous dimension scale '' to embed as a lower subclass into a ( proper class ) monoid of the form @xmath1881 ( see notation [ not : kappa+ ] ) , for pairwise disjoint complete boolean spaces @xmath37 , @xmath38 , and @xmath39 . we shall now state the new axioms defining continuous dimension scales . of course , our definition is modeled on definition [ d : dimint ] and corollary [ c : altax ] . [ d : dimintclass ] a _ continuous dimension scale _ is a partial commutative monoid @xmath13 which satisfies the following axioms . * @xmath13 has refinement , and the algebraic preordering on @xmath13 is antisymmetric . * every nonempty subset of @xmath13 admits an infimum . * @xmath596 , @xmath597 such that @xmath598 , @xmath599 , and @xmath9 . * @xmath600 , for all @xmath237 ( _ where @xmath669 means , of course , that @xmath1882 , while @xmath1883 means that @xmath1884 for any @xmath1885 _ ) . * @xmath574 exists , for all @xmath124 , @xmath228 such that @xmath131 . * every element @xmath124 of @xmath13 can be written @xmath812 , where @xmath5 is directly finite and @xmath6 is purely infinite . * let @xmath124 , @xmath125 be purely infinite elements of @xmath13 . if @xmath687 , then the set of all purely infinite elements @xmath5 of @xmath13 such that @xmath896 and @xmath897 has a least element . * the class @xmath185={{\{x\in s\midx\leq a\}}}$ ] is a set , for all @xmath237 . * there exists a dense lower _ subset _ @xmath1090 of @xmath13 . ( _ we will call @xmath1090 a _ generating lower subset _ of @xmath13 . _ ) axiom @xmath1886 is there to ensure that the `` infinity '' in does not exceed the class of all ordinals . axiom @xmath1887 is there to ensure that the `` base spaces '' @xmath37 , @xmath38 , @xmath39 in are sets ( as opposed to proper classes ) . we emphasize that we require no condition on _ subclasses _ of @xmath13 , lest this might pave the way to undesirable set - theoretical paradoxes . in fact , since the axioms defining continuous dimension scales are requirements on either elements or subsets of @xmath13 , we obtain the following result . [ p : dimintclass ] let @xmath13 be a partial commutative monoid . then @xmath13 is a continuous dimension scale if and only if every lower subset of @xmath13 is a continuous dimension scale and @xmath13 satisfies both @xmath1886 and @xmath1887 . if @xmath13 is a continuous dimension scale , then every lower subset of @xmath13 is a continuous dimension scale : the proof is _ mutatis mutandis _ the same as for lemma [ l : segdi ] . conversely , suppose that every lower subset of @xmath13 is a continuous dimension scale and @xmath13 satisfies both @xmath1886 and @xmath1887 . every subset @xmath78 of @xmath13 is contained in a generating lower subset @xmath215 of @xmath13 : namely , take @xmath1888\midx\in x\}}}.\ ] ] the rest of the proof is similar to the proof of lemma [ l : dirun ] . observe , in particular , that _ every generating lower subset of a continuous dimension scale is a continuous dimension scale_. we also obtain the following extension of theorem [ t : c(o , k)dimint ] . [ c : dimintclass ] let @xmath76 be a complete boolean space , written as a disjoint union @xmath1103 , for clopen subsets @xmath37 , @xmath38 , @xmath39 of @xmath76 . then the monoid @xmath1889 is a continuous dimension scale . everything is now ready for the proof of our general embedding theorem for continuous dimension scales . [ t : genembdi ] let @xmath13 be a continuous dimension scale , let @xmath201 be a finitary unit of @xmath13 . let @xmath1090 be a generating lower subset of @xmath13 containing @xmath201 , let @xmath76 be the ultrafilter space of @xmath1890 , with the decomposition @xmath1103 as given in section . then there exists a unique lower embedding @xmath1874 such that @xmath1733 , for all @xmath330 and all @xmath500 , and @xmath1875 takes its values in @xmath1876 , for all @xmath1550 . let @xmath1891 be the class of all lower subsets @xmath241 of @xmath13 containing @xmath1090 . in particular , for all @xmath1892 , @xmath1090 is a generating lower subset of @xmath241 , thus , by lemma [ l : segdimint ] , @xmath1893 defines an isomorphism from @xmath1894 onto @xmath1890 . let @xmath1895 denote its inverse . therefore , the ultrafilter space @xmath1896 of @xmath1894 is homeomorphic to @xmath76 , _ via _ the map @xmath1897 let the projections of @xmath1090 act on @xmath241 , by defining @xmath1898 , for any @xmath1523 and @xmath1899 . hence , by carrying the structure of @xmath1896 to @xmath76 _ via _ this isomorphism and then applying theorem [ t : uneps ] to the continuous dimension scale @xmath241 with the finitary unit @xmath201 , we obtain that there exists a unique lower embedding @xmath1900 where @xmath1901 , such that @xmath1902 , for all @xmath1523 and all @xmath1899 , and @xmath1903 takes its values in @xmath1876 , for any @xmath1550 . furthermore , for elements @xmath1904 , @xmath1905 of @xmath1891 such that @xmath1906 , the restriction of @xmath1907 to @xmath1904 satisfies the requirements of @xmath1908 . hence , by the uniqueness statement of theorem [ t : uneps ] , @xmath1907 extends @xmath1908 . let @xmath1669 denote the union of all the maps @xmath1909 , for @xmath1892 . it follows from axiom @xmath1887 that the union of all the elements of @xmath1891 is @xmath13 , thus @xmath1669 is a map from @xmath13 to @xmath1705 . it obviously satisfies the required conditions . this concludes the `` existence '' part . if @xmath1751 is another lower embedding satisfying the conditions of the conclusion of theorem [ t : genembdi ] , then , by the uniqueness statement of theorem [ t : uneps ] applied to @xmath241 , the restriction of @xmath1754 to @xmath241 equals @xmath1909 , for all @xmath1892 ; whence @xmath1753 . this concludes the `` uniqueness '' part . we shall now give the fundamental lattice - theoretical definition underlying the whole paper , the definition of an _ espalier_. this definition will consist of a list of simple axioms , numbered from ( l1 ) to ( l8 ) . interspersed between these axioms , we shall also list some very elementary properties of espaliers . the role of each of these comments will also be to prepare for the formulation of the axioms that follow them . [ d : measchlatt ] an _ espalier _ is a structure @xmath35 , where @xmath1910 is a partially ordered set , @xmath2 is a binary relation on @xmath3 , and @xmath4 is an equivalence relation on @xmath3 , subject to the following axioms : * every nonempty subset of @xmath3 has an infimum . equivalently , every majorized subset of @xmath3 has a supremum . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in particular , @xmath3 has a smallest element , that we shall denote by @xmath188 . for @xmath124 , @xmath130 , the meet @xmath162 is always defined , while the join @xmath1911 is defined if and only if the pair @xmath163 is majorized . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * for all @xmath124 , @xmath125 , @xmath1912 , the following statements hold : 1 . @xmath1913 . 2 . if @xmath150 , then @xmath1914 . 3 . if @xmath131 and @xmath357 , then @xmath356 . if @xmath1915 is majorized , @xmath150 , and @xmath1916 , then @xmath1917 . if @xmath1918 , then @xmath267 . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we can then define in @xmath3 a partial binary operation @xmath135 , by putting @xmath127 if and only if @xmath128 and @xmath150 . so , above means exactly that @xmath12 is a partial commutative monoid . _ we say that a family @xmath72 of elements of @xmath3 is _ orthogonal _ , if it is majorized and @xmath1919 is defined for every finite subset @xmath277 of @xmath56 . we then define @xmath1920 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * for all @xmath124 , @xmath130 , if @xmath131 , then there exists @xmath132 such that @xmath133 . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ since @xmath1921 , the converse of axiom is , of course , trivial . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * let @xmath68 , let @xmath69 be an orthogonal family of elements of @xmath3 . if @xmath1922 , for all finite @xmath1923 , then @xmath1924 . * @xmath65 implies that @xmath66 , for all @xmath132 . * the relation @xmath4 is _ unrestrictedly refining _ , that is , for every @xmath68 and every orthogonal family @xmath69 of elements of @xmath3 , if @xmath1925 , then there exists a decomposition @xmath1597 such that @xmath73 for all @xmath62 . * the relation @xmath4 is _ unrestrictedly additive _ , that is , for all orthogonal families @xmath72 and @xmath69 of elements of @xmath3 , if @xmath73 for all @xmath62 , then @xmath1926 . * ( the _ parallelogram rule _ ) for all @xmath124 , @xmath125 , @xmath5 , @xmath24 such that @xmath1911 is defined , @xmath1927 an espalier is _ bounded _ , if it has a largest element . the following result makes it possible to create new espaliers from old ones . we leave the straightforward proof to the reader . [ p : lspesp ] 1 . for any espalier @xmath35 , any lower subset @xmath216 of @xmath3 , endowed with the restrictions of @xmath149 , @xmath2 , and @xmath4 , is an espalier . 2 . let @xmath1928 be a family of espaliers . then the product @xmath1929 , endowed with the componentwise @xmath149 , @xmath2 , @xmath4 , is an espalier . in the context of proposition [ p : lspesp](i ) , we shall say that @xmath216 is a _ lower subespalier _ of @xmath3 . in the context of proposition [ p : lspesp](ii ) , we shall say that @xmath3 is the _ direct product _ of the family @xmath1930 of espaliers . if @xmath216 and @xmath3 are espaliers , we shall say that a map @xmath1931 is a _ lower embedding _ of espaliers , if it is an isomorphism from @xmath216 onto a lower subespalier of @xmath3 . the verification of the following lemma is straightforward . [ l : lowembesp ] let @xmath216 and @xmath3 be espaliers , let @xmath1931 be a map . then @xmath244 is a lower embedding if and only if the following conditions hold : 1 . the range of @xmath244 is a lower subset of @xmath3 . 2 . @xmath1932 if and only if @xmath1933 , for all @xmath5 , @xmath1934 . 3 . @xmath1935 if and only if @xmath1936 , for all @xmath5 , @xmath1934 . 4 . @xmath1937 if and only if @xmath1938 , for all @xmath5 , @xmath1934 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ for the remainder of section , we shall fix an espalier@xmath35 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we start up with elementary properties of orthogonal families . [ l : modpair ] for all @xmath124 , @xmath130 such that @xmath150 , the following holds : 1 . 2 . @xmath1939 , for all @xmath666 . as in ( * lemma 1.1 ) . [ c : relcomp ] let @xmath124 , @xmath125 , @xmath1912 . 1 . if @xmath1940 is orthogonal , then @xmath1941 . 2 . if @xmath1942 , then there exists @xmath132 such that @xmath1943 and @xmath1944 . \(i ) apply lemma [ l : modpair ] to the pair @xmath1945 and to @xmath1946 . \(ii ) by axiom ( l3 ) , there are @xmath621 , @xmath824 such that @xmath1947 and @xmath1948 . so @xmath1949 . by ( i ) , @xmath1950 satisfies the required conditions . by using axiom ( l4 ) , it is easy to prove the following result ( see also theorem 1.2 of @xcite ) . [ l : lattassoc ] let @xmath56 and @xmath277 be sets , let @xmath278 be a surjective map , let @xmath72 be a family of elements of @xmath3 , and let @xmath237 . then the following are equivalent : 1 . 2 . for all @xmath279 , the family @xmath1951 is orthogonal , and , if we denote its join by @xmath281 , then @xmath1952 . another useful elementary orthogonality property of espaliers is the following . [ l : aperpjjx ] let @xmath68 , let @xmath78 be a majorized subset of @xmath3 . if @xmath1953 for all finite @xmath1954 , then @xmath1955 . for finite @xmath78 , this is trivial . now suppose that @xmath78 is infinite . write @xmath1956 , where @xmath1088 is the cardinality of @xmath78 , and put @xmath882 . we argue by induction on @xmath1088 . put @xmath1957 , for all @xmath1958 . observe that @xmath1959 . for all @xmath1958 , there exists , by axiom ( l3 ) , @xmath1960 such that @xmath1961 . it follows easily that @xmath1962 for all @xmath1958 , while @xmath1963 . furthermore , it follows from the induction hypothesis that @xmath1964 for all @xmath1958 , whence @xmath1965 for every finite subset @xmath277 of @xmath1088 . by axiom ( l4 ) , it follows that @xmath1966 , that is , @xmath150 . for @xmath124 , @xmath130 , let @xmath1967 hold , if @xmath1968 for some @xmath1660 . [ l : dotjoinsim ] let @xmath124 , @xmath125 , @xmath1912 . 1 . if @xmath1969 and @xmath1970 , then @xmath136 . _ ( that is , if @xmath124 and @xmath125 are perspective , then @xmath136 . ) _ 2 . if @xmath1971 , then @xmath136 . 3 . if @xmath1972 and @xmath1973 , then @xmath1967 . \(i ) we put @xmath1974 , @xmath1975 . let @xmath5 , @xmath6 , @xmath7 such that @xmath1976 by the parallelogram rule , @xmath1977 and @xmath1978 . thus @xmath25 , so @xmath1979 \(ii ) this is , by lemma [ l : modpair](i ) , a particular case of ( i ) . \(iii ) by using axiom ( l3 ) , there are @xmath621 , @xmath824 , @xmath5 , @xmath6 , @xmath7 such that @xmath1980 it follows that @xmath1981 , thus , by axiom ( l8 ) , @xmath1982 , thus , by axiom ( l6 ) , there are @xmath1983 and @xmath1984 such that @xmath1985 . moreover , it also follows from axiom ( l8 ) that @xmath1986 , thus @xmath1987 . from @xmath1988 , @xmath1973 , and @xmath1989 it follows ( by axiom ( l2 ) ) that @xmath1990 , thus , since @xmath1991 , @xmath1992 , @xmath1993 is defined and below @xmath125 . therefore , by using axiom ( l7 ) , @xmath1994 . we observe that lemma [ l : dotjoinsim](iii ) is stronger than axiom ( iii ) in definition 2.2 of @xcite . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ from now on , we denote by @xmath1995 , or @xmath1996 if there is any ambiguity on @xmath3 , the @xmath4-equivalence class of @xmath124 , for every @xmath68 . furthermore , we denote by @xmath13 the range of @xmath1997 , and we call it the _ dimension range _ of @xmath3 , in notation , @xmath1998 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we endow @xmath13 with the partial binary operation @xmath230 defined by @xmath1999 for all @xmath54 , @xmath85 , @xmath2000 . the fact that @xmath230 is indeed well - defined follows from the finite case of axiom ( l7 ) , namely , if @xmath2001 and @xmath2002 are defined and @xmath2003 and @xmath2004 , then @xmath2005 . we denote @xmath2006 by @xmath188 . [ p : spcm ] @xmath229 is a partial commutative monoid . only the verification of the associativity of @xmath230 is not completely trivial . given @xmath124 , @xmath125 , @xmath1912 such that @xmath2007 is defined , there exist @xmath5 , @xmath24 such that @xmath2008 and @xmath2009 . then @xmath2010 for some @xmath2011 and @xmath2012 in @xmath3 , while @xmath2013 for some @xmath2014 and @xmath2015 in @xmath3 . by the finite case of axiom ( l6 ) , @xmath2016 for some @xmath2017 and @xmath2018 in @xmath3 . axiom ( l2 ) then implies that @xmath2019 , and therefore , @xmath2020 is defined , and equal to @xmath2021 since @xmath230 is obviously commutative , this implies that @xmath230 is associative as well . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the dimension range @xmath13 will always be endowed with its _ algebraic _ preordering @xmath149 , see definition . hence @xmath2022 if and only if @xmath1967 , for all @xmath124 , @xmath130 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ l : delvmeas ] the following assertions hold : 1 . let @xmath54 , @xmath2023 and let @xmath1912 such that @xmath2024 . then there are @xmath124 , @xmath130 such that @xmath127 while @xmath2025 and @xmath2026 . 2 . let @xmath68 and let @xmath2027 . if @xmath2028 , then there exists @xmath666 in @xmath3 such that @xmath2029 . \(i ) by the definition of @xmath2030 , there are orthogonal @xmath621 , @xmath2031 such that @xmath2032 while @xmath2033 and @xmath2034 . so @xmath2035 , thus , by the finite case of axiom ( l6 ) , there are @xmath124 , @xmath130 such that @xmath2036 , @xmath2037 , and @xmath127 . observe that @xmath2038 and @xmath2039 . \(ii ) by the definition of the algebraic preordering of @xmath13 , there exists @xmath2040 such that @xmath2041 . the conclusion follows from ( i ) . proposition [ p : spcm ] and lemma [ l : delvmeas ] are particular cases of more general results in chapter 4 of @xcite . [ p : schbern ] the algebraic preordering on @xmath13 is antisymmetric . that is , if @xmath1967 and @xmath2042 , then @xmath136 , for all @xmath124 , @xmath130 . similar to the proof of theorem 41 of @xcite , see also lemma 6.1.3 of @xcite . we give the details here for convenience . put @xmath2043 and @xmath2044 . by induction on @xmath264 , we construct @xmath2045 , @xmath2046 , @xmath2047 , @xmath2048 such that the following relations hold : @xmath2049 since @xmath2050 and @xmath2051 , this is easy to satisfy for @xmath2052 . now the induction step . since @xmath2053 , there are @xmath2054 and @xmath2055 in @xmath3 such that @xmath2056 similarly , we obtain elements @xmath2057 , @xmath2058 in @xmath3 such that @xmath2059 this completes the induction step for and . observe that all the @xmath2047 ( resp . , all the @xmath2060 ) are mutually @xmath4-equivalent . but then , the family consisting of all the @xmath2061 and @xmath2062 , for @xmath264 , is orthogonal and majorized by @xmath206 . thus there exists @xmath1912 such that @xmath2063 since @xmath2064 , we obtain the equality @xmath2065 since @xmath2066 for all @xmath176 , it follows from , , and axiom ( l7 ) that @xmath2067 . but @xmath2068 ; whence @xmath136 . of course , it follows that @xmath13 is _ conical_. however , a direct proof of the conicality of @xmath13 is immediate from axiom ( l5 ) . [ p : ref ] the partial commutative monoid @xmath229 satisfies the refinement property . let @xmath54 , @xmath2069 , @xmath85 , @xmath2070 such that @xmath2071 . by the definition of the addition of @xmath13 and the finite version of axiom ( l6 ) , there are @xmath124 , @xmath257 , @xmath125 , @xmath2072 such that @xmath2025 , @xmath2073 , @xmath2026 , @xmath2074 , and @xmath2075 . we put @xmath2076 . we observe that @xmath1911 is defined . there are elements @xmath621 , @xmath2077 , @xmath824 , @xmath2078 , @xmath2079 in @xmath3 such that @xmath2080 by axiom ( l8 ) , @xmath2081 and @xmath2082 . furthermore , @xmath2083 thus , by lemma [ l : dotjoinsim](ii ) , @xmath2084 . hence @xmath2085 , for some @xmath2086 ( thus @xmath2087 ) and some @xmath2088 . similarly , @xmath2089 , for some @xmath2090 and some @xmath2091 . hence we have obtained the following refinement matrix : @xmath2092 this completes the proof . from propositions [ p : ref ] and [ p : schbern ] , we deduce immediately the following . [ c : shasm0 ] the partial commutative monoid @xmath13 satisfies axiom . in particular , by proposition [ p : projba ] , the set @xmath445 of all projections of @xmath13 , endowed with the ordering @xmath149 given by @xmath482 if and only if @xmath483 ( see lemma [ l : pleqortq ] ) , is a boolean algebra . we shall also refer to the projections of @xmath13 as the _ projections of @xmath3_. they operate also on @xmath3 in a projection - like manner ( up to equivalence)see lemmas [ l : baspcdotl ] and [ l : ppbotonl ] below . for the remainder of section [ s : lattaxioms ] , we shall analyze in some detail the pairs @xmath2093 of elements of @xmath3 such that @xmath2094 . observe that the conditions @xmath150 ( in @xmath3 ) and @xmath2094 ( in @xmath13 ) are _ a priori _ unrelated . for @xmath124 , @xmath130 , @xmath2094 if and only if the only element @xmath132 such that @xmath2095 is @xmath66 . [ c : ref ] let @xmath124 , @xmath125 , @xmath1912 such that @xmath2096 and @xmath2097 . if @xmath163 is majorized , then @xmath2098 . let @xmath2072 such that @xmath2099 . in particular , @xmath2100 . it follows from lemma [ l : dotjoinsim](iii ) that @xmath2101 , thus , by assumption , @xmath2102 . therefore , by proposition [ p : ref ] and lemma [ l : xbotid](i ) , @xmath2103 , that is , @xmath2098 . for @xmath124 , @xmath125 , @xmath1912 , we define @xmath2104 to mean that @xmath128 and @xmath2094 . note , in particular , that @xmath2104 implies that @xmath129 . much more is true , see proposition [ p : aboxplusb ] . [ l : aboxbplusb ] let @xmath124 , @xmath257 , @xmath125 , @xmath1912 . if @xmath2105 and if @xmath259 , then @xmath2106 , so @xmath150 . let @xmath2072 such that @xmath2107 . since @xmath128 with @xmath2108 , it follows from lemma [ l : dotjoinsim](i ) that @xmath2004 . let @xmath824 , @xmath2109 such that @xmath2110 let @xmath2111 such that @xmath2112 . then @xmath2113 , so that @xmath2114 . therefore , by lemma [ l : dotjoinsim](ii ) , @xmath2115 , thus @xmath2116 . so @xmath819 and @xmath2117 , hence , by assumption , @xmath2118 . it follows that @xmath2119 . in particular , @xmath150 . we obtain the following important tool , proposition [ p : aboxplusb ] . it is an analogue of axiom ( d ) in @xcite and of axiom @xmath2120 in @xcite . it also holds in the `` cardinal lattices '' considered in @xcite , as lemma 2.7 of @xcite shows . however , the proof of lemma 2.7 of @xcite can not be applied here , because there is no `` orthocomplement '' in our axiom system for espaliers . [ p : aboxplusb ] let @xmath124 , @xmath130 . if @xmath2094 and @xmath163 is majorized , then @xmath150 ; so @xmath2121 . by axiom ( l1 ) , @xmath128 exists . let @xmath2122 such that @xmath2123 . by lemma [ l : dotjoinsim](i ) , @xmath2003 . since @xmath124 , @xmath2124 , there exists @xmath2125 in @xmath3 , but @xmath2126 , so , by corollary [ c : ref ] , @xmath2127 . so , @xmath2128 , with @xmath2129 . hence , by lemma [ l : aboxbplusb ] , @xmath2130 . since @xmath2131 , it follows from axiom ( l2)(iii ) that @xmath150 . we now extend proposition [ p : aboxplusb ] to arbitrary families of elements of @xmath3 . [ d : strongorth ] a family @xmath72 is _ strongly orthogonal _ , if it is majorized and @xmath2132 , for all @xmath383 in @xmath56 . [ c : aistrorth ] every strongly orthogonal family of elements of @xmath3 is orthogonal . it suffices to prove the result for @xmath56 finite . we argue by induction on the cardinality of @xmath56 . pick @xmath62 . by the induction hypothesis , @xmath2133 is orthogonal , and , by corollary [ c : ref ] , @xmath2134 . hence , by proposition [ p : aboxplusb ] , @xmath2135 in @xmath3 , that is , @xmath2136 is orthogonal . [ c : aboxplusb ] let @xmath124 , @xmath125 , @xmath5 , @xmath6 , @xmath1912 . if @xmath2137 with @xmath2138 and @xmath2139 , then @xmath2140 and @xmath2141 . since @xmath2142 is defined ( and @xmath2143 ) and @xmath1995 , @xmath2138 , it follows from corollary [ c : ref ] that @xmath2144 . so , @xmath2145 . since @xmath127 ( by proposition [ p : aboxplusb ] ) and @xmath2146 , it follows from lemma [ l : aboxbplusb ] that @xmath2147 , that is , @xmath666 . similarly , @xmath2148 . let @xmath2149 , @xmath2150 such that @xmath2151 and @xmath2152 . then @xmath2153 whence @xmath2154 . therefore , @xmath2140 and @xmath2141 . [ c : delperp ] let @xmath68 , let @xmath78 be a majorized subset of @xmath3 . if @xmath2155 for all @xmath183 , then @xmath2156 . put @xmath882 . it follows from corollary [ c : ref ] that @xmath2157 let @xmath2158 $ ] . then @xmath2159 , thus , by lemma [ l : aperpjjx ] , @xmath2160 for some finite @xmath1954 . but @xmath2161 is majorized ( by @xmath125 ) , thus , by proposition [ p : aboxplusb ] , @xmath2162 . therefore , by , @xmath2163 , that is , @xmath2164 . so we have proved that @xmath2094 . [ c : projonl ] let @xmath68 and let @xmath500 . then there exists a largest element @xmath621 of @xmath2165 $ ] such that @xmath2166 . furthermore , @xmath2167 . by lemma [ l : oplusid ] , @xmath2168 . hence , by corollary [ c : delperp ] , the supremum @xmath621 of the set @xmath78 of all elements @xmath5 of @xmath2165 $ ] such that @xmath2169 belongs to @xmath78 . from @xmath819 it follows that @xmath2170 , thus , since @xmath2166 , @xmath2171 . conversely , @xmath2172 , thus , by lemma [ l : delvmeas](ii ) , there exists @xmath2173 such that @xmath2174 . but @xmath2175 , so @xmath709 , and thus @xmath2176 . finally , by proposition [ p : schbern ] , @xmath2177 . we shall denote by @xmath2178 the element @xmath621 of corollary [ c : projonl ] , and we shall repeatedly use the properties @xmath2179 and @xmath2180 , for all @xmath68 and all @xmath500 . we shall also put @xmath2181 . note that @xmath2182 is a lower subset of @xmath3 : indeed , if @xmath68 and @xmath2183 , then @xmath2184 , so @xmath2185 . we gather up various elementary properties of the map @xmath2186 in the following lemmas [ l : plls ] and [ l : baspcdotl ] . [ l : plls ] let @xmath500 . then the following assertions hold : 1 . @xmath2187 $ ] . 2 . @xmath2182 is a lower subset of @xmath2188 . 3 . @xmath2182 is a lower subset of @xmath1910 . @xmath2182 is closed under majorized suprema . \(i ) is an immediate consequence of corollary [ c : projonl ] . the assertions ( ii ) , ( iii ) follow immediately . \(iv ) follows immediately from ( i ) , corollary [ c : delperp ] , and the fact that @xmath2189 . [ l : baspcdotl ] let @xmath124 , @xmath130 , let @xmath26 , @xmath447 . then the following assertions hold : 1 . @xmath131 implies that @xmath2190 . @xmath1967 implies that @xmath2191 . @xmath136 implies that @xmath2192 . @xmath482 implies that @xmath2193 . \(i ) @xmath2194 and @xmath2195 , thus @xmath2190 by the definition of @xmath2196 . \(ii ) it follows from @xmath1967 that @xmath2022 , thus @xmath2197 , that is , @xmath2191 . \(iii ) follows immediately from ( ii ) and from proposition [ p : schbern ] . \(iv ) since @xmath482 , @xmath2198 , and so , since @xmath2179 , it follows that @xmath2193 . [ p : piaorth ] let @xmath716 be an orthogonal family of elements of @xmath445 and let @xmath68 . then the family @xmath2199 is orthogonal in @xmath3 . for @xmath383 in @xmath56 , @xmath2200 and @xmath2201 , thus , since @xmath2202 , @xmath2203 . the result follows then from corollary [ c : aistrorth ] . [ l : ppbotonl ] let @xmath68 and let @xmath500 . then the following assertions hold : 1 . @xmath2204 . 2 . let @xmath2205 and @xmath2206 such that @xmath2207 . then @xmath2208 and @xmath2209 . \(i ) since the join @xmath2210 is defined ( and @xmath259 ) and since @xmath2211 , it follows from proposition [ p : aboxplusb ] that @xmath2212 . since @xmath2213 and by axiom ( l6 ) , there are @xmath2214 and @xmath2215 such that @xmath2216 . since @xmath2217 and @xmath819 , we have @xmath2218 . likewise , @xmath2219 , thus @xmath2220 , whence @xmath2221 . \(ii ) by assumption , @xmath2222 . by proposition [ p : aboxplusb ] , @xmath2223 . by corollary [ c : aboxplusb ] and by ( i ) , @xmath2208 and @xmath2209 . [ p : shasgc ] @xmath13 has general comparability . we prove the two following claims . [ cl : abot+abotbot = s ] @xmath2224 , for all @xmath2225 . let @xmath2226 , @xmath2227 . pick @xmath124 , @xmath132 such that @xmath2228 and @xmath2229 . by corollary [ c : delperp ] , there exists a largest element @xmath999 such that @xmath2230 . let @xmath2031 such that @xmath2231 . so @xmath2232 , with @xmath2233 . for any @xmath2234 such that @xmath2235 , the inequality @xmath832 holds by the definition of @xmath621 , but @xmath2234 , so @xmath530 since @xmath823 . hence @xmath2236 . [ cl : abcxyperp ] for all @xmath124 , @xmath130 , there are @xmath621 , @xmath824 , @xmath5 , @xmath24 such that @xmath2237 and @xmath2238 while @xmath1986 and @xmath2239 . an easy application of zorn s lemma yields a subset @xmath2240 of @xmath2241\times(0,b]$ ] which is maximal with respect to the following properties : 1 . both families @xmath72 and @xmath69 are orthogonal . @xmath73 for all @xmath62 . put @xmath2242 and @xmath2243 . by axiom ( l7 ) , @xmath1986 . pick @xmath5 and @xmath6 such that @xmath2237 and @xmath2238 . if @xmath2244 , then there are nonzero @xmath2245 and @xmath2246 such that @xmath2247 . but then , @xmath2248 still satisfies ( i ) and ( ii ) above , which contradicts the maximality of @xmath78 . hence @xmath2239 . by lemma [ l : gencompax ] , general comparability follows from claims [ cl : abot+abotbot = s ] and [ cl : abcxyperp ] . we recall that general comparability is also axiom ( m3 ) . note that general comparability in @xmath13 can be stated as the following property of @xmath3 , which we also refer to as _ general comparability _ : for any @xmath124 , @xmath130 , there exists @xmath500 such that @xmath2249 and @xmath2250 . [ l : p(aopplusb ) ] let @xmath68 , let @xmath72 be a family of elements of @xmath3 , let @xmath500 . 1 . if @xmath553 and @xmath536 , then @xmath2251 . 2 . if @xmath71 , then @xmath2252 . \(i ) it is clear that @xmath2253 for all @xmath173 . let @xmath130 such that @xmath2254 for all @xmath62 . in particular , @xmath2255 for all @xmath173 , thus @xmath562 . in addition , since @xmath553 , it follows from lemma [ l : plls](iii ) that @xmath2256 , so @xmath2257 . \(ii ) the equality @xmath2258 holds for all @xmath62 , so @xmath2259 by lemma [ l : plls](iv ) , @xmath2260 and @xmath2261 . therefore , by lemma [ l : ppbotonl ] , @xmath2252 and @xmath2262 . [ p : b(l)cba ] the boolean algebra @xmath445 is complete . it suffices to prove that every orthogonal family @xmath716 of elements of @xmath445 admits a supremum . by proposition [ p : piaorth ] , the family @xmath2199 is orthogonal , for all @xmath68 . furthermore , if @xmath130 such that @xmath136 , then , by lemma [ l : baspcdotl](iii ) , @xmath2263 for all @xmath62 , thus , by axiom ( l7 ) , @xmath2264 hence , we can define a map @xmath2265 by the rule @xmath2266 it is obvious that @xmath419 . let @xmath2226 , @xmath2267 such that @xmath2268 is defined . so @xmath2269 , for some @xmath2270 and @xmath2271 such that @xmath150 . so @xmath2272 and @xmath2273 , where @xmath257 and @xmath1675 are defined by @xmath2274 in particular , @xmath259 and @xmath260 , so @xmath2275 , and , by using lemma [ l : p(aopplusb)](ii ) , @xmath2276 whence @xmath2277 so @xmath26 is an endomorphism of @xmath229 . now let @xmath2225 . pick @xmath2270 , and pick @xmath2111 such that @xmath2278 . for all @xmath709 , if @xmath2279 , then @xmath2280 , so @xmath2281 . hence @xmath2282 . we infer that @xmath2283 . indeed , let @xmath2227 . pick @xmath5 such that @xmath2229 . so @xmath2284 by the definition of @xmath26 . but @xmath2285 for all @xmath62 , thus , by corollary [ c : delperp ] , @xmath2286 . hence @xmath2287 , with @xmath2288 . it follows that @xmath26 is a projection of @xmath13 . it is clear that @xmath2289 for all @xmath62 . let @xmath447 such that @xmath2290 for all @xmath62 . let @xmath68 . then @xmath2291 , for all @xmath62 , thus @xmath2292 . taking the image under @xmath1997 of both sides yields that @xmath2293 . this holds for all @xmath68 , whence @xmath482 . so we have verified that @xmath1278 . as a consequence , we obtain that @xmath13 satisfies axiom ( m4 ) ( observe that @xmath2294 if and only if @xmath2295 ) . [ p : boolvall ] for all @xmath124 , @xmath130 , there exists a largest @xmath500 such that @xmath2294 . put @xmath2296 , and put @xmath2297 . let @xmath716 be a maximal orthogonal family of elements of @xmath78 . for all @xmath62 , let @xmath990 such that @xmath2298 . in particular , @xmath2299 , so @xmath2300 . by corollary [ c : aistrorth ] , the family @xmath69 is orthogonal , and by proposition [ p : piaorth ] , the family @xmath2199 is orthogonal . hence , @xmath2301 [ not : bvalsmb ] for @xmath124 , @xmath130 , we put @xmath2302 . that is , in accordance to definition [ d : boolval ] , @xmath2303 is the largest projection @xmath26 of @xmath13 such that @xmath2294 . in a similar spirit as for notation [ not : bva = b ] , we put @xmath2304 , which by proposition [ p : schbern ] is the largest @xmath500 such that @xmath2192 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ standing hypotheses : @xmath3 is an espalier , @xmath13 is the dimension range of @xmath3 , and @xmath2305 is the canonical map . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ following definition [ d : purinf ] , we say that an element @xmath124 of @xmath3 is _ purely infinite _ , if @xmath1995 is purely infinite in @xmath13 . this occurs if and only if @xmath2306 for some @xmath257 , @xmath2307 in @xmath3 . purely infinite elements are also , in some references , called _ idempotent _ , or , as in @xcite , _ idem - multiple_. similarly , following definition [ d : dirfinmon ] , we say that an element @xmath124 of @xmath3 is _ directly finite _ , if @xmath1995 is directly finite in @xmath13 . observe that @xmath124 is directly finite if and only if @xmath2308 implies that @xmath2309 , for any @xmath130 . [ d : homogseq ] a family @xmath72 of elements of @xmath3 is _ homogeneous _ , if it is orthogonal and @xmath2310 , for all @xmath173 , @xmath174 . a homogeneous family @xmath72 is _ trivial _ , if @xmath2311 , for all @xmath173 ; equivalently , @xmath2311 for _ some _ @xmath62 . [ l : charpurinf ] let @xmath68 . then the following are equivalent : 1 . @xmath124 is purely infinite . 2 . there exists a homogeneous sequence @xmath2312 such that @xmath2313 . ( i)@xmath426(ii ) there are @xmath257 and @xmath2314 such that @xmath2306 and @xmath2315 . by using axiom ( l6 ) , it is then easy to construct inductively sequences @xmath2316 and @xmath2317 such that @xmath2043 , @xmath2318 , and @xmath2319 , for all @xmath176 . so @xmath2317 is a homogeneous sequence whose join , @xmath257 , belongs to @xmath2165 $ ] . since @xmath2320 , we obtain that @xmath2003 by proposition [ p : schbern ] . hence , the conclusion follows from axiom ( l6 ) . ( ii)@xmath426(i ) put @xmath2321 and @xmath2322 . by axiom ( l7 ) , @xmath2315 . since @xmath2306 , @xmath124 is purely infinite . [ l : chardf ] let @xmath68 . the following are equivalent : 1 . @xmath124 is directly finite . 2 . there is no nontrivial purely infinite element below @xmath124 . the interval @xmath2165 $ ] has no infinite nontrivial homogeneous sequence . ( i)@xmath426(ii ) let @xmath2323 $ ] be a purely infinite element , and let @xmath132 such that @xmath2324 . then @xmath2325 . since @xmath124 is directly finite , @xmath2326 , whence @xmath2327 . ( ii)@xmath2328(iii ) follows immediately from lemma [ l : charpurinf ] . ( iii)@xmath426(i ) let @xmath132 such that @xmath2329 . so @xmath2330 , for some @xmath2011 and @xmath2014 . it is then easy to construct , by induction ( and axiom ( l6 ) ) , sequences @xmath2316 and @xmath2312 of elements of @xmath3 such that @xmath2331 , @xmath2332 , @xmath2333 , and @xmath2334 for all @xmath176 . in particular , the sequence @xmath2312 is homogeneous , thus , by assumption , @xmath2335 for all @xmath176 . therefore , @xmath66 . so @xmath124 is directly finite . we deduce from this that @xmath13 satisfies axiom ( m5 ) . [ p : dfpidec ] for all @xmath68 , there are @xmath125 , @xmath1912 such that @xmath125 is purely infinite , @xmath126 is directly finite , and @xmath2336 . let @xmath50 be a maximal orthogonal family of nonzero purely infinite elements of @xmath2165 $ ] . we put @xmath2337 . for all @xmath62 , there exists a decomposition @xmath2338 where @xmath2339 . put @xmath2340 and @xmath2341 . by axiom ( l7 ) , since @xmath2343 , @xmath125 is purely infinite . let @xmath1912 such that @xmath2336 . suppose that @xmath126 is not directly finite . then , by lemma [ l : chardf ] , there exists a purely infinite element @xmath5 such that @xmath2344 . but then , enlarging the family @xmath50 by @xmath5 yields an orthogonal family of nonzero purely infinite elements of @xmath2165 $ ] , which contradicts the maximality of @xmath50 . so , @xmath126 is directly finite . we can then reformulate proposition [ p : bsminusa ] in the language of lattices . [ p : bsminusal ] let @xmath124 , @xmath130 such that @xmath1967 . then there exists @xmath1912 such that @xmath872 and @xmath2345 . propositions [ p : schbern ] , [ p : ref ] , [ p : shasgc ] , [ p : boolvall ] , and [ p : dfpidec ] establish the hypotheses of proposition [ p : bsminusa ] . thus , @xmath2346 exists in @xmath13 . since this element lies below @xmath2347 , there exists @xmath2348 $ ] such that @xmath2345 . the following important definition involves both the lattice structure and the dimension function . it is the key to proving the existence of majorized suprema in @xmath13 . [ d : trim ] 1 . let @xmath124 , @xmath130 . we write @xmath2349 , if there exists @xmath1912 such that @xmath2350 and @xmath2345 . 2 . let @xmath1088 be an ordinal . a @xmath1088-sequence @xmath2351 of elements of @xmath3 is _ trim _ , if the following conditions hold : 1 . @xmath2352 for all @xmath2353 such that @xmath2354 . 2 . for any limit ordinal @xmath2355 , the sequence @xmath2356 is majorized , and @xmath2357 . [ l : trimins ] let @xmath124 , @xmath130 such that @xmath131 , let @xmath2227 such that @xmath2358 . then there exists @xmath132 such that @xmath2359 and @xmath2229 . let @xmath1912 such that @xmath2350 . so @xmath2360 , thus @xmath2361 . hence there exists @xmath2362 such that @xmath2363 . now we put @xmath2364 . then @xmath2365 , @xmath2366 , and @xmath2367 . so , @xmath2368 . in the statement of the following lemma [ l : trimseq ] , a _ lifting _ of a family @xmath2369 of elements of @xmath13 is a family @xmath72 of elements of @xmath3 such that @xmath2370 for all @xmath62 . [ l : trimseq ] let @xmath1088 be an ordinal . 1 . for all @xmath130 , every increasing @xmath1088-sequence of elements of @xmath2371 $ ] has a trim lifting in @xmath2372 $ ] . 2 . for any majorized trim sequences @xmath2373 and @xmath2374 of elements of @xmath3 , @xmath2375 3 . for every majorized trim lifting @xmath2351 of a @xmath1088-sequence @xmath2376 of elements of @xmath13 , @xmath2377 we argue by transfinite induction on @xmath1088 . the result is vacuous for @xmath1424 . suppose that we have proved the lemma for all ordinals @xmath2378 , with @xmath2379 . \(i ) let @xmath2376 be an increasing @xmath1088-sequence of elements of @xmath2371 $ ] . we construct inductively elements @xmath2380 of @xmath2372 $ ] , for @xmath1958 . for @xmath2381 , pick any element @xmath205 of @xmath2372 $ ] such that @xmath2382 . suppose we have constructed @xmath2383 such that @xmath2384 , with @xmath2354 . by lemma [ l : trimins ] , there exists @xmath2385 such that @xmath2352 and @xmath2386 . suppose finally that @xmath2355 is a limit ordinal and that @xmath2387 is a trim lifting of @xmath2388 in @xmath2372 $ ] . we put @xmath2389 now , we observe that @xmath2390 for all @xmath2391 . since @xmath2387 is trim and majorized , it follows from ( iii ) of the induction hypothesis that @xmath2392 . by applying once again lemma [ l : trimins ] , we obtain @xmath2393 such that @xmath2394 and @xmath2395 . by the definition of a trim sequence , @xmath2351 is a trim lifting of @xmath2376 in @xmath2372 $ ] . \(ii ) we construct inductively elements @xmath2396 , for @xmath1958 , of @xmath3 , as follows . we put @xmath2397 . if @xmath2354 , then @xmath2398 , so there exists @xmath2399 such that @xmath2400 and @xmath2401 . if @xmath2355 is a limit ordinal , we put @xmath2402 . since @xmath2403 , there exists @xmath2404 such that @xmath2405 and @xmath2406 . it follows that @xmath2407 for all @xmath1958 . in particular , @xmath2408 . let @xmath2409 be constructed from @xmath2374 the same way @xmath2410 is constructed from @xmath2373 . so @xmath2411 . let @xmath2353 such that @xmath2354 . since @xmath2412 and @xmath2413 , @xmath2414 let @xmath2355 be a limit ordinal . by ( iii ) of the induction hypothesis , @xmath2415 . since @xmath2416 , we obtain that @xmath2417 hence we have proved that @xmath2418 for all @xmath1958 . hence , by axiom ( l7 ) , @xmath2419 , that is , @xmath2420 . \(iii ) let @xmath2351 be a majorized trim lifting of @xmath2376 . we put @xmath2421 . so , @xmath2422 for all @xmath1958 . now let @xmath2267 such that @xmath2423 for all @xmath1958 , and let @xmath130 such that @xmath2424 . by ( i ) , @xmath2376 has a trim lifting @xmath2425 in @xmath2372 $ ] . in particular , @xmath2426 . however , by ( ii ) , @xmath2427 ; whence @xmath2428 . so @xmath2429 . [ c : shasm1 ] @xmath13 satisfies axiom . we prove that every majorized subset @xmath78 of @xmath13 has a supremum . by proposition [ p : schbern ] , proposition [ p : shasgc ] , and lemma [ l : meetjoins ] , every majorized finite subset of @xmath13 has a supremum . so it remains to conclude in case @xmath78 is infinite . we argue by induction on the cardinality of @xmath78 . write @xmath2430 , where @xmath1088 is the cardinality of @xmath78 . by the finite case and the induction hypothesis , for all @xmath1958 , the set @xmath2431 has a supremum , say , @xmath2432 . since @xmath78 is majorized , so is @xmath2433 , that is , there exists @xmath130 such that @xmath2434 for all @xmath1958 . by lemma [ l : trimseq](i ) , the family @xmath2435 has a trim lifting in @xmath2372 $ ] , say , @xmath2436 . put @xmath2437 . by lemma [ l : trimseq](iii ) , @xmath2347 is the supremum of @xmath2433 , that is , the supremum of @xmath78 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ standing hypotheses : @xmath3 is an espalier , @xmath13 is the dimension range of @xmath3 , and @xmath2438 is the canonical map . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ at this point , what remains to do in order to conclude the proof of theorem a is to establish that @xmath13 satisfies axiom ( m6 ) . we shall devote section [ s : shas(m6 ) ] to this . in accordance with definition [ d : remov ] , we state the following definition . [ d : removl ] let @xmath124 , @xmath130 . we say that @xmath124 is _ removable _ from @xmath125 , in notation @xmath2439 , if @xmath2440 in @xmath13 . equivalently , @xmath2439 , if @xmath1967 , and @xmath2441 implies that @xmath2442 , for all @xmath330 . [ not : cardelt ] let @xmath1088 be a cardinal number , let @xmath124 , @xmath130 . 1 . let @xmath2443 be the statement that there exists a homogeneous @xmath1088-sequence @xmath2351 such that @xmath2444 2 . let @xmath2445 be the statement that there exists a homogeneous @xmath1088-sequence @xmath2351 such that @xmath2446 for example , @xmath2447 ( resp . , @xmath2448 ) means that @xmath136 ( resp . , @xmath1967 ) . another example is that @xmath2449 if and only if @xmath124 is purely infinite . [ l : locinfquot ] let @xmath124 , @xmath2450 , let @xmath85 be an infinite cardinal number . if @xmath2451 , then there exist an infinite cardinal number @xmath2452 and a projection @xmath26 of @xmath13 such that @xmath2453 and @xmath2454 . we start with a homogeneous family of @xmath85 elements of @xmath2372 $ ] all equivalent to @xmath124 ( modulo @xmath4 ) , and enlarge it to a maximal such family , say , @xmath2455 , where @xmath2452 is an infinite cardinal number . let @xmath2072 such that @xmath2456 by general comparability , there exists @xmath500 such that @xmath2457 and @xmath2458 . by the maximality of @xmath2459 , @xmath2460 , hence @xmath2453 . now we put @xmath2461 since @xmath40 is an infinite cardinal and by axiom ( l7 ) , @xmath2462 . moreover , @xmath2463 , so , by lemmas [ l : baspcdotl ] and [ l : p(aopplusb ) ] , @xmath2464 hence , by using proposition [ p : schbern ] and lemma [ l : p(aopplusb ) ] , @xmath2465 [ l : purinfal0 ] @xmath2466 , for all purely infinite @xmath68 . by lemma [ l : charpurinf ] , we have @xmath2313 for some homogeneous sequence @xmath2312 . let @xmath2467 be an infinite partition of @xmath175 , with all the @xmath2468 infinite . put @xmath2469 , for all @xmath264 . by axiom ( l7 ) , @xmath2332 for all @xmath176 . the proof is concluded by the observation that @xmath2470 . by replacing a bijection from @xmath2471 onto @xmath175 by a bijection from @xmath2472 onto @xmath1088 , for any infinite cardinal @xmath1088 , in the proof above , we easily obtain the following result . [ l : purinfkappa ] let @xmath124 , @xmath130 , let @xmath1088 be an infinite cardinal number . if @xmath2473 , then @xmath2474 . for @xmath68 , we put @xmath2475 ( see definition [ d : cc(a ) ] ) . so , @xmath2476 , for all @xmath68 . in view of lemma [ l : ppbotonl ] , @xmath768 is the smallest projection @xmath26 of @xmath13 such that @xmath2477 . [ l : locm5 ] let @xmath124 , @xmath130 purely infinite such that @xmath2439 and @xmath2478 . then there exists a purely infinite @xmath2479 such that 1 . @xmath2480 and @xmath2481 . 2 . @xmath2482 , for all purely infinite @xmath872 such that @xmath2483 and @xmath2484 . we put @xmath2485 . if @xmath2486 , then @xmath2042 , thus , since @xmath687 , @xmath2327 , a contradiction . so , @xmath2487 . for all @xmath2488 , we denote by @xmath2489 the least infinite cardinal number @xmath54 such that @xmath2490 for all @xmath2491 . by lemma [ l : purinfal0 ] , @xmath2492 for all @xmath2488 . we pick @xmath2493 such that @xmath2494 is minimum , and we prove that this @xmath611 satisfies the required conditions . of course , ( i ) holds since @xmath2493 . let @xmath1912 be purely infinite such that @xmath2495 and @xmath2484 . note that @xmath2496 ( otherwise , @xmath2497 , a contradiction ) . [ cl : qcbetaqc ] for all @xmath2498 $ ] and for every infinite cardinal number @xmath1035 , there exists @xmath1393 $ ] such that @xmath2499 . if @xmath2500 , then , since @xmath2501 ( lemma [ l : projtr](i ) ) , @xmath2502 , which is impossible since @xmath2503 . so , @xmath2504 , so @xmath2505 . in particular , @xmath2506 , so , by the definition of @xmath2507 , there exists @xmath2508 such that @xmath2509 . note that @xmath2510 . hence , by lemma [ l : locinfquot ] , there are @xmath2511 and an infinite cardinal number @xmath2452 such that @xmath2512 and @xmath2513 . in particular , @xmath2514 , so we may replace @xmath31 by @xmath2515 , and then @xmath2516 . therefore , by lemma [ l : purinfkappa ] , @xmath2517 , with @xmath2452 , thus , since @xmath2452 and by proposition [ p : schbern ] , @xmath2518 . [ cl : qelsmqc ] for all @xmath2498 $ ] , there exists @xmath1393 $ ] such that @xmath2519 . by general comparability , there exists a decomposition @xmath2520 ( in @xmath445 ) such that @xmath2521 and @xmath2522 . if @xmath2523 , then we may take @xmath2524 . so suppose that @xmath2525 , so @xmath2526 . since @xmath2527 is purely infinite , there exist , by lemmas [ l : locinfquot ] and [ l : purinfal0 ] , @xmath447 and an infinite cardinal @xmath85 such that @xmath2528 and @xmath2529 . after replacing @xmath31 by @xmath2515 , we have @xmath1393 $ ] , with @xmath2530 and @xmath2531 . since @xmath2532 , we must have @xmath2533 , whence @xmath2534 , and so @xmath2535 . hence @xmath2536 . hence , by claim [ cl : qcbetaqc ] , there exists @xmath1437 $ ] such that @xmath2537 . therefore , by , @xmath2538 . by claim [ cl : qelsmqc ] and by proposition [ p : boolvall ] , @xmath2539 . however , by assumption , @xmath2484 , thus @xmath2540 ( see lemma [ l : cc(a ) ] ) , so @xmath2482 . and now , axiom ( m6 ) ( recall that @xmath2541 if and only if @xmath2542 ; see definition [ d : cc(a ) ] ) . [ p : shas(m6 ) ] let @xmath124 , @xmath130 be purely infinite such that @xmath2439 . then there exists a purely infinite @xmath2480 such that 1 . @xmath2543 and @xmath2544 . 2 . @xmath2482 , for all purely infinite @xmath1912 such that @xmath2483 and @xmath2542 . we first claim that it will suffice to find a purely infinite element @xmath2480 satisfying ( i ) and the the statement * @xmath2482 , for all purely infinite @xmath872 such that @xmath2483 and @xmath2542 . indeed , suppose that ( i ) and ( ii@xmath2545 ) are satisfied . let @xmath1912 such that @xmath2483 and @xmath2542 . then @xmath2546 , and so @xmath2547 by corollary [ c:2 - 5 . ? there exists @xmath2548 such that @xmath2549 , whence @xmath2550 . moreover , @xmath639 is purely infinite by lemma [ l:2 - 5 . ? ] , and @xmath2551 by lemma [ l : basiccc](ii ) . since any element @xmath2552 would then satisfy @xmath2482 , the claim is proved . let @xmath180 be the set of all pairs @xmath2553 such that @xmath5 is purely infinite and the following conditions hold : * @xmath2554 and @xmath2555 . * @xmath2556 and @xmath2557 . * for all purely infinite @xmath2558 , the conditions @xmath2559 and @xmath2560 imply that @xmath2561 . let @xmath2562 be a subset of @xmath180 , maximal with the property that the @xmath1253 are nonzero and pairwise orthogonal . we observe that since @xmath52 is majorized ( by @xmath125 ) and since the @xmath1253 are pairwise orthogonal , it follows from corollary [ c : aistrorth ] that @xmath50 is an orthogonal family of @xmath3 . we put @xmath2563 the pair @xmath2564 belongs to @xmath180 . observe that @xmath2565 and @xmath2555 . since all the @xmath57 are purely infinite , @xmath5 is purely infinite . furthermore , @xmath2566 for all @xmath173 , thus , by lemma [ l : trleqtr ] , @xmath2567 . this holds for all @xmath173 , thus , by lemma [ l : projtr](ii ) and proposition [ p : b(s)cba ] , @xmath2556 . let @xmath2558 be a purely infinite element of @xmath3 such that @xmath2559 and @xmath2560 . for all @xmath62 , @xmath2568 and , by lemma [ l : basiccc](ii ) , @xmath2569 , so @xmath2570 . this holds for all @xmath173 , thus @xmath2571 . so , it suffices to prove that @xmath2572 . until the end of the proof , we suppose otherwise . put @xmath2573 . since @xmath2574 and @xmath2439 , the relation @xmath2575 holds . since @xmath2576 , there exists , by lemma [ l : locm5 ] , a purely infinite @xmath2577 such that * @xmath2578 ; * @xmath2579 , for all purely infinite @xmath2580 such that @xmath2581 and @xmath2582 . now put @xmath2583 . so , by definition , @xmath2584 . if @xmath2585 , then @xmath2586 , but @xmath2587 , so @xmath2588 , which contradicts ( @xmath54 ) above . so , @xmath2589 , thus , since @xmath2590 , the projection @xmath2591 is nonzero . now , from @xmath29 it follows that @xmath2592 , so @xmath2593 , that is , @xmath2594 . hence , @xmath2595 . moreover , @xmath2596 , whence , taking complements in @xmath2597 , @xmath2598 . in particular , @xmath2599 . by general comparability , there exists @xmath2600 in @xmath445 such that @xmath2601 and @xmath2602 . then @xmath2603 , whence @xmath2604 , and so @xmath2605 . by the definition of @xmath449 and of @xmath2606 , @xmath2607 , for all @xmath2608 $ ] , thus , by lemma [ l : bvtr ] , @xmath2609 . consider a purely infinite @xmath2610 such that @xmath2611 and @xmath2612 . since @xmath2610 and @xmath2613 , the element @xmath2614 is defined and @xmath2615 . furthermore , since @xmath773 , @xmath2616 . since @xmath2611 and @xmath2439 , it follows from lemma [ l : projtr ] that @xmath2617 . therefore , by part ( @xmath85 ) of the definition of @xmath2618 , @xmath2619 , thus @xmath2620 . so we have proved that @xmath2621 , with @xmath2606 nonzero and orthogonal to all the @xmath1253 for @xmath62 , which contradicts the maximality of @xmath2562 . so , @xmath2572 . proposition [ p : shas(m6 ) ] concludes the proof of theorem a. a more complete form of theorem a is the following . [ t : dimesp ] let @xmath35 be an espalier . then the quotient @xmath2622 can be endowed with a partial addition @xmath230 , defined by the rule @xmath2623 that makes it a continuous dimension scale , with zero element @xmath2006 . one of the questions that we shall regularly encounter throughout the study of various classes of espaliers , in chapter [ ch : clesp ] , will be what are the possible dimension ranges of members of a given class of espaliers . [ d : duniv ] a class @xmath2624 of espaliers is _ d - universal _ , if every continuous dimension scale admits a lower embedding into the dimension range of some member of @xmath2624 . we recall that the class of espaliers is closed under so - called _ lower subespaliers _ , and also under _ direct products _ of espaliers , see proposition [ p : lspesp ] . the following lemma records some elementary facts about these notions . we leave its easy proof to the reader . [ l : basicdrng ] 1 . let @xmath216 and @xmath3 be espaliers , let @xmath2625 be a lower embedding . then the rule @xmath2626 defines a lower embedding from @xmath2627 into @xmath2628 . 2 . let @xmath35 be an espalier , let @xmath13 be a lower subset of @xmath2628 , put @xmath2629 then @xmath216 is a lower subespalier of @xmath3 , and the rule @xmath2630 defines an isomorphism from @xmath2627 onto @xmath13 . 3 . let @xmath1930 be a family of espaliers , let @xmath1929 be its direct product . then the rule @xmath2631 defines an isomorphism from @xmath2632 onto @xmath2628 . in the context of lemma [ l : basicdrng](i ) , we shall of course write @xmath2633 to denote the map that sends @xmath2634 to @xmath2635 , for every @xmath2636 . as a consequence of lemma [ l : basicdrng ] , the dimension ranges of members of d - universal classes of espaliers can be nearly anything reasonable . [ p : duniv ] let @xmath2624 be a d - universal class of espaliers . 1 . if every bounded lower subespalier of every member of @xmath2624 belongs to @xmath2624 , then every bounded continuous dimension scale is isomorphic to the dimension range of some bounded member of @xmath2624 . if every lower subespalier of every member of @xmath2624 belongs to @xmath2624 , then every continuous dimension scale is isomorphic to the dimension range of some member of @xmath2624 . \(i ) let @xmath13 be a bounded continuous dimension scale , denote by @xmath2226 the largest element of @xmath13 . since @xmath2624 is d - universal , there exists @xmath2637 such that @xmath13 is ( isomorphic to ) a lower subset of @xmath2628 . let @xmath68 such that @xmath2638 , put @xmath2639 $ ] , a lower subespalier of @xmath3 . it follows from the assumption and lemma [ l : basicdrng](i ) that @xmath216 belongs to @xmath2624 and @xmath2627 is isomorphic to @xmath13 . observe that @xmath216 is bounded . \(ii ) let @xmath13 be a continuous dimension scale . since @xmath2624 is d - universal , there exists @xmath2637 such that @xmath13 is ( isomorphic to ) a lower subset of @xmath2628 . put @xmath2640 . it follows from the assumption and lemma [ l : basicdrng](ii ) that @xmath216 belongs to @xmath2624 and @xmath2627 is isomorphic to @xmath13 . the following result gives us a sufficient condition for d - universality . [ l : duniv ] let @xmath2624 be a class of espaliers satisfying the following conditions : 1 . @xmath2624 is closed under finite direct products . 2 . for every ordinal @xmath40 and every complete boolean space @xmath76 , there are @xmath2641 , @xmath2642 , @xmath2643 such that @xmath2644 has a lower embedding into @xmath2645 , @xmath2646 has a lower embedding into @xmath2647 , and @xmath1470 has a lower embedding into @xmath2648 . then @xmath2624 is d - universal . it follows from lemma [ l : basicdrng ] and assumptions ( i ) , ( ii ) above that for every ordinal @xmath40 and any complete boolean spaces @xmath37 , @xmath38 , and @xmath39 , there exists @xmath2637 such that the continuous dimension scale @xmath2649 embeds into @xmath2628 . the conclusion follows from theorem [ t : embdimint ] . the results of section [ s : duniv ] will make it possible to prove further results of d - universality . taking account of the various examples of espaliers discussed in the introduction , one is led to the conjecture that the appearance of large cardinal values in the functional representation of the dimension range of an espalier should be closely related to the existence of certain large orthogonal sums within the espalier . ( see also the proof of lemma [ l : locm5 ] . ) moreover , in the construction of espaliers of different types , we will need to know what ingredients will ensure that the dimension range of an example will be as large as desired . in the present section , we provide some answers to the above questions . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ standing hypotheses : @xmath3 is an espalier , @xmath13 is the dimension range of @xmath3 , and @xmath2438 is the canonical map . moreover , @xmath76 , @xmath37 , @xmath38 , @xmath39 are as in section . _ let @xmath40 be the ordinal and @xmath1594 the dimension function defined in section . we put @xmath2650 . we pick a lower embedding @xmath2651 as in proposition . let @xmath2652 be the corresponding lower embedding defined in section . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ l : homogrem ] let @xmath124 , @xmath130 be purely infinite elements with @xmath1967 . suppose that there is an infinite cardinal @xmath1088 such that @xmath2653 but @xmath2654 for all nonzero @xmath1912 . then @xmath2439 . suppose @xmath132 and @xmath2441 . because of general comparability in @xmath3 ( proposition [ p : shasgc ] and comment following ) , there is some @xmath1758 such that @xmath2655 and @xmath2656 . now @xmath2657 by lemmas [ l : baspcdotl ] and [ l : p(aopplusb ) ] . moreover , these lemmas imply @xmath2658 , and so we have @xmath2659 . our assumptions on @xmath124 now imply that @xmath2660 , and hence @xmath2661 by lemma [ l : ppbotonl ] . since @xmath2178 is purely infinite and @xmath2655 , we have @xmath2662 , and so @xmath2663 therefore @xmath2439 . in many examples of espaliers , the orthogonality relation coincides with disjointness in the ( partial ) lattice : @xmath2664 . let us abbreviate this condition by the symbol @xmath2665 . [ l : orthocount ] assume @xmath2665 . let @xmath132 be purely infinite , and let @xmath2666 be an infinite cardinal such that @xmath5 is not equal to any orthogonal sum of more than @xmath2666 nonzero elements . let @xmath24 and let @xmath2667 be a cardinal number such that @xmath2668 . then @xmath6 does not majorize any orthogonal sum of more than @xmath85 nonzero elements . in particular , @xmath2669 for all @xmath2670 and all nonzero @xmath2111 . by assumption , @xmath2671 with @xmath2672 and each @xmath2673 . suppose that @xmath2674 where @xmath2675 and all @xmath2676 . since @xmath2677 , we have @xmath2678 . axiom ( l4 ) then yields a finite subset @xmath2679 such that @xmath2680 . since the set of finite subsets of @xmath56 has cardinality @xmath85 , the fibres of the map @xmath2681 can not all have cardinality at most @xmath85 . hence , there exist a subset @xmath2682 with @xmath2683 and a finite subset @xmath2684 such that @xmath2685 for all @xmath2686 . thus , the element @xmath2687 satisfies @xmath2688 , and so @xmath2689 , for all @xmath2686 , because of @xmath2665 . consequently , @xmath2690 majorizes an orthogonal sum of more than @xmath85 nonzero elements , and after adjoining an additional element if necessary , we may assume that @xmath2690 equals such an orthogonal sum . however , @xmath2691 because @xmath5 is purely infinite , and so axiom ( l6 ) implies that @xmath5 is an orthogonal sum of more than @xmath85 nonzero elements . this contradicts our hypotheses . [ l : largedelta ] assume @xmath2665 . let @xmath132 be purely infinite , put @xmath1762 , and let @xmath272 be an ordinal such that @xmath5 is not equal to any orthogonal sum of more than @xmath2692 nonzero elements . let @xmath24 and let @xmath2693 be an ordinal with @xmath2694 . then @xmath2695 is defined , and @xmath2696 . we proceed by induction on @xmath2693 . assume first that @xmath2697 . the set @xmath2698 is nonempty , as it contains @xmath2699 . since the element @xmath1294 is removable from any element of @xmath3 , the element @xmath1398 is defined as the least element of @xmath78 thus , @xmath2700 . next , suppose that @xmath2701 for some ordinal @xmath2702 . there is some @xmath2703 such that @xmath2704 . by induction , @xmath2705 is defined , and @xmath2706 . now @xmath2707 for some purely infinite @xmath2708 , and lemma [ l : orthocount ] shows that @xmath2709 for all nonzero @xmath2710 . on the other hand , since @xmath2694 , we have @xmath2711 . hence , @xmath2712 by lemma [ l : homogrem ] , and so @xmath2713 . therefore @xmath2714 . since @xmath2715 is a purely infinite element with central cover @xmath26 , it follows that @xmath2695 is defined and majorized by @xmath2715 . finally , suppose that @xmath2693 is a limit ordinal . for each ordinal @xmath2716 , there exists @xmath2717 such that @xmath2718 . by induction , @xmath2705 is defined and @xmath2719 . therefore @xmath2695 is defined , and @xmath2720 . [ p : existconstant ] assume @xmath2665 . let @xmath5 , @xmath24 be purely infinite elements such that @xmath2721 , and let @xmath272 , @xmath2693 be ordinals , such that @xmath5 is not equal to any orthogonal sum of more than @xmath2692 nonzero elements , and @xmath2694 . then the following statements hold : 1 . @xmath2722 for all @xmath1411 . 2 . there exists a purely infinite element @xmath2723 such that @xmath2724 equals the constant function with value @xmath2725 . 3 . set @xmath2726\subseteq l$ ] , and restrict @xmath2727 , @xmath2 , @xmath4 from @xmath3 to @xmath2728 . then @xmath2728 is an espalier , and @xmath2729 . \(i ) in view of lemma [ l : largedelta ] , @xmath2730 is defined and majorized by @xmath2715 . since @xmath2731 for all @xmath1411 , we get @xmath2722 for all @xmath1456 . \(ii ) because of ( i ) , the lower embedding @xmath2732 sends @xmath2715 to a function @xmath1631 with @xmath2733 for all @xmath1411 . in particular , @xmath2734 , and @xmath1705 contains the constant function @xmath2735 with @xmath2736 for all @xmath1411 . since @xmath2737 is a lower embedding , there is some @xmath2738 such that @xmath2739 . note that @xmath2740 is purely infinite , because @xmath2735 is . hence , @xmath2741 . it just remains to note that @xmath2742 for some purely infinite element @xmath2723 . \(iii ) that @xmath2728 is an espalier follows from proposition [ p : lspesp](i ) . it is clear that @xmath2743 is isomorphic to the submonoid @xmath2744 \subseteq s$ ] . since @xmath2737 is a lower embedding , it maps @xmath2745 isomorphically onto @xmath2746 . in case @xmath2665 does not hold , it is not clear whether large orthogonal sums are sufficient to imply large constants . for use in that situation , we record the following more elementary approach . [ l : buildpkappa ] let @xmath1088 be an infinite cardinal , and @xmath2747 a family of purely infinite elements of @xmath13 . set @xmath2748 . for all infinite cardinals @xmath2749 , assume that @xmath2750 but @xmath2751 for all nonzero projections @xmath674 . then @xmath1293 is defined , and @xmath2752 . we show , by induction on @xmath2353 , that @xmath2753 is defined and majorized by @xmath2754 , for all infinite cardinals @xmath2755 . since @xmath2756 is a purely infinite element with central cover @xmath26 , it is clear from the definition that @xmath1398 is defined and @xmath2757 . next , suppose that @xmath2353 is an infinite cardinal less than @xmath1088 , such that the element @xmath2758 is defined and @xmath2759 . note that the elements @xmath124 and @xmath2760 both have central cover @xmath26 . by assumption , @xmath2761 for all nonzero projections @xmath674 , whence corollary [ c : remequiv ] implies that @xmath2762 . thus , @xmath2763 is defined and @xmath2764 . finally , if @xmath2353 is a limit cardinal less than or equal to @xmath1088 , such that @xmath2765 is defined and majorized by @xmath2766 for all infinite cardinals @xmath2767 , then @xmath2768 for all @xmath2666 , whence @xmath2769 is defined and @xmath2770 . [ p : secondexistconstant ] let @xmath2693 be an ordinal and @xmath2771 a family of purely infinite elements of @xmath3 with central cover @xmath2772 . for all ordinals @xmath2773 , assume that @xmath2774 but @xmath2775 for all nonzero projections @xmath2776 . then the following statements hold : 1 . @xmath2777 for all @xmath1411 . 2 . there exists a purely infinite element @xmath2723 such that @xmath2724 equals the constant function with value @xmath2725 . 3 . set @xmath2726\subseteq l$ ] , and restrict @xmath2727 , @xmath2 , @xmath4 from @xmath3 to @xmath2728 . then @xmath2728 is an espalier , and @xmath2729 . \(i ) set @xmath2778 for all ordinals @xmath2779 . then @xmath2780 is a family of purely infinite elements of @xmath13 with central cover @xmath2772 , such that for all infinite cardinals @xmath2781 , we have @xmath2750 but @xmath2782 for all nonzero @xmath2783 . thus , by lemma [ l : buildpkappa ] , @xmath2730 is defined and @xmath2784 . since @xmath2731 for all @xmath1411 , we get @xmath2777 for all @xmath1456 . \(ii ) and ( iii ) follow from ( i ) just as in proposition [ p : existconstant ] . of course , the underlying boolean algebra of a boolean espalier is complete . for a boolean algebra @xmath14 , we will denote by @xmath2785 the canonical orthogonality relation of @xmath14 , that is , @xmath2786 if and only if @xmath16 , for all @xmath5 , @xmath15 . we say that a family @xmath2787 of elements of @xmath14 is _ disjoint _ , if @xmath2788 for all @xmath383 in @xmath56 , and then we let @xmath2789 denote its join . * @xmath65 implies that @xmath66 , for all @xmath67 . * the binary relation @xmath4 is unrestrictedly refining , that is , for every @xmath2791 and every disjoint family @xmath2792 of elements of @xmath14 , if @xmath1925 , then there exists a decomposition @xmath1597 such that @xmath73 for all @xmath62 . * the binary relation @xmath4 is unrestrictedly additive , that is , for all disjoint families @xmath2787 and @xmath2792 of elements of @xmath14 , if @xmath73 for all @xmath62 , then @xmath1926 . we observe that the underlying boolean algebra of a boolean pre - espalier need not be complete . we recall ( see , for example , t. jech @xcite ) that for every boolean algebra @xmath14 , there exists a unique ( up to isomorphism ) complete boolean algebra , that we shall denote by @xmath2794 and call the _ completion _ of @xmath14 , such that @xmath14 is dense in @xmath2794 . the following result makes it possible to extend to @xmath2794 any boolean pre - espalier structure on @xmath14 . [ l : extpreesp ] let @xmath2793 be a boolean pre - espalier . define a binary relation @xmath2795 on @xmath2794 by the rule @xmath2796 for all @xmath5 , @xmath2797 . then @xmath2798 is a boolean espalier . furthermore , @xmath2795 is the smallest equivalence relation @xmath2799 on @xmath2794 containing @xmath4 such that @xmath2800 is an espalier . it is clear that every equivalence relation @xmath2799 on @xmath2794 containing @xmath4 , such that @xmath2800 is an espalier , also contains @xmath2795 , hence it suffices to prove that @xmath2798 is an espalier . since every element of @xmath2794 can be written @xmath2801 , where all the @xmath57-s belong to @xmath14 , the binary relation @xmath2795 is reflexive . it is obviously symmetric . now let @xmath124 , @xmath125 , @xmath2802 such that @xmath2803 and @xmath2804 . there are decompositions of the form @xmath2805 with @xmath2806 in @xmath14 , for all @xmath62 , and @xmath2807 , for all @xmath279 . for any @xmath62 , @xmath2808 , thus , since @xmath4 satisfies ( b1 ) , there exists a decomposition @xmath2809 with @xmath2810 , for all @xmath279 . for @xmath279 , since @xmath2811 and by ( b1 ) , there exists a decomposition @xmath2812 such that @xmath2813 , for all @xmath62 . therefore , @xmath2814 , for all @xmath2815 , and @xmath2816 and @xmath2817 ; whence @xmath2818 . therefore , @xmath2795 is an equivalence relation on @xmath2794 . it is obvious that @xmath2795 satisfies ( b0 ) . now let @xmath2819 in @xmath2794 . by definition , there are decompositions @xmath2820 and @xmath2821 such that @xmath2822 , for all @xmath279 . for @xmath279 , since @xmath2823 , there exists a decomposition @xmath2824 such that @xmath2825 , for all @xmath62 . observe that @xmath2816 ; put @xmath2809 , for all @xmath62 . thus @xmath1597 , and , by the definition of @xmath2795 , @xmath2826 , for all @xmath62 . therefore , @xmath2795 satisfies ( b1 ) . finally let @xmath1597 and @xmath2827 with @xmath2828 , for all @xmath62 . by definition , for all @xmath62 , there are decompositions @xmath2829 and @xmath2830 such that @xmath2831 , for all @xmath62 and all @xmath2832 . put @xmath2833 , then @xmath2834 and @xmath2835 , whence @xmath2803 . therefore , @xmath2795 satisfies ( b2 ) . for a boolean espalier @xmath2793 and a set @xmath56 , we let the permutation group @xmath2836 of @xmath56 act on the boolean algebra @xmath2837 by translation : namely , @xmath2838 next , let @xmath2839 be the equivalence relation on @xmath2837 associated with this action and @xmath4 , that is , @xmath2840 since @xmath2836 acts on @xmath2837 by automorphisms ( of the boolean algebra @xmath2837 ) and @xmath2793 is an espalier , it is easy to see that @xmath2841 is a boolean pre - espalier . since @xmath2837 is already complete , the espalier closure of @xmath2841 is an equivalence relation on @xmath2837 , that we shall denote by @xmath2842 . for @xmath62 , we denote by @xmath2843 the canonical map , that is , @xmath2844 is equal to @xmath5 if @xmath2845 , to @xmath188 otherwise , for all @xmath67 and all @xmath279 . \(i ) it is obvious that @xmath2846 is a @xmath2849-isomorphism from @xmath14 onto a lower subset of @xmath2837 , and that @xmath25 implies that @xmath2850 , for all @xmath5 , @xmath15 . now suppose that @xmath2850 . there are decompositions of the form @xmath2851 and @xmath2852 in @xmath14 such that @xmath2853 , for all @xmath279 . hence @xmath2854 , for all @xmath279 , whence , since @xmath2793 satisfies ( b2 ) , @xmath25 . this completes the proof of ( i ) . let @xmath2855 be nonzero ; so @xmath2856 , for some @xmath2857 . since @xmath124 has a nonzero component , there are @xmath2858 and @xmath2859 such that @xmath2860 . hence , @xmath2861 which completes the proof of ( ii ) . for a boolean algebra @xmath14 , we define a cardinal number @xmath2862 by @xmath2863 furthermore , for a set @xmath56 , we put @xmath2864 the coming set of lemmas , from [ l : prescard ] to [ l : remgext ] , is aimed at constructing boolean espaliers whose dimension ranges have large constants . instead of accomodating the results of section [ s : lrgconst ] to the present context , we propose direct proofs , probably of more interest to the boolean algebra - oriented reader . put @xmath2869 . since @xmath2866 , there are decompositions of the form @xmath2851 and @xmath2852 such that @xmath2870 for all @xmath279 . by decomposing further the @xmath2871-s and the @xmath2872-s as disjoint sums of elements of @xmath2873 $ ] , we may assume , without loss of generality , that both @xmath2871 and @xmath2872 belong to @xmath2874 , for all @xmath279 . hence , for all @xmath279 , there are @xmath2875 , @xmath2876 and @xmath2877 , @xmath2878 such that @xmath2879 and @xmath2880 . we observe that @xmath2881 put @xmath2882 , for all @xmath2883 . since the family @xmath2884 is the image under @xmath2846 of the antichain @xmath2885 of @xmath14 , the inequality @xmath2886 holds . therefore , @xmath2887 let @xmath2897 be a bijection . if @xmath78 is finite , then @xmath272 can be extended to a permutation @xmath2693 of @xmath56 , thus @xmath2898 . suppose now that @xmath78 is infinite . there exists a partition @xmath2899 of @xmath78 such that @xmath2900 . put @xmath2901 $ ] , for @xmath289 . then the restriction of @xmath272 from @xmath97 onto @xmath2902 can be extended to a permutation @xmath2903 of @xmath56 , for all @xmath289 . therefore , @xmath2904 let @xmath5 , @xmath2910 such that @xmath2911 and @xmath2912 , we prove that @xmath2913 . it follows from lemma [ l : prescard ] that @xmath2914\leq\alpha$ ] , thus , putting @xmath2915 , we obtain that @xmath2916 and @xmath2917 . from lemma [ l : x1simy1 ] it follows that @xmath2918 , whence @xmath2919 . since @xmath2920 and by lemma [ p : schbern ] , it follows that @xmath2913 . denote by @xmath2874 the complete boolean algebra of clopen subsets of @xmath76 . let @xmath2922 be an ordinal such that @xmath2923 , put @xmath2924 , endow the direct power @xmath2925 with the previously introduced @xmath2926 defined from the espalier @xmath2927 . we consider the map @xmath2928 introduced earlier . it follows from lemma [ l : basicphii ] that @xmath2929 is a lower embedding of espaliers and @xmath2930 is a lower embedding of continuous dimension scales with dense image . in particular , @xmath2931 . since @xmath76 is isomorphic to the ultrafilter space of @xmath2874 , it follows from theorems [ t : embdimint ] and [ t : dimesp ] that @xmath2932 has a lower embedding into an espalier of the form @xmath2933 for some ordinal @xmath54 and a partition @xmath1103 of @xmath76 into clopen sets . however , the continuous dimension scale @xmath2934 has a lower embedding into @xmath2932 , thus @xmath2935 and @xmath2936 . finally , it follows from lemma [ l : remgext ] that the @xmath2937-sequence @xmath2938 is @xmath1168-increasing in @xmath2932 , but all the members of this sequence have central cover @xmath2772 , thus the image of @xmath2932 in @xmath1705 contains a function whose values are all above @xmath2939 , in particular , @xmath2644 has a lower embedding into @xmath2932 . the argument of proposition [ p : duniv ] shows then that there exists a bounded lower subespalier @xmath2793 of @xmath2940 such that @xmath63 is isomorphic to @xmath2644 . the proof of lemma [ l : lotsofii ] requires some familiarity with forcing and complete boolean algebras , in particular , the random real extension and two - step iterated forcing , see @xcite . we denote by @xmath2942 the boolean algebra of all borel subsets of the real unit interval @xmath2943 $ ] modulo null sets , the _ random algebra _ , and by @xmath2944 $ ] the lebesgue measure on @xmath2942 . furthermore , let @xmath25 hold , if @xmath55 , for all @xmath5 , @xmath2945 . let @xmath2946 , for a family @xmath2947 of elements of @xmath2943 $ ] , mean that @xmath54 is the supremum over all finite subsets @xmath277 of @xmath56 of @xmath2948 . we need a couple of standard facts on the measure @xmath43 , summed up in the following claims . 1 . @xmath2949 , for any disjoint family @xmath2950 of elements of @xmath2942 . 2 . let @xmath2951 and let @xmath2947 be a family of elements of @xmath2943 $ ] . if @xmath2952 , then there exists a decomposition @xmath2953 in @xmath2942 such that @xmath2954 , for all @xmath62 . \(ii ) again , @xmath2956 is countable , so we may assume without loss of generality that @xmath2957 . it is then easy to construct inductively a nondecreasing sequence @xmath2958 of elements of @xmath2943 $ ] satisfying the conditions @xmath2959)=\alpha_0 $ ] and @xmath2960)=\alpha_{n+1}$ ] , for all @xmath264 . put @xmath2961 , then @xmath2962 $ ] and @xmath2963 $ ] , for all @xmath264 , satisfy the desired conclusion . now we work under the assumptions of lemma [ l : lotsofii ] . denote by @xmath2965 the complete boolean algebra of clopen subsets of @xmath76 . we consider the quotient of the scott - solovay @xmath2965-valued universe @xmath2966 of set theory under the equivalence relation that identifies names @xmath2967 and @xmath2968 if and only if@xmath2969 . we still denote by @xmath2966 the quotient , endowed with its natural boolean value , so , now , @xmath2970 if and only if @xmath2971 , for all @xmath2972 , @xmath2973 . furthermore , let @xmath2974 denote the ( equivalence class of the ) @xmath2965-valued name for @xmath2942 , and put @xmath2975 , the two - step iterated forcing of @xmath2965 by the random algebra of @xmath2966 . hence @xmath2874 is the complete boolean algebra of all @xmath2976 such that @xmath2977 , the partial ordering @xmath149 being defined by @xmath2978 if and only if@xmath2979 . hence , orthogonality in @xmath2874 is defined by @xmath9 if and only if @xmath2980 . let @xmath4 be the binary relation defined on @xmath2874 by @xmath2981 if and only if @xmath2982 , for all @xmath2972 , @xmath2983 . it is obvious that @xmath4 satisfies ( b0 ) . now let @xmath2985 in @xmath2874 , with @xmath2986 decomposed as @xmath2987 . if @xmath2988 denotes the canonical embedding from @xmath2965 into @xmath2989 , the relation @xmath2990 is equivalent to @xmath2991 , for all @xmath2972 , @xmath2983 . it is an easy exercise to deduce from this the relation @xmath2992 , where the symbol @xmath2993 denotes the canonical name in @xmath2966 for @xmath56 . moreover , @xmath2994 and @xmath2966 is a boolean - valued model of set theory , in particular , @xmath2966 satisfies claim [ cl:2 ] . therefore , @xmath2995 since @xmath2966 is full and the notion of function is absolute , there exists a family @xmath2996 of elements of @xmath2874 such that @xmath2997 , for all @xmath62 , and @xmath2998 hence @xmath2999 and @xmath3000 , for all @xmath62 . [ cl:4 ] there exists an isomorphism of partial commutative monoids from @xmath3006)$ ] onto the set of @xmath3002 such that @xmath3004 , endowed with the addition defined by @xmath3007 if and only if @xmath3008 . for @xmath2226 , @xmath3010 , @xmath3011 if and only if@xmath3012 , if and only if @xmath3013 , if and only if@xmath2985 . if @xmath3014 and @xmath3015 , then @xmath3016 , thus @xmath3017 , thus @xmath3018 . finally , let @xmath3019)$ ] . there exists a unique @xmath3020 such that @xmath3021\right\|}=1 $ ] , thus @xmath3001 and @xmath3022 , so @xmath1669 is surjective . the rest of the proof proceeds like in the proof of lemma [ l : lotsofi ] , of which we shall keep the notation . it follows from claim [ cl:5 ] that the boolean algebra of projections of @xmath2984 is isomorphic to @xmath2965 , see claim [ cl : projpu ] in the proof of theorem [ t : c(o , k)dimint ] . if @xmath2922 is an ordinal such that @xmath2923 and we put @xmath2924 , then @xmath2932 has a @xmath1168-increasing chain of length @xmath3023 . furthermore , observe that this time , since @xmath3006)$ ] has a lower embedding into @xmath2932 , @xmath3024 while @xmath3025 , and the image of @xmath2932 in @xmath1705 contains a function with values above @xmath2939 . hence , there exists a bounded lower subespalier @xmath2793 of @xmath2940 such that @xmath2941 . in order to give a proof of lemma [ l : lotsofiii ] , it is also convenient to be familiar with forcing and complete boolean algebras . we denote by @xmath3027 the boolean algebra of all borel subsets of the cantor space @xmath3028 modulo meager sets , the _ cohen algebra_. furthermore , for @xmath3029 , we define @xmath3030 if @xmath3031 while @xmath3032 . let @xmath25 hold , if @xmath3033 , for all @xmath5 , @xmath3034 . every nonzero element of @xmath3027 can be decomposed as a disjoint union of two ( resp . , @xmath175 ) nonzero elements of @xmath3027 . furthermore , if @xmath3037 in @xmath3027 , then @xmath3038 is countable . it follows easily that @xmath4 satisfies ( b1 ) . it obviously satisfies ( b0 ) and ( b2 ) . since every nonzero element of @xmath3027 can be decomposed as a disjoint union of two nonzero elements of @xmath3027 , every element of @xmath3027 is purely infinite . it follows that @xmath3039 . now we work under the assumptions of lemma [ l : lotsofiii ] . denote by @xmath2965 the complete boolean algebra of clopen subsets of @xmath76 . we define the ( quotiented ) scott - solovay universe @xmath2966 of set theory as in the proof of lemma [ l : lotsofii ] . furthermore , let @xmath3040 denote the ( equivalence class of the ) @xmath2965-valued name for @xmath3027 , and put @xmath3041 , the two - step iterated forcing of @xmath2965 by the cohen algebra of @xmath2966 . hence @xmath2874 is the complete boolean algebra of all @xmath2976 such that @xmath3042 , the partial ordering @xmath149 being defined by @xmath2978 if and only if@xmath2979 . hence , orthogonality in @xmath2874 is defined by @xmath9 if and only if @xmath2980 . let @xmath4 be the binary relation defined on @xmath2874 by @xmath2981 if and only if @xmath2982 , for all @xmath2972 , @xmath2983 . [ cl:3b ] there exists an isomorphism of partial commutative monoids from @xmath3045 onto the set of @xmath3002 such that @xmath3046 , endowed with the addition defined by @xmath3007 if and only if @xmath3008 . the proof that @xmath1669 is an embedding for @xmath149 and for @xmath230 is the same as in the proof of claim [ cl:4 ] of lemma [ l : lotsofii ] . let @xmath3047 . there exists @xmath3020 such that @xmath3048 , thus @xmath3001 and @xmath3022 , so @xmath1669 is surjective . the rest of the proof proceeds like in the proof of lemma [ l : lotsofii ] . it follows from claim [ cl:4b ] that the boolean algebra of projections of @xmath2984 is isomorphic to @xmath2965 . if @xmath2922 is an ordinal such that @xmath2923 and we put @xmath2924 , then @xmath2932 has a @xmath1168-increasing chain of length @xmath3023 . since @xmath3045 has a lower embedding into @xmath2932 , @xmath3049 while @xmath3050 , and the image of @xmath2932 in @xmath1705 contains a function with values above @xmath2939 . hence , there exists a bounded lower subespalier @xmath2793 of @xmath2940 such that @xmath3026 . [ rk : iineedsforc ] since the boolean algebra @xmath3027 has an absolute ( in set - theoretical sense ) dense subalgebra , namely , the boolean algebra @xmath3051 of clopen subsets of @xmath3028 , the forcing could , in principle , have been eliminated from the proof of lemma [ l : lotsofiii ] : for example , one could have taken for @xmath2874 the completion of @xmath3052 ( the tensor product for boolean algebras is just the coproduct ) . such an argument would not have worked for lemma [ l : lotsofii ] , because @xmath2942 of the ground universe may not be dense in the @xmath2942 of a generic extension . we first recall some basic lattice - theoretical definitions , see @xcite . a lattice @xmath3053 is _ modular _ , if @xmath3054 implies that @xmath3055 , for all @xmath5 , @xmath6 , @xmath7 . we say that @xmath3 is * _ complemented _ , if it has a least element @xmath188 , a largest element @xmath2772 , and every @xmath132 has a complement , that is , @xmath24 such that @xmath16 and @xmath3056 . * _ sectionally complemented _ , if it has a least element @xmath188 and every sublattice of the form @xmath2165 $ ] , for @xmath68 , is complemented ; * _ relatively complemented _ , if every sublattice of the form @xmath3057 $ ] , for @xmath131 in @xmath3 , is complemented . in general , these notions are unrelated . however , in the _ modular _ case , the following implications hold : @xmath3058 we say that the lattice @xmath3 is _ complete _ , if every subset of @xmath3 has an infimum equivalently , every subset of @xmath3 has a supremum . we say that @xmath3 is _ conditionally complete _ , if every nonempty bounded subset of @xmath3 has an infimum equivalently , the interval @xmath3057 $ ] is complete , for all @xmath131 in @xmath3 . we say that @xmath3 is _ meet - continuous _ , if for every @xmath68 and every upward directed subset @xmath78 of @xmath3 admitting a supremum , the equality @xmath3059 holds , where we put @xmath3060 . if the dual condition holds , @xmath3 is called _ join continuous _ , and if both conditions hold , @xmath3 is _ continuous_. ( this definition of continuity is not equivalent to the one presented in g. gierz _ et al . _ @xcite , which is nowadays often called `` scott continuity '' . ) a _ continuous geometry _ is any complete , complemented , modular , continuous lattice . ( this is the current terminology ; von neumann s original definition included hypotheses of irreducibility and lack of chain conditions . what we have called a continuous geometry was called a `` reducible continuous geometry '' or a `` continuous geometry in the general sense '' in some of the older literature . ) it is an open problem whether the dimension monoid of an arbitrary lattice is always a refinement monoid , however , this is solved in a few important particular cases : the case of _ finite lattices _ , of which the dimension monoids are so - called _ primitive monoids _ , and the case of _ modular lattices _ , for which an alternative presentation of the dimension monoid is given that implies refinement . we shall concentrate here on the latter . in a modular lattice @xmath3 with zero , let @xmath9 hold , if @xmath16 , for any @xmath5 , @xmath24 , and then we define @xmath3065 . the following result is folklore , see , for example , ( * ? ? ? * proposition 8.1 ) ; it says , essentially , that @xmath135 is associative in modular lattices . we state it in a way that relates it to the axioms defining espaliers . in a modular lattice @xmath3 , we define the binary relations @xmath4 ( _ perspectivity _ ) , @xmath3066 ( _ bi - perspectivity _ ) , @xmath139 ( _ projectivity _ ) , and @xmath3067 ( _ projectivity by decomposition _ ) as follows : @xmath3068 in case @xmath3 is a relatively complemented lattice with zero , the dimension monoid @xmath143 of @xmath3 is generated by the elements @xmath3069 , for @xmath132 ( see ( * ? ? ? * proposition 9.1 ) ) . we define the _ dimension range _ of @xmath3 as @xmath3070 . in case @xmath3 is also modular , @xmath2628 can be endowed with the partial addition defined by @xmath3071 furthermore , the partial commutative monoid @xmath2628 determines the commutative monoid@xmath143 , and any equality of the form @xmath3072 can be tested in a very simple way , see corollaries 9.4 and 9.5 in @xcite and proposition [ p : prmrm ] of the present paper . this is the point where the theory of espaliers and continuous dimension scales comes in . our plan is to associate , with a sectionally complemented , modular lattice @xmath3 , an espalier @xmath3075 such that the dimension range of @xmath3 , as defined above , is the dimension range of @xmath3075 . the structure @xmath3075 is simply defined as @xmath3076 , for those @xmath2 and @xmath3067 defined above , so all we need to do is find sufficient conditions for it to be an espalier . the following lemma sums up some of the hardest ( in particular item ( i ) ) and most useful results of @xcite . 1 . @xmath3077 if and only if there are decompositions @xmath3078 and @xmath3079 in @xmath3 such that @xmath3080 and @xmath3081 , for all @xmath5 , @xmath24 . 2 . let @xmath124 , @xmath130 and let @xmath69 be a family of elements of @xmath3 . if @xmath1925 , then there exists a decomposition @xmath1597 such that @xmath73 , for all @xmath62 . 3 . let @xmath72 and @xmath3082 be families of elements of @xmath3 such that @xmath3083 . then there are families @xmath3084 and @xmath3085 of elements of @xmath3 such that @xmath3086 and @xmath3087 , for all @xmath2815 . \(ii ) there exists @xmath1912 such that @xmath1971 . let @xmath3088\twoheadrightarrow[0,a]$ ] be the perspective mapping with axis @xmath126 , that is , @xmath3089 , for all @xmath3090 $ ] . put @xmath3091 , for all @xmath62 . since @xmath2693 is an isomorphism from @xmath2372 $ ] onto @xmath2165 $ ] , the equality @xmath1597 holds . furthermore , @xmath274 is perspective to @xmath3092 with axis @xmath126 , for all @xmath62 . the proof of ( iii ) is virtually the same as the one of ( * ? ? ? * lemma 12.17 ) , except that we replace countable families by transfinite ones . we give the proof here for the convenience of the reader . so , let @xmath1298 and @xmath1088 be ordinals , let @xmath3093 and @xmath3094 be orthogonal families of elements of @xmath3 such that @xmath3095 . we put @xmath3096 furthermore , we put @xmath3097 and @xmath3098 , for all @xmath1299 and all @xmath3099 . since @xmath3100 and @xmath3 is sectionally complemented , there exists @xmath3101 such that @xmath3102 . we prove the conclusion by induction on @xmath85 . it is trivial for @xmath3105 . for @xmath85 a limit ordinal , it follows easily from the meet - continuity of @xmath3 and the induction hypothesis . now suppose having proved the statement at step @xmath85 , we prove it at step @xmath3106 . it follows from the induction hypothesis that @xmath3107 , thus @xmath3108 that is , @xmath3109 . furthermore , @xmath3110 whence @xmath3111 in particular , using claim [ cl : ineqmc1 ] for @xmath3114 yields that @xmath3115 , thus @xmath3116 . thus , by item ( ii ) above , there exists a decomposition of the form @xmath3117 such that @xmath3118 , for all @xmath3119 . similarly , for all @xmath3099 , there exists a decomposition of the form @xmath3120 such that @xmath3121 , for all @xmath2391 . it follows that @xmath3122 , for all @xmath3123 and @xmath3124 , and the @xmath3125-s and @xmath3126-s are as desired . let @xmath124 , @xmath125 , @xmath3092 ( for @xmath62 ) be elements of @xmath3 such that @xmath3073 and @xmath2827 . by lemma [ l : lattinfref](i ) , there are @xmath257 , @xmath2314 , @xmath1675 , @xmath3129 such that @xmath2306 , @xmath2343 , @xmath3130 , and @xmath3131 . applying lemma [ l : lattinfref](iii ) to the equality @xmath3132 , we obtain decompositions @xmath2340 , @xmath2341 , @xmath3133 , for all @xmath62 , such that @xmath3134 and @xmath3135 , for all @xmath62 . since @xmath3136 and @xmath3137 , there are , by lemma [ l : lattinfref](ii ) , decompositions @xmath3138 and @xmath3139 such that @xmath3140 and @xmath3141 , for all @xmath62 . observe that @xmath3142 , where we put @xmath3143 , for all @xmath62 . furthermore , @xmath3144 , for all @xmath62 . hence @xmath3067 is unrestrictedly refining ( axiom ( l6 ) ) . by proposition [ p : l2l*esp ] , @xmath3075 is an espalier and @xmath3128 , thus , by theorem [ t : dimesp ] , @xmath1998 is a continuous dimension scale . by corollary [ c : embdimint ] , @xmath291 is also a continuous dimension scale , but by proposition [ p : d(a)=d(b ) ] , @xmath143 is isomorphic to @xmath291 , thus it is a continuous dimension scale . hence we can complete the program of determining the dimension theory of conditionally complete , meet - continuous , relatively complemented , modular lattices initiated by i. halperin and j. von neumann in @xcite . let @xmath3 be a conditionally complete , meet - continuous , relatively complemented modular lattice . the interval @xmath3057 $ ] is a conditionally complete , meet - continuous , complemented modular lattice , for all @xmath131 in @xmath3 , thus , by corollary [ c : dimmonmcslatt ] , its dimension monoid @xmath3145 $ ] is a continuous dimension scale . furthermore , if @xmath3146 in @xmath3 , then the natural map from @xmath3145 $ ] to @xmath3147 $ ] is , by ( * ? ? ? * corollary 13.5 ) , a lower embedding of commutative monoids . express the lattice @xmath3 as the direct limit of the direct system @xmath3148 of its closed intervals . since the @xmath3149 functor preserves direct limits , it follows from lemma [ l : dirun ] that @xmath3150\in\mathcal{i}}\operatorname{dim}[a , b]$ ] is a continuous dimension scale . we say that a ( von neumann ) regular ring @xmath17 is _ right continuous _ , if the lattice @xmath3151 of all principal right ideals of @xmath17 ( cf . * theorem 2.3 ) ) is complete and meet - continuous . in particular , every right self - injective regular ring is right continuous , see ( * ? ? ? * corollary 13.5 ) . the connection between the present section and the upcoming section [ s : rsireg ] is made possible by the following immediate consequence of ( * ? ? ? * corollary 13.4 ) . 1 . the monoid @xmath3152 of isomorphism classes of finitely generated projective right @xmath17-modules is isomorphic to the dimension monoid @xmath3153 of @xmath3151 . 2 . two principal right ideals @xmath56 and @xmath277 of @xmath17 are isomorphic if and only ifthere are decompositions @xmath3154 and @xmath3155 such that @xmath3156 and @xmath3157 . in particular , @xmath3158 if and only if@xmath3159 in the lattice @xmath3151 . it follows from theorem [ t : l(r)duniv ] below that every continuous dimension scale admits a lower embedding into the dimension range of an espalier of the form @xmath3161 , for some regular , right self - injective ring @xmath17 . it follows from proposition [ p : latt2ring](ii ) that the relations @xmath94 and @xmath3067 on @xmath3151 coincide . since @xmath3151 is a complete , meet - continuous , complemented , modular lattice , the conclusion follows . in particular , it follows from propositions [ p : l2l*esp ] and [ p : latt2ring ] and theorem [ t : l(r)duniv ] that every continuous dimension scale admits a lower embedding into @xmath3153 , for some regular , right self - injective ring @xmath17 . observe again that @xmath3151 is a complete , meet - continuous , complemented , modular lattice . for notation , terminology , and standard results on the topics of this section , we refer to @xcite . throughout the section , let @xmath17 denote a ( von neumann ) regular ( unital ) ring ; after some preliminary results , we shall assume that @xmath17 is also right self - injective , that is , @xmath17 is injective as a right module over itself . let @xmath3151 denote the collection of principal right ideals of @xmath17 . regularity implies that @xmath3151 is a complemented modular lattice , in which finite suprema and infima are given by sums and intersections , respectively ( e.g. , ( * ? ? ? * theorem 2.3 ) ) . define orthogonality in @xmath3151 to mean lattice disjointness : @xmath20 if and only if @xmath3162 . for an equivalence relation on @xmath3151 , we shall use @xmath94 , that is , isomorphism of right @xmath17-modules . a small amount of category - theoretical notation will be helpful in dealing with @xmath17-modules . we write @xmath3163 for the category of all right @xmath17-modules , and @xmath3164 for the full subcategory of @xmath3163 whose objects are all direct summands of finite direct sums of copies of a given module @xmath18 . in particular , the objects of @xmath3165 are precisely the finitely generated projective right @xmath17-modules . an expression such as `` @xmath3166 '' will abbreviate the assertion that @xmath18 is an object in the category @xmath3163 . we write @xmath3167 to stand for an arbitrary injective hull of a module @xmath18 , and if @xmath1088 is a cardinal , @xmath3168 stands for a direct sum of @xmath1088 copies of @xmath18 . for @xmath3169 , write @xmath3170 to mean that @xmath18 is isomorphic to a submodule of @xmath14 . if @xmath3171 and @xmath3170 , then regularity of @xmath17 implies that @xmath18 is isomorphic to a direct summand of @xmath14 ( * ? ? ? * theorem 1.11 ) . we use the notation @xmath3152 for the monoid of isomorphism classes of objects from @xmath3165 ( in which the addition operation is induced from the direct sum operation on modules ) . to match our notation for dimension ranges , we shall denote elements of @xmath3152 in the form @xmath3172 , rather than using a more common notation like @xmath3173 $ ] . this involves a slight but unproblematic abuse of notation in case @xmath3174 , since the element @xmath3175 stands for the isomorphism class of @xmath18 within the class of all right @xmath17-modules , whereas once we have made @xmath3151 into an espalier , the notation @xmath3172 will also be used for the image of @xmath18 in @xmath3176 , and in the latter case @xmath3172 stands for the isomorphism class of @xmath18 within @xmath3151 . [ l : v(r ) ] for any regular ring @xmath17 , the monoid @xmath3152 is a refinement monoid , the interval @xmath3177\subseteq v(r)$ ] is a partial refinement monoid , and @xmath3152 is the universal monoid of @xmath3177 $ ] . refinement in @xmath3152 is given by ( * ? ? ? * theorem 2.8 ) , and it is clear that @xmath3178 $ ] is a partial submonoid of @xmath3152 , hence a partial refinement monoid in its own right . since every object in @xmath3165 is isomorphic to a finite direct sum of principal right ideals of @xmath17 ( * ? ? ? * proposition 2.6 ) , every element of @xmath3152 is a sum of elements of @xmath13 . let @xmath1090 denote the universal monoid of @xmath13 , with canonical map @xmath3179 . there exists a unique homomorphism @xmath3180 such that @xmath3181 is the inclusion map . given @xmath3182 , write @xmath3183 for some elements @xmath3184 , and set @xmath3185 ; this is well defined by refinement . hence , we obtain a homomorphism @xmath3186 . obviously @xmath3187 is the identity map on @xmath3152 , and @xmath3188 , whence @xmath3189 is the identity map on @xmath1090 . therefore @xmath3190 is an isomorphism . we next determine the projections on @xmath3152 and on @xmath3177 $ ] . this requires working with orthogonality in @xmath3152 ( as defined in section [ s : directdecomprefmon ] ) , which is determined as follows ( * ? ? ? * proposition 2.21 ) : for any @xmath3191 , we have @xmath3192 let @xmath3193 denote the set of all central idempotents in @xmath17 ; this is a boolean algebra whose operations are given by the rules @xmath3194 @xcite . if @xmath17 is right self - injective , @xmath3193 is complete ( * ? ? ? * proposition 9.9 ) . \(i ) it is clear that there is an endomorphism @xmath3196 of @xmath3152 such that @xmath3197 for all @xmath3204 . moreover , @xmath3205 , and @xmath3206 for all @xmath3207 , so that @xmath3208 . therefore @xmath3209 . similarly , @xmath3210 , and we observe that @xmath3211 . therefore @xmath3212 . \(ii ) let @xmath611 , @xmath3213 . if @xmath3214 , then @xmath3215 , whence @xmath3216 and so @xmath3217 . conversely , if @xmath3217 , then @xmath3218 for all @xmath3182 ( lemma [ l : pleqortq](i ) ) , whence @xmath3219 ( taking @xmath3220 ) . consequently , @xmath3221 , and so @xmath3214 . this shows that the map @xmath3222 is an order - embedding of @xmath3193 into @xmath3223 . given @xmath3224 , we have @xmath3225 , and so @xmath3226 for some right ideals @xmath56 , @xmath277 such that @xmath3227 and @xmath3228 . there is an idempotent @xmath3229 such that @xmath3230 and @xmath3231 . moreover , @xmath3232 , and thus @xmath3233 . this homomorphism group being isomorphic to @xmath3234 , we conclude that @xmath3195 ( * ? ? ? * lemma 3.1 ) . in particular , we can now write @xmath3235 and @xmath3236 . any @xmath3237 is isomorphic to a direct summand of @xmath3238 for some positive integer @xmath176 , whence @xmath3239 . on the other hand , @xmath3240 is isomorphic to a direct summand of @xmath3241 , and so @xmath3242 . since @xmath3243 and @xmath3244 are ideals of @xmath3152 , it follows that they are equal , and therefore @xmath3245 . recall that we have defined orthogonality in @xmath3151 by the rule @xmath3246 . when this occurs , the right ideal @xmath3247 is both the orthogonal sum of @xmath18 and @xmath14 within @xmath3151 and the module - theoretic direct sum of @xmath18 and @xmath14 , so that the two uses of the expression @xmath3248 coincide . however , infinite orthogonal sums in @xmath3151 ( when they exist ) can not be module - theoretic direct sums , since the direct sum of an infinite family of nonzero modules is not finitely generated . to distinguish these cases , let us write @xmath3249 for the orthogonal sum of a family @xmath3250 of elements of @xmath3151 and @xmath3251 for the module - theoretic direct sum . ( for either to exist , the family @xmath3250 must be independent . ) [ p : l(r ) ] let @xmath3151 be the lattice of principal right ideals of a regular , right self - injective ring @xmath17 . then @xmath3161 is an espalier , and its dimension range is isomorphic to the interval @xmath3177 \subseteq v(r)$ ] . consequently , both @xmath3177 $ ] and @xmath3152 are continuous dimension scales . in case @xmath17 is purely infinite , @xmath3252 . by ( * * corollary 13.5 ) , @xmath3151 is complete and upper continuous (= meet - continuous ) . in particular , axiom ( l1 ) holds . as shown in the proof of ( * ? ? ? * proposition 13.3 ) , arbitrary infima in @xmath3151 are given by intersections , while the supremum of a family @xmath3250 of elements of @xmath3151 is the unique principal right ideal of @xmath17 which contains @xmath3253 as an essential submodule . since @xmath17 is right self - injective , @xmath3254 . hence , if @xmath3250 is an orthogonal family , @xmath3255 . axioms ( l2 ) , ( l3 ) , ( l5 ) , and ( l8 ) are clear , ( l2)(iv ) and ( l8 ) being standard properties of submodules of arbitrary modules . axioms ( l4 ) , ( l6 ) , and ( l7 ) are basic properties of injective hulls . therefore @xmath3151 is an espalier . it is clear that @xmath3256 $ ] . if @xmath17 is purely infinite , then @xmath3257 for all positive integers @xmath176 ( * ? ? ? * theorem 10.16 ) , in which case @xmath3177= v(r)$ ] . for the remainder of the section , assume that @xmath17 is right self - injective . before applying theorem [ t : embdimint ] , we show that the type decomposition of @xmath17 ( see ( * ? ? ? * chapter vii ) or ( * ? ? ? * chapter 10 ) ) matches the type decomposition of @xmath3152 ( definition [ d : si , ii , iii ] ) . here it is natural to work with type decompositions of modules from @xmath3165 , as in ( * ? ? ? * theorem 7.2 ) and ( * ? ? ? * theorem 10.31 ) . @xmath18 is an abelian , directly finite , or purely infinite module , respectively , if and only if @xmath3172 is a multiple - free , directly finite , or purely infinite element , respectively , of @xmath3152 . \(i ) the equivalence for directly finite modules is clear from the definitions , and the other two equivalences follow from ( * ? ? ? * theorems 2.1 , 6.2 ) . ( ii ) first , @xmath3261 if and only if @xmath3262 for all directly finite @xmath3263 , if and only if @xmath18 has no nonzero directly finite direct summands , if and only if @xmath18 is of type iii @xcite . similarly , @xmath3264 if and only if @xmath3262 for all abelian @xmath3265 , if and only if @xmath18 has no nonzero abelian direct summands . consequently , @xmath3266 if and only if every nonzero direct summand of @xmath18 contains a nonzero abelian direct summand , while @xmath3267 if and only if @xmath18 has no nonzero abelian direct summands , but every nonzero direct summand of @xmath18 contains a nonzero directly finite direct summand . thus , the remainder of part ( ii ) follows from theorems 5.1 and 5.5 of @xcite . [ t : isomv(r ) ] let @xmath17 be a regular , right self - injective ring , and write @xmath3268 where @xmath3269 is of type @xmath3270 . let @xmath3271 be the ultrafilter space of @xmath3272 . then there exists an ordinal @xmath40 such that @xmath3152 is isomorphic to a lower subset of @xmath3273 . since @xmath3274 , it follows from corollary [ c : typev(r ) ] that each @xmath3275 . by proposition [ p : l(r ) ] , @xmath3152 is a continuous dimension scale , and using lemma [ l : b(r)](ii ) we see that each @xmath3271 is homeomorphic to the ultrafilter space of @xmath3276 . therefore the theorem follows from theorem [ t : embdimint ] . we now turn our attention to nonsingular injective modules , which allows us to extend the above results to proper continuous dimension scales , and which will allow us to show that the espaliers of the form @xmath3151 form a d - universal class . let @xmath1880 denote the full subcategory of @xmath3163 whose objects are all nonsingular injective right @xmath17-modules . note that if @xmath3277 , then @xmath3278 . let @xmath3279 be the monoid of isomorphism classes of objects in @xmath3164 , where as above we use @xmath3280 to denote the isomorphism class of an object @xmath14 . note that for @xmath14 , @xmath3281 , we have @xmath3282 if and only if @xmath3283 . for @xmath3284 , let @xmath3285 denote the collection of those submodules of @xmath18 which are direct summands . then @xmath3285 is a complete , complemented , modular lattice , with infima and suprema given just as in @xmath3151 ( * ? ? ? * propositions 1.3 , 1.6 ) . we define @xmath2 in @xmath3285 as in @xmath3151 . [ l : endinject ] let @xmath18 be a nonsingular injective right @xmath17-module , and set @xmath3286 . then @xmath241 is a regular , right self - injective ring , and @xmath3287 . consequently , @xmath3279 is a dimension interval . for the first statement , see , for example , ( * ? ? ? * corollary 1.23 ) . it is well known that @xmath3164 is equivalent to the category of finitely generated projective right @xmath241-modules ( e.g. , ( * ? ? ? * theorem 18.59 ) ) , and thus @xmath3291 . therefore , proposition [ p : l(r ) ] implies that @xmath3279 is a continuous dimension scale . according to ( * ? ? ? * proposition 1.8 ) , there is a lattice isomorphism @xmath3292 given by the rule @xmath3293 . any pair of right ideals of @xmath241 is given by @xmath3294 , @xmath3295 for some pair @xmath611 , @xmath323 of idempotents in @xmath241 , and it is well known that @xmath3296 if and only if @xmath3297 ( cf . the proof of ( * ? ? ? * proposition 2.4 ) ) . hence , @xmath3298 is an isomorphism of espaliers . it is clear that @xmath3299 $ ] . a major advantage of working with nonsingular injective modules is that any set of such modules can be combined to form a new one , by taking the injective hull of the direct sum . consequently , we can pass from the category @xmath1880 to a proper continuous dimension scale which contains `` arbitrarily large '' elements . thus , let @xmath3300 denote the ( proper ) monoid consisting of all isomorphism classes @xmath3172 of objects @xmath3277 , with addition induced by direct sum . ( to help keep set - theoretic difficulties at bay , one might wish to pass from @xmath1880 to an equivalent skeletal category a category in which isomorphic objects are equal before forming this monoid . ) for any @xmath3284 , the ideal of @xmath3300 generated by @xmath3172 equals the monoid @xmath3279 ; in particular , this ideal is a set . if @xmath13 is a lower subset of @xmath3300 , then @xmath3301 for some set @xmath3302 of objects from @xmath1880 . form @xmath3303 , and observe that @xmath3304 . by lemmas [ l : endinject ] and [ l : segdi ] , @xmath3305 and @xmath13 are continuous dimension scales . for any element @xmath3306 , the class @xmath185 $ ] is contained in the set @xmath3279 and so it is a set . thus , axiom @xmath1886 is satisfied in @xmath3300 . since every object in @xmath3165 is injective ( being a direct summand of some injective module @xmath3238 ) , we have @xmath3307 and @xmath3308 . it is then clear that @xmath3152 is a lower subset of @xmath3300 . given any nonzero object @xmath3277 , choose a nonzero element @xmath32 . by ( * ? ? ? * theorem 9.2 ) , the cyclic module @xmath3309 is both projective and injective . on the one hand , this means that @xmath3310 and @xmath3311 , while on the other , @xmath3312 . thus , @xmath3152 is dense in @xmath3300 , and therefore @xmath3300 satisfies axiom @xmath1887 . as observed in the proof of theorem [ t : isomv(r ) ] , @xmath3271 is homeomorphic to the ultrafilter space of @xmath3276 for @xmath3315 , @xmath1570 , @xmath1571 . let @xmath201 be a finitary unit of @xmath3316 , and observe that since @xmath3152 is dense in @xmath3300 , the set @xmath201 is dense in @xmath3317 . thus , @xmath201 is a finitary unit of @xmath3300 . by theorem [ t : genembdi ] and its proof , there is a lower embedding @xmath3318 ( unique with respect to our choice of @xmath201 ) such that whenever @xmath3284 and @xmath3319 , the restriction of @xmath2737 to @xmath3279 matches the embedding given in theorem [ t : embdimint ] . to see that every function in @xmath3320 lies in the image of @xmath2737 , it suffices to show that for any infinite cardinal @xmath3321 , the constant function @xmath3322 with @xmath3323 for all @xmath3324 lies in the image of @xmath2737 . set @xmath3325 , let @xmath2692 be the cardinality of @xmath14 , and set @xmath3326 . in particular , @xmath14 contains no direct sums of more than @xmath2692 nonzero submodules , and @xmath3327 . we may assume that @xmath14 is an actual submodule of @xmath18 . by lemma [ l : segdimint ] , restriction from @xmath3279 to @xmath3152 provides an isomorphism @xmath3328 . according to lemma [ l : endinject ] , @xmath3285 is an espalier whose dimension range is isomorphic to @xmath3279 ( we have @xmath3329= v(a)$ ] because @xmath18 is purely infinite ) . now @xmath18 and @xmath14 are purely infinite elements of @xmath3285 with central cover @xmath2772 , and @xmath14 is not equal to any orthogonal sum of more than @xmath2692 nonzero elements . the module - theoretic statement @xmath3330 , when written in the symbolism of espaliers , says that @xmath3331 . thus , proposition [ p : existconstant ] implies that there exists a purely infinite element @xmath3332 such that @xmath3333 equals the constant function with value @xmath2725 . therefore we have @xmath3334 with @xmath3335 , which completes the proof of the theorem . the following corollary is an immediate consequence of theorem [ t : vnsir ] , in view of the fact that @xmath3279 is a lower subset of @xmath3300 for any @xmath3284 . if the reader wishes to avoid proper continuous dimension scales , this result can be proved directly , using the same methods employed in the theorem . [ c : vnsir ] let @xmath17 be a regular , right self - injective ring , and write @xmath3313 , where @xmath3269 is of type @xmath3270 . let @xmath3271 be the ultrafilter space of @xmath3272 . given any ordinal @xmath40 , there exists a nonsingular injective right @xmath17-module @xmath18 such that @xmath3336 to show that every continuous dimension scale appears as a lower subset of some @xmath3279 , it only remains to construct regular , right self - injective rings of types i , ii , iii having arbitrary complete boolean algebras as their boolean algebras of central idempotents . we shall make use of the concept of a _ maximal quotient ring _ ( see , for example , ( * ? ? ? * chapter 2 ) , @xcite ) in part of the process . the quickest way to obtain a type i example is to take @xmath3337 to be @xmath14 itself , made into a ring in the canonical way . then @xmath3337 is a commutative , regular , self - injective ring in which all elements are idempotent , and @xmath3340 . for later use , we note that since @xmath3337 is commutative , @xmath3341 . the self - injectivity of @xmath3337 implies that @xmath3337 is a continuous regular ring , thus yielding von neumann s well - known result that @xmath14 is continuous ( see ( * ? ? ? * lemma ii.4.10 ) ) . since @xmath3337 is commutative , it is abelian , and hence is of type i. as a ring , @xmath14 has characteristic @xmath3342 , while the reader may prefer examples having characteristic 0 . we can construct examples which are algebras over any field @xmath3343 , as follows . let @xmath78 be the ultrafilter space of @xmath14 , so that @xmath14 is isomorphic to the boolean algebra of clopen subsets of @xmath78 . let @xmath13 be the ring of all locally constant functions from @xmath78 to @xmath3343 ( that is , functions @xmath3344 such that each point of @xmath78 has a neighborhood on which @xmath323 is constant ) . observe that @xmath13 is a commutative regular ring , with @xmath3345 . as noted above , @xmath14 is a continuous lattice ; thus @xmath13 is a continuous regular ring . finally , let @xmath3337 be the maximal ( right ) quotient ring of @xmath13 . since @xmath13 is regular , it is a nonsingular ring , and so @xmath3337 is regular and right self - injective ( ( * ? ? ? * corollary 2.31 ) , ( * ? ? ? * theorem 13.36 ) ) . moreover , since @xmath13 is commutative , so is @xmath3337 ( see ( * ? ? ? * lemma 14.15 ) ) . therefore @xmath3337 is of type i. by ( * ? ? ? * theorem 13.13 ) , all the idempotents of @xmath3337 lie in @xmath13 ( this is not hard to prove directly in the present case ) . therefore @xmath3346 . similar methods , worked out by busqu @xcite , can be applied in the type iii case . first choose a commutative , regular , self - injective ring @xmath3337 with @xmath3340 . by ( * theorem 2.5 ) , there exists a regular , right self - injective ring @xmath3338 of type iii whose center is isomorphic to @xmath3337 . therefore @xmath3347 . it appears that the constructions used in proposition [ p : existb(r)i , iii ] do not always produce rings of type ii . we approach the type ii existence problem lattice - theoretically , _ via _ von neumann s _ coordinatization theorem _ ( e.g. , ( * ? ? ? * theorem 14.1 ) , ( * ? ? ? * chapter xi , satz 3.2 ) ) . let @xmath3 be an irreducible ( i.e. , indecomposable ) continuous geometry such that the ( unique ) dimension function @xmath2874 on @xmath3 is positive on all nonzero elements of @xmath3 and the range of @xmath2874 is the unit interval @xmath2943 $ ] . such continuous geometries were constructed by von neumann @xcite . alternatively , one could choose a simple , regular , right self - injective ring @xmath13 of type @xmath3348 ( see ( * ? ? ? * corollary 11.10 ) and ( * ? ? ? * example 10.7 , theorem 10.27 ) for existence ) and take @xmath3350 . indecomposability of @xmath3 then follows from indecomposability of @xmath13 , and the properties of @xmath2874 follow from those of the unique rank function @xmath115 on @xmath13 ( see ( * ? ? ? * corollary 16.15 ) ) , since @xmath2874 is given by the formula @xmath3351 for @xmath3352 . next , let @xmath3353 be the ( reducible ) continuous geometry constructed from @xmath3 and @xmath14 by halperin in ( * ? ? ? * theorem 1 ) . the center of @xmath3353 ( i.e. , the sublattice of neutral elements ) is isomorphic to @xmath14 by ( * ? ? ? * theorem 2 ) , and @xmath3353 contains a sublattice ( with the same largest element ) isomorphic to @xmath3 ( * ? ? ? * remark 2 , p. 351 ) . for any positive integer @xmath176 , the largest element @xmath3354 can be written as the supremum of @xmath176 independent pairwise perspective elements ( e.g. , because there exist elements @xmath132 with @xmath3355 ) , and so the same occurs in @xmath3353 . consequently , @xmath3353 has order @xmath176 ( in von neumann s sense ) for all @xmath176 . in particular , since @xmath3353 has order 4 , von neumann s coordinatization theorem implies that there exists a regular ring @xmath17 such that @xmath3356 . since @xmath3151 is thus a continuous lattice , @xmath17 is a continuous regular ring . now @xmath17 is unit - regular ( * ? ? ? * corollary 13.23 ) , and hence perspectivity in @xmath3151 is given by module isomorphism ( * ? ? ? * corollary 4.23 ) . consequently , for each positive integer @xmath176 , the module @xmath17 is a direct sum of @xmath176 pairwise isomorphic right ideals . in particular , there are no nonzero central idempotents @xmath3229 such that the ring @xmath3357 is abelian , and therefore @xmath17 is right and left self - injective ( * ? ? ? * corollary 13.18 ) . since @xmath17 is unit - regular , it is directly finite ( * ? ? ? * proposition 5.2 ) . hence , ( * ? ? ? * theorems 10.13 , 10.24 ) show that @xmath3358 where each @xmath3359 is of type i@xmath3360 and @xmath3361 is of type @xmath3348 . since the dimension theory of @xmath3362 takes values in @xmath3363 , the module @xmath3359 can not be a direct sum of @xmath3364 nonzero pairwise isomorphic right ideals ( cf . * theorem 10.10 ) or ( * ? ? ? * corollary 11.18 ) ) . thus , all @xmath3365 , and @xmath3366 is of type @xmath3348 . let @xmath13 be an arbitrary continuous dimension scale . by propositions [ p : existb(r)i , iii ] and [ p : existb(r)ii ] , together with theorems [ t : embdimint ] and [ t : isomv(r ) ] , there exists a regular , right self - injective ring @xmath3367 such that @xmath13 is isomorphic to a lower subset @xmath3368 of @xmath3369 . set @xmath3370 and @xmath3371 . by lemma [ l : endinject ] , @xmath17 is a regular , right self - injective ring and @xmath3372 $ ] . since all finitely generated projective right @xmath3367-modules are isomorphic to direct summands of @xmath18 , we see that @xmath3369 is a lower subset of @xmath3329 $ ] . thus , @xmath3368 is isomorphic to a lower subset of the dimension range of the espalier @xmath3151 . [ c : determinev(r ) ] let @xmath93 be a commutative monoid . then @xmath3373 for some regular , right self - injective ring @xmath17 if and only if @xmath93 is a continuous dimension scale containing an order - unit . if @xmath17 is a regular , right self - injective ring , then @xmath3374 is an order - unit in @xmath3152 , and @xmath3152 is a continuous dimension scale by proposition [ p : l(r ) ] . conversely , let @xmath93 be a continuous dimension scale which contains an order - unit @xmath621 . by theorem [ t : l(r)duniv ] and proposition [ p : l(r ) ] , there exists a regular , right self - injective ring @xmath3367 such that @xmath93 is isomorphic to a lower subset @xmath3375 of @xmath3369 . let @xmath2077 denote the image of @xmath621 under this isomorphism ; then @xmath3375 equals the ideal of @xmath3369 generated by @xmath2077 . now @xmath3376 for some @xmath3377 , and it is clear that @xmath3378 . by lemma [ l : endinject ] , @xmath3379 is a regular , right self - injective ring and @xmath3380 . our main references for w*-algebras will be the texts by j. dixmier @xcite , r.v . kadison and j.r . ringrose @xcite , b .- li @xcite , and s. sakai @xcite ; for aw*-algebras , we rely on the text by s.k . berberian @xcite and the monograph by i. kaplansky @xcite . a _ w*-algebra _ ( also called a _ von neumann algebra _ ) can be defined as any c*-algebra which is isomorphic ( _ qua _ c*-algebra ) to a * -subalgebra of @xmath3381 ( the algebra of all bounded linear operators ) on some ( complex ) hilbert space @xmath967 which is closed in the strong operator topology ( the topology of pointwise convergence ) . kaplansky introduced the concept of an _ aw*-algebra _ ( abbreviating `` abstract w*-algebra '' ) in order to obtain a more general class of c*-algebras defined ( and analyzed ) by purely algebraic properties . before giving the definition , we recall a few basic concepts . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ all w*- and aw*-algebras that we consider here will be assumed to be _ unital_. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ a _ projection _ in a c*-algebra @xmath18 is any self - adjoint idempotent , that is , any element @xmath3382 with @xmath146 . the _ right annihilator _ of a subset @xmath13 of @xmath18 is the right ideal @xmath3383 finally , @xmath18 is said to be an _ aw*-algebra _ if the right annihilator of any subset @xmath13 of @xmath18 is a principal right ideal generated by a projection , that is , @xmath3384 for some ( necessarily unique ) projection @xmath3382 . every w*-algebra is aw * ( * ? ? ? * , proposition 9 ) , but not conversely . for example , the ( unital ) commutative aw*-algebras are precisely ( up to isomorphism ) the algebras @xmath3385 of continuous complex - valued functions on complete boolean spaces @xmath78 ( * ? ? ? * , theorem 1 ) ; such an algebra is w * if and only if @xmath78 is _ hyperstonian _ ( ( * ? ? ? * thorme 2 ) ; cf . * theorems 5.3.3 , 5.3.4 ) ) . by definition , @xmath78 is hyperstonian ( cf . * dfinition 3 ) ) , if for any nonempty open subset @xmath1090 of @xmath78 , there exists a radon measure @xmath90 on @xmath78 , vanishing on all nowhere dense subsets of @xmath78 , such that @xmath3386 . the set @xmath3 of projections of an aw*-algebra @xmath18 is equipped with the partial ordering @xmath149 defined by @xmath482 if and only if @xmath3387 ( equivalently , @xmath3388 ) , for @xmath26 , @xmath27 . the poset @xmath1910 is a complete lattice ( * ? ? ? * , proposition 1 ) . furthermore , two projections @xmath26 , @xmath27 are _ orthogonal _ , in symbols @xmath28 , if @xmath29 ( equivalently , @xmath3389 ) . then the sum @xmath3390 is also a projection , and it is the join of @xmath480 in @xmath3 : hence @xmath3391 . finally , two projections @xmath26 and @xmath31 of @xmath18 are _ murray - von neumann equivalent _ , in symbols @xmath30 , if there exists @xmath32 such that @xmath3392 and @xmath3393 . equivalently , @xmath145 and @xmath3394 are isomorphic as right @xmath18-modules , that is , there are @xmath5 , @xmath246 such that @xmath3395 and @xmath3396this equivalence is nontrivial and contained in ( * ? ? ? * theorem 27 ) . a projection @xmath3382 is said to be _ @xmath272-finite _ ( or _ countably decomposable _ , or _ orthoseparable _ ) if @xmath26 does not majorize any uncountable orthogonal family of nonzero projections ; if the projection @xmath3397 has this property , then the algebra @xmath18 itself is called @xmath272-finite . this same terminology is also used for boolean algebras and their elements . let us say that a boolean algebra @xmath14 is _ locally @xmath272-finite _ provided every element of @xmath14 is a supremum of @xmath272-finite elements . furthermore , a boolean space @xmath78 is locally @xmath272-finite , if its boolean algebra of clopen subsets is locally @xmath272-finite . let @xmath14 denote the ultrafilter space of a hyperstonian boolean space @xmath78 . for a radon measure @xmath90 on @xmath78 , we say that a borel subset @xmath18 of @xmath78 is _ @xmath90-self - supporting _ , if @xmath3398 whenever @xmath3399 is open and @xmath3400 . then every borel subset @xmath18 of @xmath78 of positive measure contains a @xmath90-self - supporting compact subset @xmath216 of positive measure , see @xcite . if @xmath90 vanishes on all nowhere dense subsets , then @xmath3401 , hence @xmath3402 is a @xmath90-self - supporting open subset of @xmath216 with positive measure . as @xmath90 is a radon measure , @xmath216 contains a @xmath90-self - supporting _ clopen _ subset with positive measure . let @xmath2874 denote the set of elements of @xmath14 whose associated clopen set is @xmath90-self - supporting with respect to some finite radon measure @xmath90 on @xmath78 vanishing on all nowhere dense subsets . it follows from the assumption that @xmath78 is hyperstonian and the paragraph above that every element of @xmath14 is a supremum of elements of @xmath2874 . but every element of @xmath2874 is clearly @xmath272-finite . [ ex : sfnonhs ] as in section [ s : ameasth ] , we denote by @xmath3027 the boolean algebra of all borel subsets of the cantor space @xmath3028 modulo meager sets . let @xmath78 denote the ultrafilter space of @xmath3027 . then @xmath78 is clearly @xmath272-finite . however , there is no nontrivial radon measure on @xmath78 , as shown , for example , by the argument on pages 8283 in ( * ? ? ? * chapter 21 ) . in particular , @xmath78 is not hyperstonian . axiom ( l4 ) : for @xmath26 , @xmath3405 , the element @xmath3406 is also a projection , and @xmath3407 is equivalent to @xmath3408 . now let @xmath3409 be an orthogonal family of elements of @xmath3 such that @xmath3410 , for all finite @xmath1923 . this means that @xmath3411 , for all finite @xmath1923 , thus @xmath3412 , that is , @xmath3413 . the `` projections '' of the espalier @xmath3 are not the projections of @xmath18 ( which are the elements of @xmath3 ) , but they correspond to the _ central _ projections of @xmath18 . in fact , all central idempotents of @xmath18 are projections ( * ? ? ? * , exercise 1 ) , and so we may use without ambiguity the notation @xmath3414 of section [ s : rsireg ] to stand for the boolean algebra of central projections in @xmath18 . [ l : projoflattproj ] let @xmath3 be the lattice of projections of an aw*-algebra @xmath18 . for each @xmath3415 , there is a projection @xmath3416 such that @xmath3417 for all @xmath3418 . the rule @xmath3419 defines an isomorphism of @xmath3414 onto @xmath3420 . set @xmath3421 . it is clear that for each @xmath3422 , there is a projection @xmath3423 as described , and @xmath3424 . it is also clear that @xmath3214 implies @xmath3425 , for @xmath3426 . on the other hand , if @xmath3427 , the projection @xmath3428 is nonzero . note that @xmath3429 , whence @xmath3430 . since @xmath3431 , it follows that @xmath3432 . thus , the map @xmath3433 given by @xmath3419 is an order - embedding that respects complements . it only remains to show that this map is surjective . given @xmath3434 , we have @xmath3435 , and so there exist orthogonal projections @xmath3436 such that @xmath3437 while @xmath3438 and @xmath3439 . further , @xmath3440 , and so there is no nonzero projection @xmath3418 such that @xmath3441 . we next show that @xmath611 and @xmath323 are central projections . since @xmath3442 , it is enough to show that @xmath3443 . let @xmath5 be an arbitrary element of @xmath3444 , and let @xmath3445 and @xmath3446 be the _ right _ and _ left projections _ of @xmath5 , respectively ( ( * ? ? ? * , definition 4 ) , @xcite ) , that is , the unique projections such that @xmath3447 and @xmath3448 generate , respectively , the right and left annihilators of @xmath5 . since @xmath3449 , we find that @xmath3450 and @xmath3451 , that is , @xmath3452 and @xmath3453 . however , @xmath3454 ( see ( * ? ? ? * , theorem 3 ) or ( * ? ? ? * theorem 63 ) ) , whence @xmath3455 and so @xmath3456 . thus @xmath3457 , proving that @xmath3443 , as desired . consequently , @xmath611 and @xmath323 are central , as claimed . now for any @xmath3405 , we have @xmath3458 with @xmath3459 and @xmath3460 , whence @xmath3461 and @xmath3462 . hence , we conclude that @xmath3463 . therefore @xmath3464 , completing the proof . in the context of proposition [ p : aw*esp ] , let us denote by @xmath154 $ ] the @xmath4-equivalence class of a projection @xmath26 of @xmath18 . the _ addition _ of these equivalence classes is defined by @xmath3465+[q]=[p\oplus q]=[p+q ] , \text { for any orthogonal projections } p\text { and } q.\ ] ] the dimension range of @xmath3 is , of course , @xmath2622 , equipped with the above partial addition . just as in section [ s : rsireg ] , we can define the monoid @xmath3279 of isomorphism classes of finitely generated projective right @xmath18-modules . as noted above , projections @xmath26 , @xmath3466 satisfy @xmath30 if and only if @xmath3467 , and so we obtain an embedding of partial monoids , @xmath3468 , where @xmath154\mapsto \delta(pa)$ ] . under this embedding , @xmath3469\mapsto \delta(a)$ ] . any direct summand of the right module @xmath18 has the form @xmath3470 for an idempotent @xmath611 , and since @xmath611 is equivalent to a projection @xmath3382 ( * ? ? ? * theorem 26 ) , we have @xmath3471 and so @xmath154\mapsto \delta(ea)$ ] . similarly , any pair of orthogonal idempotents in @xmath18 is equivalent to a pair of orthogonal projections , so that any pair of elements @xmath621 , @xmath3472 such that @xmath3473 must be the image of a pair of elements of @xmath2628 whose sum is defined . thus , the embedding above maps @xmath2628 isomorphically onto the interval @xmath3329\subseteq v(a)$ ] , which we record in the theorem below . [ t : dimaw * ] let @xmath3 be the lattice of projections of an aw*-algebra @xmath18 . then the dimension range @xmath2622 is a continuous dimension scale , and @xmath3474\subseteq v(a)$ ] . if @xmath18 is a w*-algebra , then the ultrafilter space of @xmath3420 is hyperstonian . that @xmath2628 is a continuous dimension scale follows from theorem [ t : dimesp ] . we have just seen above that @xmath3475 $ ] . observe that @xmath3476 , where @xmath3477 is the center of @xmath18 . if @xmath18 is a w*-algebra , then so is @xmath3477 , whence @xmath3478 for some hyperstonian complete boolean space @xmath78 . in particular , the ultrafilter space of @xmath3420 is homeomorphic to @xmath78 and thus it is hyperstonian . in particular , when @xmath3 is the lattice of projections of an aw*-algebra @xmath18 , it follows from theorem [ t : embdimint ] that the partial commutative monoid@xmath42 embeds as a lower subset into a commutative monoid of the form @xmath165 for complete boolean spaces @xmath37 , @xmath38 , @xmath39 . theorem [ t : dimaw * ] implies that these spaces must be hyperstonian in case @xmath18 is a w*-algebra . there exists a type i , ii , iii decomposition for aw*-algebras ( see @xcite ) which parallels that for regular , right self - injective rings ; in fact , kaplansky developed much of the type i , ii , iii theory for _ baer rings _ ( rings in which the right or left annihilator of any element is generated by an idempotent ) , a class of rings which includes both aw*-algebras and regular , right self - injective rings . we shall use some of the terminology and results of this theory without explicit references . we point out that an aw*-algebra @xmath18 is called a _ factor _ provided the center of @xmath18 equals the complex field @xmath3479 ; equivalently , @xmath18 is a factor if and only if @xmath18 is nonzero and @xmath3480 . [ l : bigw*factors ] let @xmath40 be an ordinal and @xmath3481 . there exists a w*-factor @xmath3482 of type @xmath3270 which contains a family @xmath3483 of nonzero purely infinite projections such that @xmath3484 but @xmath3485 for all ordinals @xmath3486 . choose a hilbert space @xmath3487 with an orthonormal basis of cardinality @xmath2939 , and set @xmath3488 . for each ordinal @xmath1410 , choose a projection @xmath3489 such that the closed subspace @xmath3490 of @xmath3487 has an orthonormal basis of cardinality @xmath179 . the desired properties of @xmath3491 and the @xmath3492 are clear . next , choose w*-factors @xmath3493 and @xmath3494 of types ii and iii ( e.g. , ( * ? ? ? * part i , 9.4 ) , ( * ? ? ? * chapters 6 , 8) , ( * ? ? ? * chapter 4 ) ) . these factors can be chosen as subalgebras of @xmath3495 for a separable hilbert space @xmath3496 ( e.g. , ( * ? ? ? * remark , p. 155 ) , ( * ? ? ? * theorem 7.3.16 ) ) , so that they are @xmath272-finite ( * ? ? ? * proposition 1.14.3 ) . now let @xmath3497 and @xmath3498 be the w*-tensor products @xmath3499 and @xmath3500 . these algebras are of types ii and iii , respectively ( e.g. , ( * ? ? ? * propositions 11.2.21 , 11.2.26 ) , ( * ? ? ? * proposition 2.6.3 , theorem 2.6.4 ) ) , and they are factors ( * ? ? ? * proposition 2.6.7 ) . now let @xmath3501 or @xmath1571 , and set @xmath3502 for all ordinals @xmath1410 . it is clear that these @xmath3503 are purely infinite projections , and that @xmath3504 for all ordinals @xmath3505 . observe that the w*-algebra @xmath3506 is isomorphic to @xmath3507 , which is in turn isomorphic to a w*-subalgebra of @xmath3508 . since @xmath3509 has an orthonormal basis of cardinality @xmath179 , we thus see that @xmath3503 does not majorize any orthogonal family of more than @xmath179 nonzero projections . on the other hand , @xmath3510 , whence @xmath3511 majorizes an orthogonal family of @xmath3512 nonzero projections ( equivalent to itself ) . therefore @xmath3513 . we can now show that the class of projection lattices of w*-algebras , while not d - universal , is at least d - universal relative to continuous dimension scales for which the ultrafilter space of the boolean algebra of projections is hyperstonian . for @xmath3515 , choose w*-factors @xmath3516 and families @xmath3517 of purely infinite projections as in lemma [ l : bigw*factors ] . let @xmath3518 , which is a w*-algebra because @xmath3271 is hyperstonian , and note that @xmath3519 is isomorphic to the boolean algebra of clopen subsets of @xmath3271 . let @xmath3520 be the w*-tensor product @xmath3521 , which has type j by the results referenced in lemma [ l : bigw*factors ] . since @xmath3482 is a factor , the centers of @xmath3522 and @xmath3520 are isomorphic ( * ? ? ? * proposition 2.6.7 ) , and thus @xmath3523 , _ via _ the map @xmath3524 . consequently , if @xmath3525 is the lattice of projections of @xmath3520 , the ultrafilter space of @xmath3526 is homeomorphic to @xmath3271 . since @xmath3520 is of type j , it follows that @xmath3527 is of type j , that is , @xmath3528 in the notation of definition [ d : si , ii , iii ] . set @xmath3529 for all ordinals @xmath1410 , and observe that the @xmath3530 are purely infinite projections with central cover @xmath2772 , such that @xmath3531 for all ordinals @xmath3505 . for each point @xmath3534 , let @xmath3535 be the w*-algebra homomorphism obtained by tensoring the identity map on @xmath3482 with the evaluation map @xmath3536 from @xmath3522 to @xmath3479 . observe that @xmath3537 and @xmath3538 . moreover , @xmath3539 for some projection @xmath3540 , and @xmath3541 . if @xmath3542 , then @xmath3543 for all @xmath3544 . since @xmath3545 , we must have @xmath3546 for all @xmath3544 , and thus @xmath3547 . this contradiction establishes the claim . we now apply proposition [ p : secondexistconstant ] , and conclude that there exist projections @xmath3548 for each j such that the dimension ranges of the intervals @xmath3549 $ ] have the following form : @xmath3550/{\sim}\ & \cong\ { \mathbf{c}}(\omega_{{\mathrm{i}}},{\mathbb{z}}_\gamma ) \\ [ 0,r_{{\mathrm{ii}}}]/{\sim}\ & \cong\ { \mathbf{c}}(\omega_{{\mathrm{ii}}},{\mathbb{r}}_\gamma ) \\ [ 0,r_{{\mathrm{iii}}}]/{\sim}\ & \cong\ { \mathbf{c}}(\omega_{{\mathrm{iii}}},{\mathbf{2}}_\gamma).\end{aligned}\ ] ] therefore the dimension range of the lattice of projections of the w*-algebra @xmath3551 has the desired form . note that each of the projections @xmath3552 is purely infinite , whence the projection @xmath3397 is purely infinite , and consequently @xmath3329= v(a)$ ] . therefore , in view of theorem [ t : dimaw * ] , the present theorem is proved . [ c : w*hyperduniv ] let @xmath13 be a continuous dimension scale . then @xmath13 admits a lower embedding into the dimension range of the lattice of projections of some w*-algebra if and only if the ultrafilter space of @xmath1743 is hyperstonian . in order to see that the projection lattices of aw*-algebras form a d - universal class of espaliers , we need an analogue of theorem [ t : bigw*drng ] in which @xmath37 , @xmath38 , @xmath39 are arbitrary complete boolean spaces and @xmath18 is an aw*-algebra . however , there is no general theory of aw*-tensor products available to replace the w*-tensor products @xmath3553 used in our proof . p. ara has suggested that one might be able to use the _ monotone complete tensor products _ introduced by m. hamana @xcite instead . ( we thank him for making us aware of hamana s work . ) rather than developing the necessary auxiliary results about monotone complete tensor products here , we complete the picture by taking a different route . namely , we borrow the methods and results of g. takeuti @xcite and some of the subsequent results obtained in m. ozawa @xcite . these methods involve forcing , more specifically , the scott - solovay model @xmath2865of @xmath14-valued set theory ( also used in section [ s : ameasth ] ) , for any complete boolean algebra @xmath14 . we give a short summary of what we shall use from @xcite . if @xmath3554 is an aw*-algebra in @xmath2865 , the _ bounded global section algebra _ @xmath3555 of @xmath3554 is the set of all @xmath3556 such that @xmath3557 and @xmath3558 for some constant @xmath325 , endowed with its canonical structure of aw*-algebra . for example , @xmath3559 if and only if @xmath3560 . the center of @xmath3555 contains a copy of the bounded global section algebra @xmath3561 of the complex numbers . observe that @xmath3561 is isomorphic to the algebra of continuous maps from the ultrafilter space of @xmath14 to @xmath3479 . in case @xmath3554 is an aw*-factor in @xmath2865 , the center of @xmath3555 is exactly ( the canonical copy of ) @xmath3561 , see ( * ? ? ? * theorem 5 ) . in particular , for @xmath3562 , the central idempotent of @xmath3555 corresponding to @xmath621 is the unique element @xmath3563 such that @xmath3564 while @xmath3565 . by lemma [ l : projoflattproj ] , the boolean algebra of projections of @xmath3555 is also isomorphic to @xmath14 . thus , letting @xmath3 be the espalier of projections of @xmath3555 and identifying the elements of @xmath14 with the projections of @xmath2628 , we obtain that @xmath3566 has the same meaning in the present paper and in @xcite . by arguing as in the proof of theorem [ t : bigw*drng ] , it suffices to prove that for every complete boolean algebra @xmath14 , every ordinal @xmath40 , and every @xmath3567 , there exist an aw*-algebra @xmath18 of type j with algebra of central idempotents isomorphic to @xmath14 and a @xmath2937-sequence @xmath3568 of projections of central cover @xmath2772 such that @xmath3569 but @xmath3570 , for @xmath3505 . by applying lemma [ l : bigw*factors ] within @xmath2865 , we obtain a factor @xmath3554 of type @xmath3270 in @xmath2865 and a @xmath14-valued name @xmath3571 such that the following statements hold in @xmath2865 ( that is , they have boolean value @xmath2772 ) : @xmath3572 where @xmath3573 denotes the espalier of projections of @xmath3554 within @xmath2865 . now let @xmath3574 be the bounded global section algebra of @xmath3554 . it follows from ( * ? ? ? * theorem 7 ) ( see also @xcite ) that @xmath18 is type j , furthermore , its algebra of central idempotents is isomorphic to @xmath14 . for all @xmath1410 , let @xmath3575 be the unique @xmath14-valued name such that @xmath3576 . for @xmath3505 , it follows from , that @xmath3577 and @xmath3578 . in the first group , we shall mention the following . for given , `` practical '' examples , where we need to verify that a given structure is an espalier , the axiom ( l7 ) is often a source of problems . thus we may ask to what extent it is possible to remove axiom ( l7 ) from the definition of an espalier , thus defining `` pre - espaliers '' ( see also definition [ d : bpreesp ] ) . but then , in order to extend a pre - espalier @xmath35 to an espalier , we need to define a new binary relation @xmath2795 on @xmath3 by letting @xmath3579 hold , if there are decompositions @xmath2953 and @xmath3580 such that @xmath3581 , for all @xmath62 . however , proving the transitivity of the new relation @xmath2795 leads to the verification of a common refinement property , see lemma [ l : extpreesp ] for the boolean case . this problem can be formulated as follows . [ pb : pre - espref ] let @xmath35 be a structure satisfying all axioms from ( l0 ) to ( l8 ) with the possible exception of ( l7 ) , and let @xmath50 and @xmath3582 be families of elements of @xmath3 such that @xmath3583 . are there families @xmath3584 and @xmath3585 of elements of @xmath3 such that @xmath3586 ( for all @xmath62 ) , @xmath3587 ( for all @xmath279 ) , and @xmath3588 ( for all @xmath2815 ) ? the second group of questions asks for constructing further classes of espaliers , within other areas of mathematics . of course , isomorphism types of various structures are privileged , see , for example , b. jnsson and a. tarski s appendix in @xcite . in another direction , one might ask about extensions of various results of cancellation or unique decomposition , known for finite structures ( see ( * ? ? ? * chapter 5 ) ) to infinite structures subjected to completeness conditions . this would in turn yield , for example , nontrivial cancellation results for further infinite structures , of which the main result of @xcite about _ @xmath272-complete effect algebras _ would be a prototype . expecting infinite generalizations of finite results _ via _ espaliers is reasonable as long as there are enough refinement theorems around , see , again , ( * ? ? ? * chapter 5 ) . hence the lovsz cancellation theorems , see @xcite or ( * ? ? ? * section 5.7 ) , do not enter this category , as they are established by counting arguments , in contexts where refinement does not always hold . we do not know of any framework that could extend lovsz s results to infinite structures subjected to completeness conditions . bernau , _ unique representation of archimedean lattice groups and normal archimedean lattice rings _ , proc . london math . soc . * 15 * ( 1965 ) , 599631 ; _ addendum _ proc . london math . * 16 * ( 1966 ) , 384 . g. grtzer , `` general lattice theory . second edition '' , new appendices by the author with b.a . davey , r. freese , b. ganter , m. greferath , p. jipsen , h.a . priestley , h. rose , e.t . schmidt , s.e . schmidt , f. wehrung , and r. wille . birkhuser verlag , basel , 1998 . xx+663 p. r.n . mckenzie , g.f . mcnulty , and w.f . taylor , `` algebras , lattices , varieties . volume i. '' the wadsworth & brooks / cole mathematics series . monterey , california : wadsworth & brooks / cole advanced books & software , 1987 . xii+361 p. f. maeda , `` kontinuierliche geometrien '' ( german ) , die grundlehren der mathematischen wissenschaften in einzeldarstellungen mit besonderer bercksichtigung der anwendungsgebiete , bd . * 95*. springer - verlag , berlin - gttingen - heidelberg , 1958 . ( translated from japanese by s. crampe , g. pickert , and r. schauffler ) .

we develop dimension theory for a large class of structures of the form @xmath0 , where @xmath1 is a partially ordered set , @xmath2 is a binary relation on @xmath3 , and @xmath4 is an equivalence relation on @xmath3 , subject to certain axioms . we call these structures _ espaliers_. for @xmath5 , @xmath6 , @xmath7 , we say that @xmath8 holds , if @xmath9 and @xmath10 is the supremum of @xmath11 . the _ dimension theory _ of @xmath3 is the universal @xmath4-invariant homomorphism from @xmath12 to a partial commutative monoid@xmath13 . we say that @xmath13 is the _ dimension range _ of @xmath3 . particular examples of espaliers are the following : 1 . let @xmath14 be a complete boolean algebra . for @xmath5 , @xmath15 , we say that @xmath9 if @xmath16 , and we take @xmath4 to be any zero - separating , unrestrictedly additive and refining equivalence relation on @xmath14 ( for instance , equality ) . 2 . let @xmath17 be a right self - injective von neumann regular ring . we denote by @xmath3 the lattice of all direct summands of a given nonsingular injective right @xmath17-module , for instance , the lattice of finitely generated right ideals of @xmath17 . for @xmath18 , @xmath19 , we say that @xmath20 if @xmath21 , and @xmath22 if @xmath23 . 3 . more generally , let @xmath3 be a complete , meet - continuous , complemented , modular lattice . for @xmath5 , @xmath24 , we say that @xmath9 if @xmath16 , and @xmath25 if @xmath5 and @xmath6 are projective by ( finite ) decomposition . 4 . let @xmath18 be an aw*-algebra . we denote by @xmath3 the lattice of projections of @xmath18 , and take the standard orthogonality and equivalence relations on @xmath3 . for @xmath26 , @xmath27 , then , @xmath28 if @xmath29 , and @xmath30 if @xmath26 and @xmath31 are _ murray - von neumann equivalent _ , that is , there exists @xmath32 such that @xmath33 and @xmath34 . we prove that the dimension range of any espalier @xmath35 is a _ lower interval _ of a commutative monoid of the form @xmath36 where @xmath37 , @xmath38 , and @xmath39 are complete boolean spaces , and where we put , for every ordinal @xmath40 , @xmath41 endowed with their interval topology and natural addition operations . conversely , we prove that every lower interval of a monoid of the form ( * ) can be represented as the dimension range of an espalier arising from each of the contexts ( i)(iv ) above . the context of w*-algebras requires the spaces @xmath37 , @xmath38 , and @xmath39 to be _ hyperstonian _ , and no further restriction is needed . this subsumes many earlier dimension - theoretic results , and , in applications , completes theories developed for examples such as ( i)(iv ) above .

self - avoiding walks ( saws ) and polygons ( saps ) on regular lattices are combinatorial problems of tremendous inherent interest as well as serving as simple models of polymers and vesicles @xcite . they are fundamental problems in lattice statistical mechanics . @xmath1-step self - avoiding walk _ @xmath2 is a sequence of _ distinct _ vertices @xmath3 such that each vertex is a nearest neighbour of it predecessor . saws are considered distinct up to translations of the starting point @xmath4 . we shall use the symbol @xmath5 to mean the set of all saws of length @xmath1 . a self - avoiding polygon of length @xmath1 is a @xmath6-step saw such that @xmath4 and @xmath7 are nearest neighbours and a closed loop can be formed by inserting a single additional step . one is interested in the number of saws and saps of length @xmath1 , various metric properties such as the radius of gyration , and for saps one can also ask about the area enclosed by the polygon . in this paper we study the following properties : * the number of @xmath1-step self - avoiding walks @xmath8 ; * the number of @xmath1-step self - avoiding polygons @xmath9 ; * the mean - square end - to - end distance of @xmath1-step saws @xmath10 ; * the mean - square radius of gyration of @xmath1-step saws @xmath11 ; * the mean - square distance of a monomer from the end points of @xmath1-step saws @xmath12 ; * the mean - square radius of gyration of @xmath1-step saps @xmath13 ; and * the @xmath14 moment of the area of @xmath1-step saps @xmath15 . the metric properties for saws are defined by , @xmath16 , \\ { \langle r^2_m \rangle}_n = \frac{1}{c_n } \sum_{\bm{\omega}_n } \left [ \frac{1}{2(n+1 ) } \sum_{i=0}^n \left [ ( \omega_0-\omega_j)^2+(\omega_n-\omega_j)^2 \right ] \right ] , \end{aligned}\ ] ] with a similar definition for the radius of gyration of saps . it is generally believed that the quantities listed above has the asymptotic forms as @xmath17 : @xmath18 , \label{eq : asympsaw } \\ p_n = b \mu^n n^{\alpha-3}[1 + o(1 ) ] , \label{eq : asympsap } \\ { \langle r^2_e \rangle}_n = cn^{2\nu}[1 + o(1 ) ] , \label{eq : asympee } \\ { \langle r^2_g \rangle}_n = dn^{2\nu}[1 + o(1 ) ] , \label{eq : asymprg}\\ { \langle r^2_m \rangle}_n = en^{2\nu}[1 + o(1 ) ] , \label{eq : asympmd } \\ { \langle r^2 \rangle}_n = fn^{2\nu}[1 + o(1 ) ] , \label{eq : asympsaprg } \\ { \langle a^k \rangle}_n = g^{(k)}n^{2\nu k}[1 + o(1 ) ] . \label{eq : asympmom } \end{aligned}\ ] ] the critical exponents are believed to be universal in that they only depend on the dimension of the underlying lattice . the connective constant @xmath19 and the critical amplitudes @xmath20@xmath21 vary from lattice to lattice . in two dimensions the critical exponents @xmath22 , @xmath23 and @xmath24 have been predicted exactly , though non - rigorously , using coulomb - gas arguments @xcite . while the amplitudes are non - universal , there are many universal amplitude combinations . any ratio of the metric saw amplitudes , e.g. @xmath25 and @xmath26 , is expected to be universal @xcite . of particular interest is the linear combination @xcite ( which we shall call the cscps relation ) @xmath27 where @xmath28 and @xmath29 . in two dimensions cardy and saleur @xcite ( as corrected by caracciolo , pelissetto and sokal @xcite ) have predicted , using conformal field theory , that @xmath30 . this conclusion has been confirmed by previous high - precision monte carlo work @xcite as well as by series extrapolations @xcite . privman and redner @xcite proved that the combination @xmath31 is universal , cardy and guttmann @xcite proved that @xmath32 , and cardy and mussardo @xcite proved that @xmath33 is universal and gave the first theoretical estimate of the value @xmath34 . @xmath35 is an integer constant such that @xmath9 is non - zero when @xmath1 is divisible by @xmath35 . so @xmath36 for the triangular lattice and 2 for the square and honeycomb lattices . @xmath37 is the area per lattice site and @xmath38 for the square lattice , @xmath39 for the honeycomb lattice , and @xmath40 for the triangular lattice . richard , guttmann and jensen @xcite conjectured the exact form of the critical scaling function for self - avoiding polygons and consequently showed that the amplitude combinations @xmath41 are universal and predicted their exact values . the exact value for @xmath42 had previously been predicted by cardy @xcite . the validity of this conjecture was recently confirmed numerically to a high degree of accuracy using exact enumeration data for saps on the square , honeycomb , and triangular lattices @xcite . the asymptotic form ( [ eq : asympsaw ] ) only explicitly gives the leading contribution . in general one would expect corrections to scaling so , e.g , @xmath43\ ] ] in addition to `` analytic '' corrections to scaling of the form @xmath44 , there are `` non - analytic '' corrections to scaling of the form @xmath45 , where the correction - to - scaling exponent @xmath46 is nt an integer . in fact one would expect a whole sequence of correction - to - scaling exponents @xmath47 , which are both universal and also independent of the observable , that is , the same for @xmath8 , @xmath9 , and so on . in a recent paper @xcite we study the amplitudes and the correction - to - scaling exponents for two - dimensional saws , using a combination of series - extrapolation and monte carlo methods . we enumerated all self - avoiding walks up to 59 steps on the square lattice , and up to 40 steps on the triangular lattice , measuring the metric properties mentioned above , and then carried out a detailed and careful analysis of the data in order to accurately estimate the amplitudes and correction - to - scaling exponent . the analysis provides firm numerical evidence that @xmath48 as predicted by nienhuis @xcite . in this paper we give a detailed account of the algorithm used to calculate the triangular lattice series analysed in @xcite , perform some further analysis of the series and confirm to great accuracy the predicted exact values of the critical exponents , then we briefly summarise the results of the analysis from @xcite and finally study other amplitude combinations . the use of transfer - matrix methods for the enumeration of lattice objects has its origin in the pioneering work of enting @xcite who enumerated square lattice self - avoiding polygons using the finite lattice method . the basic idea of the finite lattice method is to calculate partial generating functions for various properties of a given model on finite pieces , say @xmath49 rectangles of the square lattice , and then reconstruct a series expansion for the infinite lattice limit by combining the results from the finite pieces . the generating function for any finite piece is calculated using transfer matrix ( tm ) techniques . the algorithm we use to enumerate saws and saps on the triangular lattice builds on this approach and more specifically our algorithm is based in large part on the one devised by enting and guttmann @xcite for the enumeration of saps on the triangular lattice with the generalisation to saws following the work of conway , enting and guttmann @xcite and using further recent enhancements and parallelisation as described in @xcite . in this section we give a detailed description of the saw algorithm and then briefly outline the changes required to enumerate saps . a snapshot of the boundary line ( dashed line ) during the transfer matrix calculation on the triangular lattice . saws are enumerated by successive moves of the kink in the boundary line so that one vertex ( shaded ) at a time is added to the rectangle . to the left of the boundary line we have drawn an example of a partially completed saw . ] we implement the triangular lattice as a square lattice with additional edges connecting the top - left and bottom - right vertices of each unit cell ( see fig [ fig : transfer ] ) . we use @xmath49 rectangles as our finite lattices . the most efficient implementation of the tm algorithm generally involves bisecting the finite lattice with a boundary ( this is just a line in the case of rectangles ) and moving the boundary in such a way as to build up the lattice cell by cell . the sum over all contributing graphs is calculated as the boundary is moved through the lattice . due to the symmetry of the triangular lattice we need only consider rectangles with @xmath50 . saws in a given rectangle are enumerated by moving the intersection so as to add one vertex at a time , as shown in figure [ fig : transfer ] . in most cases it is most efficient to let the boundary line cut through the edges of the lattice . however , on the triangular lattice it is more efficient to let the boundary line cut through the vertices @xcite . essentially this variation leads to only half as many intersected vertices ( as opposed to edges ) along the boundary line . for each configuration of occupied or empty vertices along the intersection we maintain a generating function for partial walks cutting the intersection in that particular pattern . if we draw a saw and then cut it by a line we observe that the partial saw to the left of this line consists of a number of loops connecting two vertices ( we shall refer to these vertices as loop - ends ) in the intersection , and pieces which are connected to only one vertex ( we call these free ends ) . the other end of the free piece is an end point of the saw so there are at most two free ends . in addition it is possible that the saw touches a vertex ( that is the saw comes in along one edge and exits along another edge both without crossing the boundary line ) . all these cases are illustrated in figure [ fig : transfer ] . in applying the transfer matrix technique to the enumeration of saws we regard them as sets of edges on the finite lattice with the properties : * a weight @xmath51 is associated with an occupied edge . * all vertices are of degree 0 , 1 or 2 . * there are at most two vertices of degree 1 and the final graph has exactly two vertices of degree 1 ( the end points of the saw ) . * apart from isolated sites , the final graph has a single connected component . * each graph must span the rectangle from left to right and from bottom to top . we are not allowed to form closed loops , so two loop - ends can only be joined if they belong to different loops . to exclude loops which close on themselves we need to label the occupied vertices in such a way that we can easily determine whether or not two loop - ends belong to the same loop . the most obvious choice would be to give each loop a unique label . however , on two - dimensional lattices there is a more compact scheme relying on the fact that two loops can never intertwine . each end of a loop is assigned one of two labels depending on whether it is the lower end or the upper end of a loop . each configuration along the boundary line can thus be represented by a set of states @xmath52 , where @xmath53 if we read from the bottom to the top , the configuration along the intersection of the partial saw in figure [ fig : transfer ] is @xmath54 . the seven possible outputs from a single iteration of the tm algorithm . depending on the states of the three vertices @xmath55 , @xmath56 , and @xmath57 in the input some of the outputs can not occur . ] in figure[fig : tmit ] we have illustrated what can happen locally as the boundary line is moved . before the move , the boundary line intersects the vertices @xmath55 , @xmath56 and @xmath57 and after the move the vertices @xmath55 , @xmath58 and @xmath57 are intersected by the boundary line . we shall refer to the boundary line configuration prior to a move as the ` source ' and after the move as the ` target ' . in a basic iteration step we can insert bonds along the edges emanating from vertex @xmath56 . since vertex @xmath56 ca nt have degree greater than 2 we can at most insert two new bonds . however , depending on the states of vertices @xmath55 and @xmath57 in the source , some of the edge configuration in figure [ fig : tmit ] may be forbidden . the updating of the partial generating function depends most crucially on the state of vertex @xmath56 and to a somewhat lesser extent on the states of the vertices @xmath55 and @xmath57 . the basic limitation on the allowed outputs are that conditions ( 2)(4 ) must be enforced . in the following we shall briefly describe how the updating rules are derived . * state of vertex @xmath59 is 0 . * since vertex @xmath56 is empty all the outputs in figure [ fig : tmit ] are possible . in the first output we insert no bonds . this is always allowed and no changes are made to the configuration . in the next three outputs we insert a single bond . this makes vertex @xmath56 of degree one and is thus only allowed if there is at most one free end in the source . there are further restrictions on the insertion of a bond to vertices @xmath55 or @xmath57 . firstly if a vertex is touched ( in state 3 ) we can not insert a bond since this would result in a vertex of degree 3 . secondly if the vertex is a free end ( in state 4 ) we join two free ends . this leads to the formation of a completed sub - graph and is only permitted if the resulting graph is a valid saw . so the configuration can not contain other pieces of a saw and the only permissible states of other vertices in the intersection are 0 and 3 . if a valid saw is created we multiply the source generating function by @xmath51 ( representing the new bond ) before adding it to the total for the saw generating function . in the last three outputs we insert a partial loop . again there are restrictions on the insertion of bonds to vertices @xmath55 and @xmath57 . as before we can not insert a bond to a vertex in state 3 . otherwise the first two outputs are always allowed . the last output is a little more complicated . if both vertices @xmath55 and @xmath57 are in state 4 we join two free ends and as before we check if the result is a valid saw and if so add this partial generating function the saw generating function ( this time we multiply the source generating function by @xmath60 ) . if vertex @xmath55 is in state 1 and vertex @xmath57 in state 2 we can not join the two vertices since this would result in a closed loop . after the insertion of new bonds we have to assign a state to vertex @xmath58 and quite possibly change the states of vertices @xmath55 and @xmath57 ( and perhaps the states of some other vertices in the target configuration ) . the state of vertex @xmath58 will be 0 ( no bond ) , 1 ( lower loop - end ) , 2 ( upper loop - end ) or 4 ( free end ) . next we consider what happens to vertices @xmath55 and @xmath57 . when these vertices are empty in the source they can take the values just listed above in the target . if they are occupied in the source they either retain their state in the target ( no bonds inserted ) or change to state 3 ( a bond is inserted ) . in the latter case we may have to change the state of other vertices in the target . if we insert a free end and it joins a lower ( upper ) loop - end we must change the matching upper ( lower ) loop - end to a free end . otherwise we may join two lower ( upper ) loop - ends and then we must change the matching upper ( lower ) loop - end of the inner most loop to the lower ( upper ) loop - end of the new joined loop . * state of vertex @xmath59 is 1 . * a lower end of a loop enters vertex @xmath56 . if we insert no further bonds a new degree 1 vertex is created . as above this is only allowed provided the source has at most one free end . the matching upper loop - end becomes a free end . otherwise the lower end has to be continued by inserting a single bond ( partial loops can not be inserted since this would make vertex @xmath56 of degree 3 ) either to vertex @xmath58 which becomes a state 1 vertex ; to vertex @xmath55 if not in state 3 or state 2 ( closed loop would be formed ) ; or to vertex @xmath57 if not in state 3 . again we have to change the states of vertices @xmath55 and @xmath57 when a bond is inserted on these vertices . if the source state of the vertices was 0 the target state becomes 1 , otherwise the target state becomes 3 and as above we may need to change the state of other vertices as well . * state of vertex @xmath59 is 2 . * an upper end of a loop enters vertex @xmath56 . if we terminate the loop - end a new degree 1 vertex is created . again this is only allowed provided the source has at most one free end . the matching lower end of the loop becomes a free end . the upper end can always be continued to vertex @xmath58 ; to vertex @xmath57 if it is not in state 3 ; and to vertex @xmath55 provided it is not in state 3 or 1 ( this would result in a closed loop ) . the state of the target vertices are changed as described above . * state of vertex @xmath59 is 3 . * this is the simplest situation . vertex @xmath56 is of degree 2 so no bonds can be inserted and only the output with all empty edges is allowed . the state of vertex @xmath58 is 0 and the states of all other vertices are unchanged . * state of vertex @xmath59 is 4 . * a free end is entering vertex @xmath56 . if we insert no further bonds a partial walk is terminated at the vertex . this is only allowed if the resulting graph is a valid saw and the source generating function is added to the saw generating function . the free end can always be continued to vertex @xmath58 and to vertices @xmath55 and @xmath57 if they are not in state 3 . as before , if we join two free ends we check if it is a valid saw and then add the partial generating function ( multiplied by @xmath51 ) to the saw generating function . otherwise the target configuration is updated as described previously . [ [ sap - updating - rules . ] ] sap updating rules . + + + + + + + + + + + + + + + + + + + the updating rules used when enumerating saps are essentially just a subset of the saw rules . obviously there are no degree 1 vertices in the sap case so we ca nt insert a single bond if vertex @xmath56 is empty . likewise if vertex @xmath56 is occupied we must continue the loop - end . completed saps are formed by closing a loop ( if there are no other loop - ends in the source ) . this happens when the local source configuration @xmath61 is @xmath62 and we insert a bond from @xmath56 to @xmath55 , @xmath63 and we insert a partial loop from @xmath55 through @xmath56 to @xmath57 , or @xmath64 and we insert a bond from @xmath56 to @xmath57 . the use of _ pruning _ to obtain more efficient tm algorithms was used for square lattice saps in @xcite . numerically it was found that the computational complexity was close to @xmath65 , much better than the @xmath66 of the original approach . we have used similar techniques for the enumerations carried out for this paper . pruning allows us to discard most of the possible configurations for large @xmath67 because they only contribute at lengths greater than @xmath68 , where @xmath68 is the maximal length to which we choose to carry out our calculations . the value of @xmath68 is limited by the available computational resources , be they cpu time or physical memory . briefly pruning works as follows . firstly , for each configuration we keep track of the current minimum number of steps @xmath69 already inserted to the left of the boundary line in order to build up that particular configuration . secondly , we calculate the minimum number of additional steps @xmath70 required to produce a valid sap or saw . there are three contributions , namely the number of steps required to connect the loops and free ends , the number of steps needed ( if any ) to ensure that the saw touches both the lower and upper border , and finally the number of steps needed ( if any ) to extend at least @xmath67 edges in the length - wise direction ( remember we only need rectangles with @xmath50 ) . if the sum @xmath71 we can discard the partial generating function for that configuration , and of course the configuration itself , because it wo nt make a contribution to the count up to the lengths we are trying to obtain . there are no principal differences between pruning saws and saps though the detailed implementation is more complicated for the saw case . we found it necessary to explicitly write subroutines to handle the three distinct cases where the tm configuration contains zero , one and two free ends , respectively . but in all cases we essentially have to go through all the possible ways of completing a saw in order to find the minimum number of steps required . this is a fairly straightforward task though quite time consuming . the time @xmath72 required to obtain the number of polygons or walks of length @xmath1 grows exponentially with @xmath1 , @xmath73 . for the algorithm without pruning the complexity can be calculated exactly . time ( and memory ) requirements are basically proportional to a polynomial ( in @xmath1 ) times the maximal number of configurations , @xmath74 , generated during a calculation . when the boundary line is straight we can find the exact number of configurations . first look at the situation for saps when there are no free ends . the configurations correspond to 2-coloured motzkin paths @xcite , since we can map empty and touched vertices to the two colours of horizontal steps , vertices in the lower state to a north - east step , and vertices in the upper state to a south - east step . the number of such paths @xmath75 with @xmath1 steps is easily derived from the generating function @xcite @xmath76/2x^2,\ ] ] which means that @xmath77 as @xmath78 . note that @xmath75 slightly over counts @xmath74 since configurations without a loop are nt permitted , but since there are only @xmath79 of these there is no change in the asymptotic growth . when the boundary line has a kink ( such as in figure [ fig : transfer ] ) @xmath74 is no longer given exactly by @xmath80 . however , it is obvious that @xmath81 so we see that asymptotically @xmath74 grows like @xmath82 . since a calculation using rectangles of widths @xmath83 yields the number of saps up to @xmath84 it follows that the complexity of the algorithm is @xmath73 with @xmath85 . the number of transfer matrix configurations in the unpruned saw algorithm is simply obtained by inserting 0 , 1 or 2 free ends into a 2-coloured motzkin path and eliminating the paths corresponding to a configurations with only empty or touched vertices . in this case a calculation using rectangles of widths @xmath83 yields the number of saws up to @xmath86 it follows that the complexity of the algorithm is @xmath73 with @xmath87 . the pruned algorithm is much too difficult to analyse exactly . so all we can give is a numerical estimate of the growth in the number of configurations with @xmath1 . that is obtained by just running the algorithm and measuring the maximal number of configurations generated for different values of @xmath1 . from the actual numbers it appears that for the sap case increasing @xmath1 by 2 increases the number of configurations by slightly less than a factor of 2 . this would mean that for the pruned sap algorithm @xmath88 . in the saw case it appears that increasing @xmath1 by 4 increases the number of configuration by a factor close to 5 . so for the pruned saw algorithm @xmath89{5}=1.495\ldots$ ] . so once again pruning results in much more efficient algorithms . the transfer - matrix algorithms used in the calculations of the finite lattice contributions are eminently suited for parallel computation . the bulk of the calculations for this paper were performed on the facilities of the australian partnership for advanced computing ( apac ) . the apac facility is an hp alpha server cluster with 125 es45 s each with four 1 ghz chips for a total of 500 processors in the compute partition . each server node has at least 4 gb of memory . nodes are interconnected by a low latency high bandwidth quadrics network . the most basic concern in any efficient parallel algorithm is to minimise the communication between processors and ensure that each processor does the same amount of work and uses the same amount of memory . in practice one naturally has to strike some compromise and accept a certain degree of variation across the processors . one of the main ways of achieving a good parallel algorithm using data decomposition is to try to find an invariant under the operation of the updating rules . that is we seek to find some property of the configurations along the boundary line which does not alter in a single iteration . the algorithm for the enumeration of saps and saws are quite complicated since not all possible configurations occur due to pruning and an update at a given set of vertices might change the state of a vertex far removed , e.g. , when two lower loop - ends are joined we have to relabel one of the associated upper loop - ends as a lower loop - end in the new configuration . however , there is still an invariant since any vertex not directly involved in the update can not change from being empty to being occupied and vice versa , likewise a touched vertex will remain unchanged . that is only the kink vertices can change their occupation or touched status . this invariant allows us to parallelise the algorithm in such a way that we can do the calculation completely independently on each processor with just two redistributions of the data set each time an extra column is added to the lattice . we have already used this scheme for saps @xcite and lattice animals @xcite and refer the interested reader to these publications for further detail . our parallelisation scheme is also very similar to that used by conway and guttmann @xcite . the generalisation of the algorithm required to calculate metric properties and area - weighted moments has been described in detail in @xcite in the square lattice case . only some minor adjustments are needed in order to apply these ideas to metric properties on the triangular lattice ( no changes are needed for the area - weighted moments ) . we have represented the triangular lattice as a square lattice with extra edges along one of the main diagonals in a unit cell . a point @xmath90 on the square lattice is the point @xmath91 on the triangular lattice where @xmath92 and @xmath93 . as shown in @xcite calculation of metric properties involves summation over products of the @xmath94 and @xmath95 coordinates of the distance vectors . to be explicit we define the radius of gyration according to the _ vertices _ of the saw . note that the number of vertices is one more than the number of steps . the radius of gyration of @xmath96 points at positions @xmath97 is @xmath98 from the triangular lattice coordinates we see that both @xmath99 and @xmath100 carry a factor @xmath101 so in order to ensure that we get integer coefficients we multiply by 4 and the algorithm will thus calculate the coefficients @xmath102 . in order to do this we maintain five partial generating functions for each possible boundary configuration , namely * @xmath103 , the number of ( partially completed ) saws . * @xmath104 , the sum over saws of the squared components of the distance vectors . * @xmath105 , the sum of the @xmath94-component of the distance vectors . * @xmath106 , the sum of the @xmath95-component of the distance vectors . * @xmath107 , the sum of the ` cross ' product of the components of the distance vectors , that is , @xmath108 . as the boundary line is moved to a new position each target configuration @xmath109 might be generated from several sources @xmath110 in the previous boundary position . the partial generating functions are updated as follows , with @xmath90 being the coordinates of the newly added vertex on the square lattice : @xmath111,\nonumber \\ x_g(u , s ) & = & \sum_{s ' } u^{n'}[x_g(u , s)+ \delta_g ( 2s+t)c(u , s ' ) ] , \\ y_g(u , s ) & = & \sum_{s ' } u^{n'}[y_g(u , s)+ \delta_g tc(u , s ' ) ] , \nonumber \\ xy_g(u , s ) & = & \sum_{s ' } u^{n ' } [ xy_g(u , s')+\delta_g ( 2s+t)x_g(u , s ' ) + \delta_g 3 t y_g(u , s ' ) ] , \nonumber \end{aligned}\ ] ] where @xmath112 is the number of steps added to the partial saw . @xmath113 if the new vertex is empty ( has degree 0 ) and @xmath114 if the new vertex is occupied ( has degree @xmath115 ) . the updating rules for the other metric properties are generalised similarly . we calculated the number of polygons up to perimeter 60 , while the radius of gyration and first 10 area - weighted moments were obtained up to perimeter 58 . we calculated the number of saws , their mean - square radius of gyration , mean - square end - to - end distance , and the mean - square distance of monomers from the end points . these quantities were obtained for walks up to length 40 . the calculations required up to 35 gb of memory using up to 40 processors at a time and in total we used about 15000 cpu hours . in table [ tab : sapser ] we list the number of saps and their radius of gyration while in table [ tab : sawser ] we list the series for the saw problem . these series and those for the area - weighted moments are available at ` www.ms.unimelb.edu.au/~iwan ` . rrrrrr @xmath1 & @xmath9 & @xmath116 & @xmath1 & @xmath9 & @xmath116 + 3 & 2 & 6 & 32 & 2692047018699717 & 25886228326621869696 + 4 & 3 & 24 & 33 & 10352576717684506 & 110846359749047031012 + 5 & 6 & 102 & 34 & 39902392511347329 & 474213717578995665624 + 6 & 15 & 468 & 35 & 154126451419554156 & 2026979522666735966994 + 7 & 42 & 2172 & 36 & 596528356905096920 & 8657009828812246231296 + 8 & 123 & 9978 & 37 & 2313198287784319026 & 36944420238568755696168 + 9 & 380 & 45816 & 38 & 8986249863419780682 & 157546885404468362432148 + 10 & 1212 & 208686 & 39 & 34969337454759091232 & 671378005865890422968520 + 11 & 3966 & 944766 & 40 & 136301962040079085257 & 2859142640844460643187642 + 12 & 13265 & 4253484 & 41 & 532093404471021533628 & 12168301979788445465498400 + 13 & 45144 & 19046580 & 42 & 2080235431107538787148 & 51756227545091330753357904 + 14 & 155955 & 84891654 & 43 & 8144154378525048003270 & 220011744770726296282498056 + 15 & 545690 & 376756392 & 44 & 31927176350778729318192 & 934740492588407244896782986 + 16 & 1930635 & 1665684774 & 45 & 125322778845662829008494 & 3969252848247139670605665948 + 17 & 6897210 & 7338822888 & 46 & 492527188641409773340797 & 16846468953704095289170900908 + 18 & 24852576 & 32233105398 & 47 & 1937931188484341585677962 & 71466199766730550647612342396 + 19 & 90237582 & 141171369444 & 48 & 7633665703654150673637363 & 303035054640652779166447899354 + 20 & 329896569 & 616694403366 & 49 & 30101946001283232799847562 & 1284380183482800747257353493532 + 21 & 1213528736 & 2687630355198 & 50 & 118823919397444557546535851 & 5441398704214816650431847144246 + 22 & 4489041219 & 11687756315940 & 51 & 469508402822449711313115200 & 23043633507948438933442640818176 + 23 & 16690581534 & 50726031551790 & 52 & 1856933773092076293566747007 & 97548735673726189271333029096494 + 24 & 62346895571 & 219753786787212 & 53 & 7351015093472721439659392448 & 412789876403022674873495520537906 + 25 & 233893503330 & 950403133411176 & 54 & 29126027071450640626653986531 & 1746140617537848477455116275581178 + 26 & 880918093866 & 4103923685277414 & 55 & 115500592701344029351721102550 & 7383765950134760244068261726914950 + 27 & 3329949535934 & 17695343555964594 & 56 & 458398255374927436357237021173 & 31212646862418768098391776139187758 + 28 & 12630175810968 & 76195720234557276 & 57 & 1820727406941365079260306390484 & 131899272021134280524854379727885732 + 29 & 48056019569718 & 327682567452126696 & 58 & 7237327695683743010999188700157 & 557209110506518251250962658184410206 + 30 & 183383553173255 & 1407546930663067986 & 59 & 28789332223533619621001538109842 & + 31 & 701719913717994 & 6039368800117995984 & 60 & 114602547490254934327469368968190 & + rrrrr @xmath1 & @xmath8 & @xmath117 & @xmath118 & @xmath119 + 1 & 6 & 1 & 1 & 1 + 2 & 30 & 12 & 22 & 17 + 3 & 138 & 97 & 282 & 178 + 4 & 618 & 654 & 2778 & 1476 + 5 & 2730 & 3977 & 23305 & 10667 + 6 & 11946 & 22624 & 175194 & 70359 + 7 & 51882 & 122821 & 1215740 & 434708 + 8 & 224130 & 644082 & 7939156 & 2557166 + 9 & 964134 & 3288739 & 49422491 & 14477823 + 10 & 4133166 & 16440648 & 295993366 & 79492861 + 11 & 17668938 & 80783857 & 1717056604 & 425633898 + 12 & 75355206 & 391310240 & 9697408184 & 2231674940 + 13 & 320734686 & 1872763387 & 53533130211 & 11494836257 + 14 & 1362791250 & 8870963422 & 289769871988 & 58310378811 + 15 & 5781765582 & 41647686501 & 1541876281342 & 291901836462 + 16 & 24497330322 & 194014270964 & 8081886977224 & 1444405248178 + 17 & 103673967882 & 897639074623 & 41801262603145 & 7074419785415 + 18 & 438296739594 & 4127904278590 & 213650877117460 & 34334678700977 + 19 & 1851231376374 & 18879838654237 & 1080407596025856 & 165283451747722 + 20 & 7812439620678 & 85930246593928 & 5411153165106856 & 789827267540498 + 21 & 32944292555934 & 389382874004291 & 26865804448156781 & 3749241090582031 + 22 & 138825972053046 & 1757383045067340 & 132328831054383256 & 17689855417349797 + 23 & 584633909268402 & 7902553525660965 & 647064413113509344 & 83004601828121876 + 24 & 2460608873366142 & 35417121500633314 & 3142945284616515512 & 387503899136724032 + 25 & 10350620543447034 & 158241760294727837 & 15172247917136636793 & 1800616777561080887 + 26 & 43518414461742966 & 705008848574456242 & 72826367061554681960 & 8330920471773661365 + 27 & 182885110185537558 & 3132749279518281223 & 347722481262776946768 & 38390978707292879316 + 28 & 768238944740191374 & 13886614514918779812 & 1652126117509776447678 & 176259763248055992656 + 29 & 3225816257263972170 & 61415827107198652263 & 7813839241496101017943 & 806446563482615080995 + 30 & 13540031558144097474 & 271046328280157919578 & 36798230598686798952874 & 3677867046530479086571 + 31 & 56812878384768195282 & 1193838903060544883615 & 172603075240086498030932 & 16722626138383080469074 + 32 & 238303459915216614558 & 5248569464050058190772 & 806559315077883801952302 & 75819788411079420147060 + 33 & 999260857527692075370 & 23034474248167644819305 & 3755672941408238341746325 & 342850281196290726391195 + 34 & 4188901721505679738374 & 100925879660029490332616 & 17429779928912903943728776 & 1546457563237807336247617 + 35 & 17555021735786491637790 & 441524252843364233569911 & 80636231608943399450377104 & 6958970268567678359172166 + 36 & 73551075748132902085986 & 1928731794198995523104424 & 371943975622752362856339418 & 31245121332848941331142166 + 37 & 308084020607224317094182 & 8413734243045682304542891 & 1710813401690158618688146075 & 139991577634597301110308061 + 38 & 1290171266649477440877690 & 36655327788277288494374240 & 7848181414990001769700643892 & 625968026891459936611240307 + 39 & 5401678666643658402327390 & 159494618902280757690831541 & 35911648943670829119431170002 & 2793684462154188994667777314 + 40 & 22610911672575426510653226 & 693174559672551318610401776 & 163929038497681452701025717812 & 12445679176337664122926617782 + the series studied in this paper have coefficients which grow exponentially , with sub - dominant term given by a critical exponent . the generic behaviour is @xmath120 and hence the coefficients of the generating function @xmath121 , where @xmath122 . to obtain the singularity structure of the generating functions we used the numerical method of differential approximants @xcite . our main objective is to obtain accurate estimates for the connective constant @xmath19 and confirm numerically the exact values for the critical exponents @xmath123 , @xmath124 and @xmath125 . estimates of the critical point and critical exponents were obtained by averaging values obtained from second and third order inhomogeneous differential approximants . the error quoted for these estimates reflects the spread ( basically one standard deviation ) among the approximants . note that these error bounds should _ not _ be viewed as a measure of the true error as they can not include possible systematic sources of error . once the exact values of the exponents have been confirmed we turn our attention to the `` fine structure '' of the asymptotic form of the coefficients . in particular we are interested in obtaining accurate estimates for the amplitudes . we do this by fitting the coefficients to the form given by ( [ eq : asympsaw])-([eq : asympmom ] ) . in this case our main aim is to test the validity of the predictions for the amplitude combinations mentioned in the introduction . the expected behaviour of the mean - square radius of gyration ( [ eq : asympsaprg ] ) and moments of area ( [ eq : asympmom ] ) of saps results in the following predictions for the generating functions which we study : @xmath126 where we have taken into account that the smallest polygon has perimeter 3 . thus we expect these series to have a critical point at @xmath127 , and as stated previously the exponents @xmath128 and @xmath129 . in table [ tab : anasap ] we have listed the results from our series analysis of the sap generating function . it is evident that the estimates for the critical exponent is in complete agreement with the expected value @xmath130 . based on the estimates we find that @xmath131 . we found no evidence that the sap generating function had any other singularities . lllll @xmath132 & & + & & & & + 0 & 0.2409175671(28 ) & 1.5000142(45 ) & 0.2409175706(62 ) & 1.500006(12 ) + 2 & 0.2409175709(14 ) & 1.5000076(29 ) & 0.2409175716(30 ) & 1.5000071(58 ) + 4 & 0.2409175714(27 ) & 1.5000061(56 ) & 0.2409175699(40 ) & 1.5000078(63 ) + 6 & 0.2409175707(29 ) & 1.5000075(58 ) & 0.2409175712(29 ) & 1.5000065(57 ) + 8 & 0.2409175724(44 ) & 1.500003(10 ) & 0.2409175662(80 ) & 1.500012(14 ) + 10 & 0.2409175717(39 ) & 1.5000051(83 ) & 0.2409175704(22 ) & 1.5000083(41 ) + if we take the conjecture @xmath23 to be true we can obtain a refined estimate for the critical point @xmath133 . in figure [ fig : sapexp ] we have plotted estimates for the critical exponent @xmath134 against the number of terms used by the approximant and against estimates for the critical point @xmath133 , respectively . each dot represents estimates obtained from a third order inhomogeneous differential approximant . the order of the inhomogeneous polynomial was varied from 0 to 10 . as can be seen from the left panel the estimates for the critical exponent clearly include the exact value and appear to settle down as the number of terms increases ( though there is a hint of a downwards trend ) . from the right panel we observe that the estimates cross the line @xmath135 at a value @xmath136 . based on this analysis we adopt the value @xmath137 and thus @xmath138 as our final estimates . plots of estimates from third order differential approximants for @xmath134 vs. , respectively , the number of terms used by the differential approximants ( left panel ) and @xmath133 ( right panel ) . ] table [ tab : anasaprg ] lists the results of our series analysis for the sap radius of gyration generating function . it is evident that the estimates for the critical exponent as obtained from third order differential approximants is in complete agreement with the expected behaviour . the estimates from the second order approximants are generally slightly lower than the expected value . one would generally expect third order differential approximants to be superior since they are better suited to represent complicated functional behaviour . we take this as clear numerical confirmation that @xmath129 . lllll @xmath132 & & + & & & & + 0 & 0.24091726(11 ) & 1.99885(27 ) & 0.24091761(30 ) & 2.0001(14 ) + 2 & 0.24091727(14 ) & 1.99891(26 ) & 0.24091728(33 ) & 1.99925(64 ) + 4 & 0.240917246(95 ) & 1.99881(14 ) & 0.24091713(30 ) & 1.99889(36 ) + 6 & 0.240917269(87 ) & 1.99884(14 ) & 0.24091741(19 ) & 1.99935(65 ) + 8 & 0.240917239(73 ) & 1.99879(11 ) & 0.24091743(24 ) & 1.99947(78 ) + 10 & 0.240917281(96 ) & 1.99888(16 ) & 0.24091737(25 ) & 1.99932(70 ) + in passing we only briefly mention that our analysis of the generating functions @xmath139 for area - weighted saps with @xmath140 clearly confirmed the expected values , @xmath141 , for the critical exponents . suffice to say that the estimates range from @xmath142 for @xmath143 to @xmath144 for for @xmath145 . from the expected behaviour ( [ eq : asympsaw ] ) of @xmath8 and the metric properties of saws ( [ eq : asympee])-([eq : asympmd ] ) we get that the generating functions : @xmath146 where the exponents @xmath147 and @xmath129 . in table [ tab : anasaw ] we list estimates of the critical point @xmath133 and exponent @xmath124 from the series for the triangular lattice saw generating function . the estimates were obtained by averaging over those approximants which used at least the first 32 terms of the series . based on these estimates we conclude that @xmath148 and @xmath149 . the estimate for @xmath133 is compatible with the more accurate estimate @xmath137 obtained above from the analysis of the sap generating function . the analysis adds further support to the already convincing evidence that the critical exponent @xmath150 exactly . however , we do observe that both the central estimates for both @xmath133 and @xmath124 are systematically slightly lower than the expected values . lllll @xmath132 & & + & & & & + 0 & 0.240917491(34 ) & 1.343637(42 ) & 0.240917538(21 ) & 1.343687(23 ) + 2 & 0.240917529(37 ) & 1.343677(36 ) & 0.240917537(13 ) & 1.343686(22 ) + 4 & 0.240917529(42 ) & 1.343682(47 ) & 0.240917534(30 ) & 1.343682(32 ) + 6 & 0.240917524(27 ) & 1.343673(27 ) & 0.240917545(24 ) & 1.343693(25 ) + 8 & 0.240917523(28 ) & 1.343668(35 ) & 0.240917543(23 ) & 1.343692(27 ) + 10 & 0.240917513(31 ) & 1.343662(29 ) & 0.240917530(22 ) & 1.343679(25 ) + as for the sap case we find it useful to plot the behaviour of the estimates for the critical exponent @xmath124 against both @xmath133 and the number of terms used by the differential approximants . this is done in figure [ fig : sawexp ] . each dot represents estimates obtained from a second order inhomogeneous differential approximant . from the left panel we observe that the estimates of @xmath124 exhibits a certain upwards drift as the number of terms increases . so the estimates have not yet settled at their limiting value , but there can be no doubt that the predicted exact value of @xmath124 is fully consistent with the estimates . in the left panel we observe that the @xmath151-estimates fall in a narrow range . note that this range does not include the intersection point between the exact @xmath124 and the precise @xmath133 estimate . this is probably a further reflection of the lack of ` convergence ' to the true limiting values . plots of estimates from second order differential approximants for @xmath124 vs. , respectively , the number of terms used by the differential approximants ( left panel ) and @xmath133 ( right panel ) . ] finally , we turn our attention to the metric properties of saws . in table [ tab : anametric ] we have listed the estimates for @xmath133 and critical exponents obtained by an analysis of the associated generating functions ( [ eq : saweegf])([eq : sawmdgf ] ) . the estimates from the radius of gyration @xmath152 and @xmath153 are in full agreement with the more accurate sap estimate for @xmath133 and the predicted exact exponent value @xmath154 . the analysis of the generating functions for the end - to - end distance and monomer distance yield estimates of @xmath133 a little below the expected value and likewise the exponent estimates @xmath155 and @xmath156 are a somewhat smaller that the exact values @xmath157 and @xmath158 , respectively . we are fully convinced that this is because the series are not long enough to allow the exponent estimates to settle at the true limiting values , as was also the case for the saw generating function as shown in figure [ fig : sawexp ] . lllllll @xmath132 & & & + & & & & & & + 0 & 0.240917330(86 ) & 2.84307(36 ) & 0.240917594(53 ) & 4.843619(70 ) & 0.240917324(92 ) & 3.84296(17 ) + 2 & 0.240917298(62 ) & 2.84295(12 ) & 0.240917600(53 ) & 4.843626(72 ) & 0.24091715(22 ) & 3.84270(35 ) + 4 & 0.240917249(39 ) & 2.84272(35 ) & 0.240917605(62 ) & 4.843631(81 ) & 0.24091722(23 ) & 3.84281(41 ) + 6 & 0.240917311(71 ) & 2.84295(16 ) & 0.240917578(71 ) & 4.843590(99 ) & 0.24091732(17 ) & 3.84299(34 ) + 8 & 0.240917328(52 ) & 2.842938(73 ) & 0.240917616(67 ) & 4.843646(89 ) & 0.240917304(43 ) & 3.842922(80 ) + 10 & 0.240917373(99 ) & 2.84303(19 ) & 0.240917612(57 ) & 4.84365(10 ) & 0.240917276(98 ) & 3.84285(18 ) + the asymptotic form of the coefficients @xmath9 of the generating function of square lattice saps has been studied in detail previously @xcite . there is now clear numerical evidence that the leading correction - to - scaling exponent for saps and saws is @xmath48 , as predicted by nienhuis @xcite . as argued in @xcite this leading correction term combined with the @xmath130 term of the sap generating function produces an _ analytic _ background term . indeed in the previous analysis of saps there was no sign of non - analytic corrections - to - scaling to the generating function ( a strong indirect argument that the leading correction - to - scaling exponent must be half - integer valued ) . one therefore finds that asymptotically @xmath9 behaves as @xmath159.\ ] ] this form was confirmed with great accuracy in @xcite . estimates for the leading amplitude @xmath160 can thus be obtained by fitting @xmath9 to the form ( [ eq : sapasymp ] ) using an increasing of number of correction terms . as in @xcite we find it useful to check the behaviour of the estimates by plotting the results for the leading amplitude vs. @xmath161 ( see figure [ fig : sapgfampl ] ) , where @xmath9 is the last term used in the fitting . in addition we also wish to check the sensitivity of the procedure to small changes in the value of @xmath19 . clearly the amplitude estimates in top panels are quite well converged . notice that as more correction terms are added the estimates exhibit less curvature and the slope becomes less steep . this is very strong evidence that ( [ eq : sapasymp ] ) indeed is the correct asymptotic form of @xmath9 . the estimates shown in the bottom panels are not so well behaved . those in the left panel are not monotonic and after initially decreasing they start to increase with @xmath1 . the estimates in the right panel while monotonic have much steeper slopes and the slopes do not appear to change much as more correction terms are used . we think this is strong evidence that @xmath162 is very close to the true value . based on the plots in the top right panel we estimate that @xmath163 . estimates of the leading amplitude @xmath160 plotted against @xmath161 using different number of terms in the asymptotic expansion . in the top panels we use the value @xmath162 . the right panel just gives a more detailed look at the data shown in the left panel . in the bottom panels we use two different values @xmath164 ( left panel ) and @xmath165 ( right panel ) chosen to be at the extreme ends of our error - bounds for @xmath19 . ] the asymptotic form of the coefficients @xmath166 in the generating function for the radius of gyration was studied in @xcite . the generating function ( [ eq : saprggf ] ) has critical exponent @xmath167 , so the leading correction - to - scaling term no longer becomes part of the analytic background term . we thus use the following asymptotic form : @xmath168.\ ] ] we find @xmath169 . this is in full agreement with the predicted exact value @xcite @xmath170 , where , for the triangular lattice , @xmath36 and @xmath40 . combining the exact expression for @xmath171 with the more accurate estimate for @xmath160 given above we estimate that @xmath172 . the amplitudes of the area - weighted moments were studied in @xcite . we fitted the coefficients to the assumed form @xmath173,\ ] ] where the amplitude @xmath174 is related to the amplitude defined in equation ( [ eq : asympmom ] ) . the scaling function prediction for the amplitudes @xmath175 is @xcite @xmath176 where the numbers @xmath177 are given by the quadratic recursion @xmath178 we obtained @xcite the results for the amplitude combinations listed in table [ tab : momampl ] . it is clear that the estimates for the first 10 area weighted moments are in excellent agreement with the predicted exact values . lllll amplitude & exact value & square & hexagonal & triangular + @xmath160 & unknown & 0.56230130(2 ) & 1.27192995(10 ) & 0.2639393(2 ) + @xmath179 & @xmath180 & @xmath181 & @xmath182 & @xmath183 + @xmath184 & @xmath185 & @xmath186 & @xmath187 & @xmath188 + @xmath189 & @xmath190 & @xmath191 & @xmath192 & @xmath193 + @xmath194 & @xmath195 & @xmath196 & @xmath197 & @xmath198 + @xmath199 & @xmath200 & @xmath201 & @xmath202 & @xmath203 + @xmath204 & @xmath205 & @xmath206 & @xmath207 & @xmath208 + @xmath209 & @xmath210 & @xmath211 & @xmath212 & @xmath213 + @xmath214 & @xmath215 & @xmath216 & @xmath217 & @xmath218 + @xmath219 & @xmath220 & @xmath221 & @xmath222 & @xmath223 + @xmath224 & @xmath225 & @xmath226 & @xmath227 & @xmath228 + the amplitude ratios @xmath25 and @xmath26 were estimated by direct extrapolation of the relevant quotient sequence , using the following method @xcite : given a sequence @xmath229 defined for @xmath230 , assumed to converge to a limit @xmath231 with corrections of the form @xmath232 , we first construct a new sequence @xmath233 defined by @xmath234 . the generating function @xmath235 . estimates for @xmath231 and the parameter @xmath56 can then be obtained from differential approximants . in this way , we obtained the estimates @xcite @xmath236 and @xmath237 . these amplitude estimates leads to a high precision confirmation of the cscps relation ( [ eq : cscps ] ) @xmath238 . the amplitudes of the saw generating function and the metric properties were also studied in @xcite by fitting of the coefficients to the assumed form @xmath239 , \\ \label{eq : saweeampl } c_n{\langle r^2_e \rangle}_n/6 \sim \mu^n n^{\gamma+2\nu-1 } \left[ac/6+\sum_{i\ge 0}a_i / n^{1+i/2 } \right ] , \\ \label{eq : sawrgampl } ( n+1)^2c_n{\langle r^2_g \rangle}_n/6 \sim \mu^n n^{\gamma+2\nu+1 } \left[ad/6+\sum_{i\ge 0}a_i / n^{1+i/2 } \right ] , \\ \label{eq : sawmdampl } ( n+1)c_n{\langle r^2_m \rangle}_n/6 \sim \mu^n n^{\gamma+2\nu } \left[ae/6+\sum_{i\ge 0}a_i / n^{1+i/2 } \right].\end{aligned}\ ] ] estimates of the leading amplitude @xmath20 for saws , @xmath240 for the end - to - end distance , @xmath241 for the radius of gyration , and @xmath242 for the monomer distance from the end - points , plotted against @xmath161 using varying number of terms in the asymptotic expansion . ] in figure [ fig : sawampl ] we have plotted the estimates for the leading amplitudes against @xmath161 while varying the number of terms used in fitting to the expressions given above . from these we estimate that @xmath243 , @xmath244 , @xmath245 , and @xmath246 . the estimate for @xmath20 is the same as that obtained in @xcite while the remaining amplitude estimates are a little lower and have smaller error - bars that those quoted in @xcite . the main reason is that here we are only interested in the leading amplitudes and base our estimates on the fits using 6 - 8 terms , while in @xcite estimates for sub - leading amplitudes were also considered and required to be stable and consequently only fits with up to 4 terms were considered . for the metric amplitudes we thus obtain the estimates @xmath247 , @xmath248 , and @xmath249 . for the universal amplitude ratios we get @xmath250 and @xmath251 . we note that these estimates of the amplitude ratios are fully consistent with the more accurate estimates given above . this gives us further confidence that this method for obtaining amplitude estimates is valid . in particular , it appears , that in order to estimate the leading amplitude , we do not have to insist that estimates for sub - leading amplitudes be well converged . the smaller error - bars obtained from the fits using 6 - 8 terms thus appear soundly based . naturally , some readers might wish to take a more cautious approach . in table [ tab : amplratio ] we have summarised estimates of various universal amplitude combinations , obtained from the work reported in this paper and elsewhere . as can be seen the estimates for all lattices are in perfect agreement strongly confirming the universality of the various combinations . lllll lattice & @xmath25 & @xmath26 & @xmath31 & @xmath252 + square @xcite & 0.140299(6 ) & 0.439647(6 ) & 0.216835(15 ) & -0.000024(28 ) + triangular @xcite & 0.140296(6 ) & 0.439649(9 ) & 0.216823(10 ) & -0.000036(34 ) + honeycomb @xcite & 0.1403(1 ) & 0.4397(2 ) & 0.2170(3 ) & -0.00013(67 ) + kagom @xcite & 0.140(1 ) & 0.440(1 ) & 0.2144(25 ) & -0.0015(47 ) + we have presented both improved and parallel algorithms for the enumeration of self - avoiding polygons and walks on the triangular lattice . these algorithms have enabled us to obtain polygons up to perimeter length 60 , their radius of gyration and area - weighted moments up to perimeter 58 , while for self - avoiding walks to length 40 we calculated the number of walks as well as the metric properties of mean - square end - to - end distance , mean - square radius of gyration and the mean - square distance of a monomer from the end points . the analysis of the polygon series enabled us to obtain a very precise estimate for the connective constant @xmath0 . we confirmed to a very high degree of accuracy the predicted exponent values @xmath128 , @xmath147 and @xmath129 . we noticed that , as is the case for the square lattice problem , the saw asymptotics is worse behaved than the sap asymptotics , i.e. , estimates for @xmath19 and the critical exponents are at least an order of magnitude more accurate in the sap case . it quite is possible that this behaviour is due to the leading correction - to - scaling exponent @xmath253 . in the sap case this correction simply becomes part of the analytic background term and the sap generating function is therefore simpler since it only has analytic corrections to scaling . we also obtained very accurate estimates for the leading amplitude of the sequence @xmath9 of sap coefficients @xmath163 and using the exact expression for the amplitude combination @xmath171 we find @xmath172 . our data for the area - weighted moments was used @xcite to confirm the correctness of theoretical predictions for the values of the amplitude combinations @xmath41 . finally we obtained accurate estimates for the critical amplitudes @xmath243 , @xmath247 , @xmath248 , and @xmath249 . the estimate for the ratio @xmath254 is in very good agreement with the theoretical estimate @xmath34 @xcite . the amplitude estimates led to a high precision confirmation of the cscps relation ( [ eq : cscps ] ) @xmath255 . the series for the problems studied in this paper can be obtained via e - mail by sending a request to i.jensen@ms.unimelb.edu.au or via the world wide web on the url http://www.ms.unimelb.edu.au/ iwan/ by following the relevant links . the calculations presented in this paper would not have been possible without a generous grant of computer time on the server cluster of the australian partnership for advanced computing ( apac ) . we also used the computational resources of the victorian partnership for advanced computing ( vpac ) . we gratefully acknowledge financial support from the australian research council . 10 jensen i 2003 counting polyominoes : a parallel implementation for cluster computing in _ computational science iccs 2003 _ ( eds . p m a sloot , d abramson , a v bogdanov , j j dongarra , a y zomaya and y e gorbachev ) ( berlin : springer ) vol . 2659 of _ lecture notes in computer science _ 203212 richard c , jensen i and guttmann a j 2003 scaling function for self - avoiding polygons in _ proceedings of the international congress on theoretical physics th2002 ( paris ) , supplement _ ( eds . d iagolnitzer , v rivasseau and j zinn - justin ) ( basel : birkhuser ) pp . 267277 _ preprint _ cond - mat/0302513

we use new algorithms , based on the finite lattice method of series expansion , to extend the enumeration of self - avoiding walks and polygons on the triangular lattice to length 40 and 60 , respectively . for self - avoiding walks to length 40 we also calculate series for the metric properties of mean - square end - to - end distance , mean - square radius of gyration and the mean - square distance of a monomer from the end points . for self - avoiding polygons to length 58 we calculate series for the mean - square radius of gyration and the first 10 moments of the area . analysis of the series yields accurate estimates for the connective constant of triangular self - avoiding walks , @xmath0 , and confirms to a high degree of accuracy several theoretical predictions for universal critical exponents and amplitude combinations .

gravitational redshift has been reported by most of the authors without consideration of rotation of a body , as detailed in our previous work dubey and sen ( 2014 ) . payandeh and fathi ( 2013 ) had obtained the gravitational redshift for a static spherically symmetric electrically charged object in isotropic reissner - nordstrom geometry . dubey and sen ( 2014 ) had obtained the expression for gravitational redshift from rotating body in kerr field . they also showed the rotation and the latitude dependence of gravitational redshift from a rotating body ( such as pulsars ) . the expression of gravitational redshift factor ( @xmath0 ) from rotating body in kerr field was given as ( eqn . 69 , dubey and sen ( 2014 ) ) : @xmath1 apsel ( 1978 , 1979 ) had discussed , that the motion of a particle in a combination of gravitational and electro and magneto static field can be determined from a variation principle of the form @xmath2 = 0 . the form of the physical time is determined from an examination of the maxwell - einstein action function . the field and motion equations are actually identical to those of maxwell - einstein theory . the theory predicted that even in a field free region of space , electro and magneto static potentials can alter the phase of wave function and the life time of charged particle . + gravitational redshift has been reported by most of the authors without consideration of static electric and / or magnetic charge present within the rotating body . with this background , the present paper is a continuation of our previous work dubey and sen ( 2014 ) , to study the influence of electro and magneto static charges of a rotating gravitating body on the redshift . + the present paper is organized as follows . in section - 2 , we derive the expression for radial component of electro and magneto static field from charged rotating body by using the associated potential of kerr - newman field . in section -3 , we derive the expression for combined gravitational and electro - magneto - static redshift from rotating body . finally , some conclusions are made in section -4 . when rotation is taken into consideration spherical symmetry is lost and off - diagonal terms appear in the metric and the most useful form of the solution of kerr family is given in terms of t , r , @xmath3 and @xmath4 , where t , and r are boyer - lindquist coordinates running from - @xmath5 to + @xmath5 , @xmath3 and @xmath4 , are ordinary spherical coordinates in which @xmath4 is periodic with period of 2 @xmath6 and @xmath3 runs from 0 to @xmath7 . covariant form of metric tensor for kerr family ( kerr ( 1963 ) , newman et al . ( 1965 ) ) in terms of boyer - lindquist coordinates with signature ( + , -,-,- ) is expressed as : @xmath8 where @xmath9 s are non - zero components of kerr family . + if we consider the three parameters : mass ( m ) , rotation parameter ( a ) and charge ( electric ( q ) and / or magnetic ( p ) ) , then it is easy to include charge in the non - zero components of @xmath9 of kerr metric , simply by replacing @xmath10 with @xmath11 . + non - zero components of @xmath9 of kerr - newman metric are given as follows ( page 261 - 262 of carroll ( 2004 ) ) : @xmath12 @xmath13 @xmath14 @xmath15sin^{2}\theta\ ] ] @xmath16 with @xmath17 and @xmath18 where @xmath19 is the schwarzschild radius . q and p are electric and magnetic charges respectively and a @xmath20 is rotation parameter of the source . if we replace @xmath11 by @xmath21 and further if we put rotation parameter of the source ( a ) equal to zero , then it reduces to reissner - nordstrom metric . also if we replace @xmath11 by @xmath10 then the kerr - newman metric reduces to kerr metric and further if we put rotation parameter of the source ( a ) equal to zero then it reduces to schwarzschild metric . + the associated potential of the kerr - newman metric are expressed as ( page 262 of carroll ( 2004 ) ) : @xmath22 @xmath23 @xmath24 @xmath25 electromagnetic field strength tensor @xmath26 can be expressed in terms of potential as ( page 65 ( eqn . 23.3 ) of landau and lifshitz ( 2008 ) ) : @xmath27 radial component of electric field @xmath28 can be related to electromagnetic field strength tensor @xmath26 by the expression ( page 254 of carroll ( 2004 ) ) : @xmath29 using equations ( 10 - 14 ) we can express the radial component of electric field @xmath30 as : @xmath31 after simplification the above equation can be written as : @xmath32 radial component of magnetic field @xmath33 can be related to electromagnetic field strength tensor @xmath26 by the expression ( page 254 of carroll ( 2004 ) ) : @xmath34 levi - civita tensor ( @xmath35 ) and levi - civita symbol ( @xmath36 ) in four - dimension are related by the expression ( page 24 and 83 of carroll ( 2004 ) ) : @xmath37 where g=@xmath38 is the determinant of metric @xmath9 and + @xmath36 = @xmath39 + radial component of magnetic field ( @xmath40 ) given by equation ( 18 ) can be now rewritten as : @xmath41 using the property of levi - civita symbol ( @xmath36 ) and anti symmetric property of component of electromagnetic field strength tensor @xmath42 the above expression can be written as : @xmath43 thus the difference in kerr and kerr - newman metric lies in replacing @xmath10 by @xmath11 . the determinant of the kerr - newman metric can be written as ( similar expression is given in case of kerr metric page 347 ( eqn . 104.5 ) of landau and lifshitz ( 2008 ) ; page 16 ( eqn . 1.70 ) of wiltshire et al . ( 2009 ) ) : @xmath44 using equations ( 10 - 14 ) we can express the component of electromagnetic field strength tensor @xmath45 as : @xmath46 after simplification the above equation can be written as : @xmath47\ ] ] combining the equations ( 21 ) , ( 22 ) and ( 24 ) , the radial component of magnetic field ( @xmath40 ) can be written as : @xmath48\ ] ] substituting @xmath49 in equations ( 17 ) and ( 25 ) , we can write the radial components of electric and magnetic field for equatorial plane as : @xmath50 and @xmath51 frame dragging is a general relativistic feature of all solutions to the einstein field equations associated with rotating masses . due to the influence of gravity , frame dragging or dragging of inertial frame arises in the kerr metric . the quantity @xmath52 is termed as angular velocity of frame dragging as given by collas and klein ( 2004 ) . + considering the ray of light emitted radially outward from the surface of a compact object ( from a rotating body with radius r ) , the general expression of angular velocity of frame dragging ( @xmath52 ) in kerr field was given as ( eqn . ( 46 ) of dubey and sen ( 2014 ) : @xmath53 if we consider a rotating body having electric charge ( q ) and magnetic charge ( p ) , then using the above equation we can write the expression for angular velocity of frame dragging in kerr - newman field ( @xmath54 ) as : @xmath55 again for any general @xmath3 and at @xmath56 , the expression for @xmath52 from above equation ( 29 ) can be written as : @xmath57 for equatorial plane where @xmath58 , the above expression ( 30 ) can be rewritten as : @xmath59 the above expression ( 31 ) is the expression for frame dragging @xmath60 on the equatorial plane . + + in general relativity , redshift ( z ) and redshift factor ( @xmath0 ) are defined as : @xmath61 this @xmath62 is the frequency measured by a distant observer in terms of proper time ( @xmath63 ) and @xmath64 is the frequency measured in terms of the world time ( t ) . a redshift ( z ) of zero corresponds to an un - shifted line , whereas @xmath65 indicates blue - shifted emission and @xmath66 red - shifted emission . a redshift factor ( @xmath0 ) of unity corresponds to an un - shifted line , whereas @xmath67 indicates red - shifted emission and @xmath68 blue - shifted emission . + + for a sphere , the photon is emitted at a location on its surface where @xmath69 , as the sphere rotates . + + as a result , we can write the expression of frequency as observed by distant observer as : @xmath70 where @xmath71 , is defined as eikonal and @xmath72 is the wave four - vector . + + following ( eqn.(69 ) of dubey and sen ( 2014 ) ) we can write the expression for gravitational redshift factor for a rotating body with electric charge ( q ) and magnetic charge ( p ) as : @xmath73 @xmath74sin^{2}\theta}{(\frac{d\phi}{c dt}})^{2 } + 2 \frac{a sin^{2}\theta ( r_{g}r - q^{2}-p^{2})}{r^{2}+ a^{2}cos^{2}\theta}(\frac{d\phi}{c dt})}\ ] ] where @xmath75 is the corresponding expression of angular velocity of frame dragging in kerr - newman field as given by equation ( 29 ) . + again for any general @xmath3 and at @xmath56 , the expression for @xmath76 from above equation ( 34 ) can be written as : @xmath77 @xmath74sin^{2}\theta}{(\frac{d\phi}{c dt}})^{2 } + 2 \frac{a sin^{2}\theta ( r_{g}r - q^{2}-p^{2})}{r^{2}+ a^{2}cos^{2}\theta}(\frac{d\phi}{c dt})}\ ] ] where @xmath78 is the corresponding expression of angular velocity of frame dragging in kerr - newman field as given by equation ( 30 ) . + + for equatorial plane where @xmath58 , the above expression ( 35 ) of @xmath76 can be written as : @xmath79 @xmath80 where @xmath81 is the corresponding expression of angular velocity of frame dragging on the equatorial plane as given by equation ( 31 ) . + + if we set magnetic charge ( p ) and rotation parameter of source ( a ) both equal to zero in the above expression of redshift factor ( 36 ) , then the coresponding redshift factor in reissner - nordstrom geometry ( @xmath82 ) can be written as : @xmath83 where @xmath84 is the gravitational redshift in reissner - nordstrom geometry . + + now the obtained expression of redshift factor ( 37 ) exactly matches with the gravitational redshift factor for a static spherically symmetric electrically charged object in reissner - nordstrom geometry ( eqn . ( 30 ) of payandeh and fathi ( 2013 ) ) . + + we can replace the values of electric charge ( q ) and magnetic charge ( p ) in terms of radial components of electric field @xmath28 and magnetic field @xmath33 by utilizing equations ( 17 ) and ( 25 ) . thus we can obtain the general expression for the redshift factor from charged rotating body in kerr - newman field , which can be considered as combined gravitational and electro - magneto - static redshift factor . + + in the above expression of redshift factor given by equations ( 34 ) to ( 36 ) : * if we substitute p=0 and a=0 then we can obtain the coresponding redshift factor in reissner - nordstrom geometry . * if we substitute q = p=0 , then we can obtain the coresponding redshift factor in kerr geometry . * if we substitute q = p=0 and a=0 , then we can obtain the coresponding redshift factor in schwarzschild geometry . * if we substitute a=0 , then we can obtain the coresponding redshift factor from a static body of same mass ( schwarzschild mass ) having static electric and magnetic charge present in the body . further for showing the physical significance of the calculations reported here , we have considered a pulsar psr j 1748 - 2446ad ( dubey and sen ( 2014 ) ; hessels et al . ( 2006 ) ) having schwarzschild radius ( @xmath85)=4.05 km , physical radius ( r)= 20.10 km , and rotation parameter ( a)=2.42 km . + + in fig . 1 , we make a plot using equation ( 35 ) , showing the variation of redshift ( @xmath86 ) with latitude ( @xmath3 ) , at different values of @xmath87 = 0 , 1.0 @xmath88 , 1.0 @xmath89 , 1.5 @xmath89 , 1.8 @xmath89 @xmath90 . there are some of arbitrary values of @xmath87 , permissible by equation ( 9 ) , such that @xmath91 . + + from fig . 1 , it is clearly seen that the value of gravitational redshift increases as the sum of square of electrostatic and magnetostatic charges ( @xmath92 ) increases . the amount of gravitational redshift also increases from pole to equatorial region ( maximum at equator ) at a fixed value of @xmath87 . + + below we apply the results of our calculations to one practical case . + + pulsars are rapidly rotating neutron stars and it is known that neutron stars have intense magnetic field . if we consider a rotating object having intense magnetic field ( such as pulsars ) , and electrostatic charge ( q ) equals to zero , then we proceed as follows : + from equation ( 25 ) , we can rewrite the radial component of magnetic field ( @xmath93 ) as : @xmath94\ ] ] after simplification the above equation ( 38 ) can be rewritten as : @xmath95 now we can write magneto static charge ( p ) in terms of radial component of magnetic field ( @xmath40 ) as : @xmath96 substituting electrostatic charge ( q ) equals to zero in the expression of redshift factor as given by equation ( 30 ) , we can write the expression of redshift factor ( @xmath97 ) of a rotating object having intense magnetic field as : @xmath98 @xmath99sin^{2}\theta}{(\frac{d\phi}{c dt}})^{2 } + 2 \frac{a sin^{2}\theta ( r_{g}r - p^{2})}{r^{2}+ a^{2}cos^{2}\theta}(\frac{d\phi}{c dt})}\ ] ] where @xmath100 can be obtained by substituting electrostatic charge ( q ) equals zero in equation ( 30 ) . @xmath101 the obtained expression ( 41 ) of redshift factor of a rotating object having intense magnetic field , also shows the dependence on latitude at which light ray has been emitted . + + in fig . 2 , we make a plot using equation ( 41 ) , showing the variation of redshift ( @xmath86 ) with latitude ( @xmath3 ) , at different values of magnetostatic field @xmath33 = 0 , 1.0 @xmath102 , 1.5 @xmath102 , 2.0 @xmath102 , 2.5 @xmath102 tesla , for a pulsar psr j 1748 - 2446ad . from fig . 2 , it is clearly seen that the value of gravitational redshift increases as the magnetostatic field @xmath33 increases . the amount of gravitational redshift also increases from pole to equatorial region ( maximum at equator ) at a fixed value of magnetostatic field @xmath33 . + + this work has high significance in the area of astrophysics research . there are many objects in nature , like neutron stars , magnetars etc which have high amount of rotation , electric and magnetic field . in general , sun - like stars have surface magnetic field . in addition one can expect a sun - like star to hold some amount of net electric charge ( @xmath103 e.s.u ) due to frequent escape of electrons than that of protons ( motz ( 1961 ) ) . thus calculations reported in this paper will have astrophysical significance . 1 . considering the three parameters : mass , rotation parameter and charge , the combined gravitational and electro - magneto - static redshift factor for a rotating body has been calculated by using kerr - newman geometry . 2 . the effect of electrostatic and magnetostatic charges on the amount of redshift of a light ray emitted at various latitudes from the rotating body has been calculated . 3 . under the boundary condition of zero electrostatic and magnetostatic charges , the calculated expression for gravitational redshift factor , reduces to the corresponding expression for gravitational redshift factor in kerr geometry . further if we consider the rotation velocity of the body to be zero , then we can obtain the corresponding gravitational redshift factor for a static body of same mass ( schwarzschild mass ) . 4 . under the boundary condition of zero magnetostatic charge and zero rotation velocity , the calculated expression for gravitational redshift factor , reduces to the corresponding expression for gravitational redshift factor in reissner - nordstrom geometry . further if we consider the electrostatic charge to be zero , then we can obtain the corresponding gravitational redshift factor for a static body of same mass ( schwarzschild mass ) . gravitational redshift increases as the electrostatic and magnetostatic charges increase , for a fixed value of latitude at which light ray has been emitted . 6 . gravitational redshift increases from pole to equatorial region ( maximum at equator ) , for a given set of values for electrostatic and magnetostatic charge . * acknowledgments * we wish to thank dr . atri deshmukhya , department of physics , assam university , silchar , india for inspiring discussions . finally we are thankful to the anonymous referee of this paper , for very useful comments . apsel , d. : international journal of theoretical physics * 17*(8 ) , 643 ( 1978 ) + + apsel , d. : general relativity and gravitation * 10*(4 ) , 297 ( 1979 ) + + carroll , s.m . : spacetime and geometry . an introduction to general relativity vol . 1 . pearson , addison wesley , p. 24 , 83 , 254 , 261 - 262 ( 2004 ) + + collas , p. , klein , d. : general relativity and gravitation * 36*(5 ) , 1197 ( 2004 ) + + dubey , a.k . , sen , a.k . : international journal of theoretical physics * 54*(7 ) , 2398 ( 2014 ) + + hessels , j.w . , ransom , s.m . , stairs , i.h . , freire , p.c . , kaspi , v.m . , camilo , f. : science * 311*(5769 ) , 1901 ( 2006 ) + + kerr , r.p . : phys . lett . * 11 * , 237 ( 1963 ) + + landau , l.d . , lifshitz , e.m . : the classical theory of fields vol . 2 . 4th revised edn . butterworth - heine- mann , indian reprint , p. 65 , 357 ( 2008 ) + + motz , l. : nature * 189 * , 994 ( 1961 ) + + newman , e.t . , couch , e. , chinnapared , k. , exton , a. , prakash , a. , torrence , r. : journal of mathematical physics * 6*(6 ) , 918 ( 1965 ) + + payandeh , f. , fathi , m. : international journal of theoretical physics * 52*(9 ) , 3313 ( 2013 ) + + wiltshire , d.l . , visser , m. , scott , s.m . : the kerr spacetime : rotating black holes in general relativity . cambridge university press , p. 8

it is well known fact that gravitational mass can alter the space time structure and gravitational redshift is one of its examples . static electric or magnetic charge can also alter the space time structure , similar to gravitational mass , giving rise to its effect on redshift . this can also be considered as electro and magneto static redshift . gravitational redshift has been reported by most of the authors without consideration of static electric and / or magnetic charges present in the rotating body . in the present paper , we considered the three parameters : mass , rotation parameter and charge to discuss their combined effect on redshift , for a charged rotating body by using kerr - newman metric . it has been found that , the presence of electrostatic and magnetostatic charge increases the value of so - called gravitational redshift . calculations have been also done here to determine the effect of electrostatic and magnetostatic charges on the amount of redshift of a light ray emitted at various latitudes from a charged rotating body . the variation of gravitational redshift from equatorial to non- equatorial region has been calculated , for a given set of values of electrostatic and magnetostatic charges .

in this paper we study the set @xmath3 where @xmath4 is an integer and @xmath5 denotes the fractional part . this set is related to badly approximable numbers in diophantine approximation , and has been studied by nilsson @xcite , who studied the hausdorff dimension of the set as a map of @xmath1 , and in more generality by urbanski @xcite who considered the orbit of an expanding map on the circle . as nilsson did we will consider @xmath6 as a subshift of finite type which enables us to see it as a problem in dynamical systems . when studied as a subshift of finite type we can find the dimension of @xmath6 using the spectral radius of the corresponding transition matrix , and this motivates the theorem of this paper which characterizes the characteristic polynomial of this matrix . the author would like to thank his phd supervisor simon kristensen and he would also like to than johan nilsson for reading and commenting on an early version of this paper . we begin with a definition of part and residue which comes from elementary integer division with residue . we let @xmath4 be an integer throughout the paper and start with a well known result . for integers @xmath7 and @xmath8 there are unique integers @xmath9 _ ( part ) _ and @xmath10 ( residue ) , with @xmath11 such that @xmath12 we note that if we write @xmath13 in base @xmath0 it is easy to find the part and the residue , since @xmath14 and @xmath15 . the matrix we will consider in this paper is defined as follows . for @xmath16 we define a 0 - 1 matrix @xmath17 of size @xmath18 by @xmath19 we let @xmath20 with @xmath21 be the @xmath22 matrix made from picking only the rows and columns from @xmath17 corresponding to the elements in @xmath23 and for @xmath24 we let @xmath25 be the @xmath26 matrix where we have removed the first @xmath27 rows and columns from @xmath17 . we will often omit the dependency on @xmath28 when it is not confusing . considering @xmath29 and @xmath30 in base @xmath0 we see that @xmath31 if and only if the first @xmath32 digits of @xmath33 are equal to the last @xmath32 digits of @xmath34 . so when @xmath35 we see that the base @xmath36 expansions of the numbers in @xmath6 can be seen as a subshift of finite type with transition matrix @xmath37 . the metric of the subshift and the unit interval are equivalent so the dimensional properties are the same . in particular , finding the hausdorff dimension of @xmath6 now boils down to finding the spectral radius @xmath38 , since @xmath39 for a proof of the first equality see @xcite . this is why we were interested in finding the characteristic polynomial of @xmath40 . the main theorem of this paper is a complete characterization of these polynomials . in order to state this theorem we need the following definition . for integers @xmath41 with @xmath42 we define @xmath43 using this definition we let @xmath44 be the _ minimal prefix _ of @xmath45 . this is well defined since @xmath46 for any @xmath45 with @xmath42 . the notion of minimal prefix is taken from nilsson @xcite , but is here defined somewhat differently since we only consider finite sequences . let us consider some examples . let @xmath47 . then @xmath48 so @xmath49 and @xmath50 if we let @xmath51 we have @xmath52 and @xmath53 but @xmath54 so @xmath55 and @xmath56 . we are now ready to state the main theorem . [ thm : char ] let @xmath29 be an integer such that @xmath57 and let @xmath58 be the characteristic polynomial of @xmath40 . then @xmath59 where @xmath60 and @xmath61 is the base @xmath0 expansion of @xmath62 . notice that this implies the equality @xmath63 and that @xmath64 where @xmath65 is the perron root of @xmath66 . first recall that we can find the characteristic polynomial @xmath67 of @xmath40 as @xmath68 where we also require that @xmath69 , or as @xmath70 the first formula is sometimes used as the definition of the characteristic polynomial , and for a proof of the latter see @xcite . we now try to outline the proof that essentially is the construction of an algorithm that calculates both the characteristic polynomial of @xmath40 and @xmath71 . * we prove that all the submatrices @xmath72 that give non - zero principal minors are permutations , so when removing rows and columns from the first to the last , we only change the characteristic polynomial when removing rows and columns corresponding to the smallest element of a cycle . * if @xmath73 then @xmath29 is the smallest element of an @xmath28-cycle and this is the only permutation of size @xmath74 that has @xmath29 as an element . so removing @xmath29 decreases the @xmath28th coefficient of the characteristic polynomial by @xmath75 and leaves all the preceding coefficients unchanged . on the other hand , if @xmath76 , then the nontrivial part of the characteristic polynomial , @xmath77 , can be found as @xmath78 since we have and can prove that @xmath79 for all @xmath80 . * if @xmath73 , then @xmath81 , and if @xmath76 then @xmath82 , so we see that @xmath83 and the characteristic polynomials follow the same pattern . * since the theorem is true for @xmath84 , we can now use induction if @xmath85 . if not , we increase @xmath29 until we have @xmath85 , which happens since @xmath86 . * the @xmath87st , @xmath88nd , , @xmath36th coefficient of @xmath58 are all zero , because we have found the first @xmath89 coefficients of the characteristic polynomial for any @xmath89 , so if we pick @xmath90 and @xmath91 such that @xmath92 and @xmath93 , then we see that @xmath94 has its @xmath87th , @xmath88th , , @xmath89th coefficients equal to zero , which will then also be true for @xmath77 . this finishes the proof of the theorem . @xmath95 \midrule l(i ) & 1 & 3 & 3 & 2 & 3 & 3 & 2 & 3 & 3 & 1 & 1 & 1 & 1 & 1 & 3 & 2 & 2 & 3 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \midrule a_{i+1,i+1 } & \mathbf{1 } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \mathbf{1 } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ a_{i+1,i+1}^2 & 1 & 0 & 0 & \mathbf{1 } & 0 & 0 & \mathbf{1 } & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & \mathbf{1 } & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & \mathbf{1 } \\ \bottomrule \end{array}\ ] ] the results in this sections explain some properties of the part and residue functions and gives a characterization of the powers of @xmath96 . we will use these results throughout the paper , often without specifically stating so . the proofs in this section are rather straightforward and may be skipped on a first read . 1 . for @xmath97 we have @xmath98 and @xmath99{\part{n , k } , j } = \part{n , k+j}.\ ] ] 2 . for @xmath100 we have @xmath101 let us first prove the two equalities in 1 . since @xmath102 is the same as @xmath103 we have the first equality . now assume that @xmath104 . now @xmath105{\part{n , k},j } + \res{\part{n , k},j}$ ] , so @xmath106{\part{n , k},j } + q^k\res{\part{n , k},j } + \res{n , k},\ ] ] but since @xmath107 and @xmath108 we have @xmath109 and by the uniqueness of the residue and parts we see that @xmath110 . now consider 2 . , so let @xmath111 . we have @xmath112{\part{n , k},j - k } + \res{\part{n , k},j - k } = q^{j - k } \part{n , j } + \res{\part{n , k},j - k}\ ] ] and @xmath113{\res{n , j},k } = q^k \part{\res{n , j},k } + \res{n , k}.\ ] ] so @xmath114 and since @xmath115 this implies that @xmath116 [ lem : powersofa ] let @xmath27 be an integer with @xmath117 . then @xmath118 if and only if @xmath119 we will prove this by induction . for @xmath120 it is the definition of @xmath96 , so assume that @xmath121 . we assume that the lemma is true for all smaller @xmath27 . if @xmath118 there must exist some @xmath45 with @xmath42 and @xmath122 and @xmath123 . using the induction hypothesis we get @xmath124 for this @xmath45 . now by part 2 . of the above proposition we have @xmath125 and using we get @xmath126{\res{i-1,m - k+1 } , m - k } = \part[\big]{\part{j-1,1},k-1},\ ] ] and using part 1 . of the proposition we get @xmath127 as desired . now assume that @xmath128 . let @xmath129 this is a positive integer smaller than @xmath36 . by the uniqueness of the residue and parts we see that @xmath130 and @xmath131 > from and the induction hypothesis we see that @xmath123 . we now want to prove that @xmath122 . recall that we assume @xmath128 , so @xmath132{\res{i-1,m - k+1},m - k } \\ & = \res{i-1,m - k } \\ & = \part{j-1,k}.\end{aligned}\ ] ] using this and we see that @xmath133{\res{n-1,m-1},k-1 } \\ & = q^{k-1 } \part{j-1,k } + \res{n-1,k-1 } \\ & = q^{k-1 } \part{j-1,k } + \res{\part{j-1,1},k-1 } \\ & = q^{k-1 } \part[\big]{\part{j-1,1},k-1 } + \res{\part{j-1,1},k-1 } \\ & = \part{j-1,1}.\end{aligned}\ ] ] this proves that @xmath123 and @xmath134 which implies that @xmath135 . now assume that there is another @xmath136 such that @xmath137 and @xmath138 . then @xmath139 and @xmath140 so @xmath141{\res{n'-1,m-1},k-1 } \\ & = q^{k-1 } \res{i-1,m - k+1 } + \res { \part{j-1,1},k-1 } \\ & = n-1,\end{aligned}\ ] ] which proves that there can be only one such @xmath45 , so @xmath118 . [ lem : ineqs ] if @xmath142 is such that @xmath143 and @xmath144 , then @xmath145 for all integers @xmath30 with @xmath146 . if @xmath144 then @xmath99{\part{a , k},j } = \part[\big]{\part{b , k},j},\ ] ] and hence @xmath147 since @xmath148 we thus have @xmath145 as desired . we now prove the following rather simple lemma which states that the only non - zero principal minors can be found as submatrices of @xmath96 which are permutations . [ lem : mustbeperm ] if @xmath149 then the corresponding matrix is a permutation matrix . assume that we choose @xmath23 such that one of the rows of @xmath72 has two ones . in other words there are @xmath150 such that @xmath151 using the definition of @xmath96 this implies that @xmath152 now let @xmath153 be arbitrary . then @xmath154 if and only if @xmath155 , which is true if and only if @xmath156 so @xmath157 for all @xmath153 , so the @xmath158th and @xmath159th column are equal and so @xmath160 . the proof is similar when we assume that there are two ones in one column . recall that if @xmath72 is a permutation , then @xmath161 where @xmath162 and @xmath163 s are all cycles . this motivates the following two theorems , where we characterize the subsets @xmath23 where @xmath72 is a cycle . we are interested in the smallest elements of cycles , since the whole cycle are removed when we remove this element , which we will prove is exactly the numbers that are minimal . we say that an integer @xmath45 with @xmath164 is @xmath28-minimal if @xmath165 or equivalently using lemma [ lem : powersofa ] if @xmath166 let @xmath167 be such that @xmath72 is a @xmath27-cycle for some @xmath117 . then @xmath168 is minimal with @xmath169 . let @xmath170 be a @xmath27-cycle with @xmath171 for @xmath172 and @xmath173 . without loss of generality we can assume that @xmath174 . using lemma [ lem : powersofa ] we get that @xmath175 for @xmath172 and @xmath176 so we need to prove that @xmath177 for @xmath178 . we have the non - strict inequality since @xmath179 . so assume for contradiction that @xmath180 now since @xmath181 we have @xmath182 and due to lemma [ lem : ineqs ] we have @xmath183 since @xmath80 . since @xmath184 we have @xmath185 . using we get @xmath186 now consider @xmath187 . since @xmath188 we have @xmath189 so @xmath190 and hence @xmath191 this implies that @xmath192 which is a contradiction against @xmath193 being the least element in @xmath23 . [ thm : uniquecycle ] assume that @xmath34 is minimal . then there is a unique @xmath21 such that @xmath194 and @xmath72 is a @xmath195-cycle . we let @xmath196 where @xmath197{\vdots}}\\ i_k-1 & = q^{k-1 } \res{i-1,m - k+1 } + \part{i-1,m - k+1}.\end{aligned}\ ] ] we now need to prove that @xmath198 and that @xmath199 for all @xmath200 . using the uniqueness of the part and residue we see that @xmath201 and @xmath202 for @xmath200 . the first of these equations implies that @xmath198 . since @xmath203 we know that @xmath204 for @xmath205 . this implies that @xmath206 since both @xmath207 and @xmath208 are smaller than @xmath209 . we now need to prove that this @xmath23 is unique . assume that we have @xmath210 , where we order the elements such that @xmath211 . this implies that @xmath212 for all @xmath200 . since @xmath72 is a @xmath27-cycle , we furthermore know that @xmath213 , so @xmath214 now we want to prove that @xmath215 , so let @xmath216 be given . we have @xmath217 and @xmath218 , so we just need to prove that @xmath219 we have @xmath220{\res{i_n'-1,m - k+n-1 } , n-1 } \\ & = \res{\part{i-1,k - n+1 } , n-1 } \\ & = \res[\big]{\res{i_n-1,m - k+n-1 } , n-1 } \\ & = \res{i_n-1 , n-1 } \\ & = \part{i-1,m - n+1}\end{aligned}\ ] ] so @xmath221 for all @xmath45 , and so @xmath222 . if @xmath223 then there is exactly one @xmath21 such that @xmath224 and @xmath72 is a @xmath28-cycle . this follows from the fact that @xmath225 for all @xmath226 . in particular we have @xmath227 for all @xmath29 . now compare this corollary with the following lemma . [ lem : minimaldecrease ] if @xmath228 , then @xmath229 . it is enough to prove that @xmath230 , since we certainly have @xmath231 . using the definition we see that this is equivalent to @xmath232 . if @xmath233 we are done , so assume that @xmath234 . now either @xmath232 , in which case we are done , or @xmath235 . now since @xmath236 we have @xmath237 since @xmath238 , but @xmath239 which is a contradiction . recalling the idea of the proof we here see that if @xmath223 and we remove the @xmath29th row and column of @xmath17 , then we remove exactly one permutation of size @xmath74 , namely an @xmath28-cycle , which increases the @xmath28th coefficient of the characteristic polynomial by one , and we also see that it increases the @xmath28th digit of the base @xmath0 expansion of @xmath83 by one . in the following chapter we will no longer suppress the dependency on @xmath28 , since we are interested in mapping permutations between matrices of different sizes while preserving cycles . we will illustrate the idea with an example . if @xmath240 , and we write all numbers in base @xmath241 we see that @xmath242 is a @xmath241-cycle in @xmath243 . we now map this up to @xmath244 which is a @xmath241-cycle in @xmath245 . on the other hand we could also map down to @xmath246 which is a @xmath241-permutation in @xmath247 . in this section we will formally define these maps , and also prove that they map cycles to cycles . we begin with the ` down ' map which is defined in the following way . for an integer @xmath29 with @xmath248 we define @xmath249 if @xmath90 and @xmath250 we let @xmath251 we now prove the following lemma . if we for integers @xmath252 have @xmath253 , then @xmath254 we have @xmath255 and @xmath256 for all @xmath257 , and we need to prove that @xmath258 for all @xmath259 . but this is clearly the case since @xmath260 , so @xmath261 for all @xmath257 . [ cor : recmin ] let @xmath29 be an integer with @xmath262 . if @xmath253 , then @xmath263 this follows from the definition of the minimal prefix . we saw earlier that the characteristic polynomial of a matrix can be found by considering the trace of the powers of the matrix . so if we can map permutations bijectively between two transition matrices we must have the same characteristic polynomials . as before we only need to consider cycles as all permutations are products of cycles . an ordered @xmath27-tuple of distinct elements , @xmath264 with @xmath265 for all @xmath266 is a @xmath27-cycle in @xmath267 if @xmath268 for all @xmath269 , and @xmath270 . in other words , if we have @xmath271 for @xmath269 and @xmath272 and @xmath273 for all @xmath266 . we have a ` down ' map , mapping from large matrices to smaller and we now define an ` up ' map , mapping from smaller to larger . let @xmath274 be a @xmath27-cycle in @xmath267 . then we let @xmath275 and for @xmath90 we let @xmath276 . let @xmath277 and let @xmath278 be a @xmath27-cycle in @xmath279 . then @xmath280 is a @xmath27-cycle in @xmath281 . furthermore , if @xmath282 is a @xmath27-cycle in @xmath281 , then @xmath283 is a @xmath27-cycle in @xmath279 . to prove that @xmath284 is a @xmath27-cycle in @xmath281 can be done by straightforward calculations . we also get that @xmath283 is a @xmath27-cycle in @xmath285 rather straightforward . the problem is to prove that it actually is a @xmath27-cycle in @xmath279 , or in other words that there are no @xmath27-cycles with their smallest element in the interval between @xmath286 and @xmath1 . recalling the definition of @xmath287 and that the least element of a cycle always is minimal we thus need to prove that if we have @xmath288 , then @xmath45 can not be minimal . we get that @xmath289 and @xmath290 so @xmath291 so if we assume that @xmath45 is minimal we get @xmath292 which is a contradiction . this finishes the proof of the theorem . these two lemmas now lead to the following theorem regarding the invariance of the traces . [ thm : powertrace ] let @xmath293 . then @xmath294 more generally we have @xmath295 whenever @xmath296 . each @xmath27-cycle contributes to the trace , and since the maps used in the lemmas map all @xmath27-cycles injectively , we get the theorem . newton s formula for the characteristic polynomial gives us , that if @xmath297 is the characteristic polynomial of @xmath25 where @xmath298 , then @xmath299 so the above theorem gives us that @xmath300 combining this with the simple lemma below gives us the proof of the main theorem . let @xmath45 be an integer with @xmath42 . then @xmath301 we see that @xmath302 so we just need to prove that @xmath303 . assume that @xmath304 . then @xmath305{\part{qn , j},1 } = q \part{qn , j+1 } = q \part{n , j } \geq q \res{n , m - j } = \res{qn , m}.\ ] ] now assume that @xmath306 for some @xmath307 . then @xmath308 so @xmath309 which is a contradiction . we are now ready to prove the main theorem . we prove this theorem using induction . if @xmath84 it is certainly true since @xmath310 for all @xmath29 with @xmath311 and @xmath312 is the all one matrix of size @xmath313 . we see that when choosing @xmath28 and @xmath314 we have two possibilities : either we have @xmath236 or @xmath315 . in the first case removing the @xmath29th column and row only removes one non - zero minor , namely the unique @xmath28-cycle with @xmath29 as its minimal element given in theorem [ thm : uniquecycle ] . in this case we also have that the last digit of @xmath316 is @xmath317 which must be non - zero , so here we just decrease @xmath318 with @xmath75 , so the first @xmath28 coefficients of the characteristic polynomial changes in the right way due to lemma [ lem : minimaldecrease ] . if we have @xmath319 we see that we can find the characteristic polynomial of the smaller matrix of size @xmath209 instead and multiply it by @xmath320 . as we see in corrolary [ cor : recmin ] this is also the case for @xmath321 . so by induction we are done . now we need to prove that the remaining coefficients are all zero . to prove this we once again use lemma [ thm : powertrace ] to see that the @xmath89th coefficient of @xmath322 must be equal to the @xmath89th coefficient of @xmath323 for any @xmath90 . and here we see that the @xmath87th , @xmath88th , , and @xmath89th coefficient all are zero , since the @xmath89th digit of the base @xmath0 expansion of @xmath324 is zero . this finishes the proof of the theorem . now define @xmath325 . recall from that when @xmath1 has finite base @xmath0 expansion we can calculate @xmath326 . nilsson @xcite proved that this function is continuous and constant almost everywhere . using the theorem we see that if we have @xmath327 such that @xmath328 then @xmath329 and since @xmath330 is a decreasing function it must be constant on the interval @xmath331.\ ] ] now let @xmath311 be given and let @xmath332 we now claim that @xmath333 to prove this we see that @xmath334 and so @xmath335 . now @xmath336 and @xmath337 which proves the claim . this gives us @xmath338 for all @xmath28 and letting @xmath339 we get that @xmath330 is constant on the interval @xmath340.\ ] ] now letting @xmath84 we find @xmath341 which has one root , @xmath342 , so we get @xmath343 on this interval . a bit more work allows us to calculate @xmath344 for @xmath345 for larger @xmath45 since we here need to solve polynomial equations of degree @xmath45 . calculating the spectral radii of @xmath346 , we can make numerical plots of the function @xmath330 . the plot in figure [ fig:2357 ] was made using gnu octave . for @xmath347 . ] we now want to consider @xmath330 as @xmath2 . we consider the function @xmath348 where @xmath349 and wish to prove that @xmath330 and @xmath350 are somewhat asymptotically similar . this can also be expressed by saying that @xmath351 behaves somewhat like @xmath352 , which is true in the starting point of the intervals where @xmath330 is constant , so we get the following theorem . let @xmath354 be given . then if we let @xmath356 we have @xmath357 now @xmath358 and likewise for @xmath350 since both functions are decreasing . due to the result we got earlier on constant intervals we have @xmath359 so recalling the definition of @xmath29 we have @xmath360 and since @xmath361 as @xmath2 , both the lower and upper bound converges to @xmath75 . this finishes the proof . www d. k. faddeev and v. n. faddeeva , _ computational methods of linear algebra _ , w.h . freeman and company , 1963 . j. nilsson , on numbers badly approximable by dyadic rationals , _ israel journal of mathematics _ 171 ( 2009 ) , pp . y. pesin , _ dimension theory in dynamical systems _ , the university of chicago press , 1997 . m. urbanski , on hausdorff dimension of invariant sets for expanding maps of the circle , _ ergorid theory and dynamical systems _ 6 ( 1986 ) , pp 295309 .

in this paper we consider the times-@xmath0 map on the unit interval as a subshift of finite type by identifying each number with its base @xmath0 expansion , and we study certain non - dense orbits of this system where no element of the orbit is smaller than some fixed parameter @xmath1 . the hausdorff dimension of these orbits can be calculated using the spectral radius of the transition matrix of the corresponding subshift , and using simple methods based on euclidean division in the integers , we completely characterize the characteristic polynomials of these matrices as well as give the value of the spectral radius for certain values of @xmath1 . it is known through work of urbanski and nilsson that the hausdorff dimension of the orbits mentioned above as a map of @xmath1 is continuous and constant almost everywhere , and as a new result we give some asymptotic results on how this map behaves as @xmath2 .

instabilities of the normal metallic state lead to a rich variety of quantum phase transitions in interacting electron systems . near a quantum critical point electronic excitations are strongly scattered by order parameter fluctuations such that fermi liquid theory breaks down.@xcite the fluctuation effects and the ensuing non - fermi liquid behavior is particularly pronounced in two - dimensional systems . it is therefore not surprising that quantum critical fluctuations are frequently invoked as a mechanism for the enigmatic strange metal behavior observed in cuprate superconductors and other layered correlated electron compounds . theoretical works have mostly focused on the quantum critical point and its extension into the quantum critical regime at finite temperature , where quantum fluctuations are particularly important . less attention has been paid to the _ ginzburg region _ near the critical temperature , which is characterized by strongly interacting classical order parameter fluctuations.@xcite this is somewhat unwarranted since classical critical fluctuations also affect electronic excitations very strongly . in two dimensions they lead to a contribution of the order @xmath2 to the quasi - particle decay rate , where @xmath3 is the diverging correlation length.@xcite in this paper we compute the size of the ginzburg region above the critical temperature near a quantum critical point in two - dimensional metals . more specifically , we consider continuous quantum phase transitions associated with the spontaneous breaking of a _ discrete _ symmetry , described by an effective hertz action @xcite for a scalar order parameter with dynamical exponents @xmath4 or @xmath5 . we calculate the transition temperature @xmath0 as a function of the non - thermal control parameter for the quantum phase transition , as well as the ginzburg temperature @xmath1 above @xmath0 . the size of the ginzburg region @xmath6 is determined to leading order in the distance from the quantum critical point . the dependence of the ginzburg temperature on the control parameter was derived already by millis.@xcite however , that study did not access the ginzburg region between @xmath1 and @xmath0 . a comprehensive analysis of all finite temperature transition and crossover lines in dimensions @xmath7 was performed by sachdev . @xcite for discrete symmetry breaking in two dimensions the critical temperature @xmath0 and the ginzburg temperature below @xmath0 were recently compared within a renormalization group study which allowed to approach the finite temperature transition.@xcite however , in that work the flow equations were solved only numerically , while we now present analytic results . the paper is organized as follows . in section ii we derive the renormalization group equations for the effective hertz action . these are solved analytically in an approximate form in section iii . in section iv we discuss the results for @xmath0 , @xmath1 and the size of the ginzburg region , before concluding in section v. our analysis is based on the hertz action @xcite @xmath8 & = & \frac{t}{2 } \sum_{\om_n } \int \frac{d^dq}{(2\pi)^d } \phi_{\bq,\om_n } \left ( \delta_0 + \bq^2 + \frac{|\omega_{n}|}{|\bq|^{z-2 } } \right ) \phi_{-\bq,\om_n } \nonumber\\ & & + \frac{u_0}{4 ! } \int_0^{\frac{1}{t } } \ ! d\tau \int d^d x \ , \phi^4(\bx,\tau ) \ ; , \label{action}\end{aligned}\ ] ] where @xmath9 is a real scalar order parameter field and @xmath10 its momentum representation ; @xmath11 with integer @xmath12 denotes the bosonic matsubara frequencies . for the dynamical exponent @xmath13 we consider the cases @xmath4 , which describes density wave transitions , and @xmath5 , relevant for a nematic transition or ising - type ferromagnetic transitions . we do not address the issue under which circumstances the hertz action provides a faithful description of quantum criticality in two - dimensional metals.@xcite before embarking on the renormalization group approach , we would like to emphasize that in two dimensions @xmath14 can not be obtained from a first order expansion in the quartic coupling @xmath15 , even if it is weak and irrelevant at the quantum critical point . to leading order in @xmath15 , the inverse susceptibility @xmath16 is given by @xmath17 where @xmath18 is a positive constant . at finite temperature the matsubara frequencies are discrete and the classical fluctuation contribution from @xmath19 diverges logarithmically in the limit @xmath20 in two dimensions . trying to treat this divergence by a self - consistent equation , replacing @xmath21 by @xmath16 under the integral , one finds that the transition temperature @xmath0 is suppressed to zero at the critical point given by @xmath22 , irrespective of @xmath21 . this behavior is reminiscent of the mermin - wagner theorem , which excludes spontaneous breaking of a continuous symmetry in two dimensions . however , the above first order calculation is essentially independent of the symmetry of the order parameter , and is therefore misleading at least in the case of a discrete symmetry . we solve the problem by using flow equations which describe the renormalization of the inverse susceptibility ( or mass ) @xmath16 and the quartic coupling @xmath23 due to fluctuations . the flow equations are derived from an approximate ansatz for the exact effective action @xmath24 $ ] , that is , the generating functional for vertex functions in the presence of an infrared cutoff @xmath25.@xcite the cutoff is implemented by adding a regulator term of the form @xmath26 to the bare action @xmath27 $ ] . the exact flow of @xmath24 $ ] is given by the wetterich equation @xcite @xmath28 = \frac{1}{2 } { \rm tr } \frac{\partial_{\lam } r^{\lam}}{\gam^{(2)\lam}[\phi ] + r^{\lam } } \ ; , \label{flow_exact}\ ] ] where @xmath29 $ ] is the matrix of second derivatives of @xmath24 $ ] with respect to @xmath30 and the trace sums over momenta and frequencies . we approximate @xmath24 $ ] by an ansatz of the form eq . ( [ action ] ) with a renormalized mass term @xmath31 and a renormalized coupling @xmath32 . inserting this ansatz in the exact flow equation for @xmath24 $ ] and comparing coefficients , one obtains @xcite @xmath33 ^ 2 } \ ; , \label{flow_delta}\ ] ] @xmath34 ^ 3 } \ ; . \label{flow_u}\ ] ] the initial conditions for the flow are @xmath35 and @xmath36 , where @xmath37 is a ( fixed ) ultraviolet cutoff . as a regulator we choose the litim @xcite function @xmath38 , with derivative @xmath39 , which restricts the momentum integrals in the flow equations to @xmath40 and replaces the @xmath41-term in the denominators by @xmath42 . from now on we fix the dimensionality to @xmath43@xmath44@xmath45 . the matsubara sums in the above flow equations can be expressed in terms of polygamma functions @xmath46 , defined as the @xmath12-th derivative of the digamma function @xmath47 . explicit @xmath25-dependencies can be removed from the right hand side of the flow equations as usual by introducing rescaled dimensionless variables @xmath48 one then obtains @xmath49 \ ; , \label{flow_tdelta}\ ] ] @xmath50 \ ; , \label{flow_tu}\ ] ] where @xmath51 . the first term in each equation is due to the factor @xmath52 in the definition of the dimensionless variables , the second one captures classical fluctuations ( @xmath19 ) , and the third one quantum fluctuations ( @xmath53 ) . for sufficiently small but finite temperature the flow passes through two distinct regimes , which are distinguished by the size of the rescaled temperature @xmath54 . initially one has @xmath55 , such that quantum fluctuations dominate , while in the final stage , for @xmath56 , the flow is governed by classical fluctuations . in the latter regime the third term on the right hand side of the flow equations ( [ flow_tdelta ] ) and ( [ flow_tu ] ) can be neglected . for @xmath55 one can use the expansion of the polygamma functions for large arguments , @xmath57 and @xmath58 , to approximate the integrals in eqs . ( [ flow_tdelta ] ) and ( [ flow_tu ] ) as @xmath59 & \approx&\!\!\ ! { \phantom - } \frac{1}{z(1+\tdelta ) \tt } \ ; , \label{int_tdelta } \\ \frac{1}{\tt^3 } \!\!\int_0 ^ 1 \!\!d\tq \ , \tq^{3z-5 } \ , \psi_2[h(\tdelta,\tq,\tt ) ] & \approx&\!\!\ ! \frac{-1}{z(1+\tdelta)^2 \tt } \ ; . \label{int_tu}\end{aligned}\ ] ] the cutoff scale corresponding to @xmath60 is given by @xmath61 . following millis @xcite we approximate the flow by its quantum contribution with the expansion eqs . ( [ int_tdelta ] ) and ( [ int_tu ] ) for @xmath62 , and we discard the quantum terms for @xmath63 . the flow equations ( [ flow_tdelta ] ) and ( [ flow_tu ] ) exhibit a fixed point at @xmath64 , @xmath65 , and @xmath66 . this fixed point is approached by the flow at the finite temperature phase transition , and it describes ( approximately ) the classical non - gaussian critical fluctuations at @xmath0 . there @xmath67 is small during the entire flow . above @xmath0 it remains small until the fluctuation contributions to the flow saturate at small @xmath25 . we therefore approximate @xmath68 in the denominators of the flow equations , which allows us to solve them analytically . note that within this approximation fluctuation contributions are symmetric under @xmath69 , while in the exact flow they are larger for @xmath70 compared to @xmath71 , which leads to a suppression of @xmath0 . in the quantum regime ( @xmath62 ) the approximate flow equations have the form @xmath72 recall that @xmath73 is also a flowing quantity . the explicit solution for @xmath13@xmath44@xmath74 reads @xmath75 where @xmath76 the integration constants @xmath77 and @xmath78 are determined by the initial conditions at @xmath37 as @xmath79 and @xmath80 . the solution for @xmath4 is given by @xmath81 where @xmath82 . the integration constants @xmath83 and @xmath84 are determined by the initial conditions : @xmath85 and @xmath86 . in the classical regime ( @xmath63 ) the approximate flow equations read @xmath87 the explicit solution has the form @xmath88 the integration constants @xmath89 and @xmath90 are determined by the boundary conditions @xmath91 and @xmath92 at the scale @xmath93 , yielding @xmath94 and @xmath95 . at @xmath96@xmath44@xmath97 one has @xmath93@xmath44@xmath97 and the flow can be obtained by taking the zero temperature limit of the solution in the quantum regime . for @xmath5 the unscaled quartic coupling @xmath32 saturates at the finite value @xmath98 for @xmath99 . note that the rescaled variable @xmath100 vanishes at @xmath101 . for a generic choice of @xmath21 the inverse susceptibility @xmath31 scales to a finite value near @xmath21 . at the quantum critical point , @xmath102 the inverse susceptibility scales to zero for @xmath99 . for @xmath4 the quartic coupling @xmath32 vanishes logarithmically for @xmath99 . the inverse susceptibility remains generically finite , except at the quantum critical point given by @xmath103 the phase transition line in the @xmath104 phase diagram is determined by the condition @xmath105 for @xmath99 . using the solution for @xmath31 in the classical regime , eq . ( [ tdelta_c ] ) , this yields a condition on the integration constants @xmath89 and @xmath90 , namely @xmath106 . the constants @xmath89 and @xmath90 can be expressed in terms of the bare variables @xmath21 and @xmath15 by matching the initial condition for the classical flow at @xmath93 to the solution of the flow in the quantum regime . for @xmath5 , one obtains @xmath107}{4c(t ) } \ ; , \label{tc_z3}\ ] ] where @xmath108 . expanding for small temperatures @xmath96 yields @xmath109 with @xmath110 . note that dependencies on the ultraviolet cutoff @xmath37 are absorbed in @xmath111 and @xmath112 . inverting eq . ( [ tc_z3_exp ] ) to leading order in @xmath96 yields @xmath113 with @xmath114 . for @xmath4 , we find @xmath115}{4 c(t ) } , \label{tc_z2}\ ] ] where @xmath116 with @xmath117 . for low temperatures this becomes @xmath118}{\ln(t_1/t ) } + \frac{2\pi}{3}\frac{t}{\ln(t_1/t ) } + \co\left ( t/ ( \ln t)^2 \right ) . \label{tc_z2_exp}\ ] ] note that the dependence on @xmath15 enters only via the scale @xmath119 . inverting eq . ( [ tc_z2_exp ] ) for small @xmath96 yields @xmath120 } \ ; , \ ] ] with @xmath121 . the functional form of the leading temperature dependencies in eqs . ( [ tc_z3_exp ] ) and ( [ tc_z2_exp ] ) is consistent with a numerical solution of the full flow equations ( [ flow_tdelta ] ) and ( [ flow_tu ] ) . however , the prefactors are not reproduced exactly , since they are affected by the approximations required for the sake of an analytic solution . in fig . 1 we show results for the critical line as obtained from eq . ( [ tc_z3 ] ) for @xmath5 and eq . ( [ tc_z2 ] ) for @xmath4 for specific choice of parameters . for @xmath5 the logarithmic correction can be clearly seen , whereas for @xmath4 the ( partially compensating ) logarithmic corrections are hardly visible , such that the critical line looks almost linear . and ginzburg temperature @xmath1 as a function of @xmath122 for @xmath5 ( a ) and @xmath4 ( b ) . the bare @xmath123-coupling is @xmath124 , and the ultraviolet cutoff @xmath125 in all cases . the parameter specifying the ginzburg criterion has been chosen as @xmath126 . the ginzburg temperature below @xmath0 has not been computed here , but has been added schematically for completeness.,scaledwidth=45.0% ] we now compute the ginzburg line in the phase diagram , which marks the boundary of the non - gaussian classical fluctuation regime above @xmath0 . scaling in the critical regime is governed by the interacting wilson - fisher fixed point @xmath127 . we determine the ginzburg line from the condition @xmath128 with @xmath129 , where @xmath130 is the scale at which @xmath131 crosses zero , that is , @xmath132 . the parameter @xmath133 is a measure for the closeness of @xmath100 to the fixed point . this criterion comes about as follows . close to criticality @xmath131 is negative at the beginning of the flow , but for @xmath134 it eventually increases and diverges with the trivial scaling dimension ( see eq . ( [ tdelta_flowapprox ] ) ) . if @xmath131 crosses zero whilst @xmath100 is still small , @xmath100 has little influence on the flow of @xmath131 also for @xmath135 such that we are in the gaussian regime with mean - field exponents . on the other hand , if @xmath100 has reached a value close to its fixed point at @xmath136 , it affects the flow of @xmath16 substantially leading to non - gaussian scaling . there is no unique choice of @xmath133 quantifying the `` closeness '' to the fixed point . this reflects the fact that the ginzburg line marks a crossover regime and not a sharp transition . the leading low-@xmath96 behavior of @xmath137 computed from the ginzburg criterion described above turns out to be the same as that for @xmath138 , with the same prefactor , irrespective of the choice of @xmath133 . however , differences appear in the first subleading term . for @xmath5 , one obtains @xmath139 and for @xmath4 , @xmath140 at low temperatures . we have used the fixed point value @xmath141 as deduced from eq . ( [ tu_flowapprox ] ) . note that the terms on the right hand sides are positive . solving for @xmath6 as a function of @xmath142 , one finds that @xmath143 is of order @xmath144 for @xmath5 , and of order @xmath145^{-1}$ ] for @xmath4 . hence , the size of the ginzburg region @xmath6 is practically of the order @xmath0 near the quantum critical point . by contrast , in three dimensions it is of order @xmath146.@xcite the results for @xmath147 are plotted in fig . [ fig1 ] for the same choice of parameters as above . we can see a substantial ginzburg regime opening between @xmath1 and @xmath0 . the @xmath0- and @xmath1-lines merge when the quantum critical point is approached , @xmath148 , since that critical point is gaussian . we have derived analytic expressions for the transition temperature @xmath0 and the ginzburg temperature @xmath1 above @xmath0 as a function of the non - thermal control parameter @xmath21 near a quantum critical point with a scalar ( ising universality class ) order parameter in a two dimensional metal . the calculations are based on flow equations derived from a perturbative renormalization group for the hertz model . the renormalization of the quartic coupling is crucial to avoid an artificial suppression of @xmath0 to zero in two dimensions . both @xmath0 and @xmath1 are essentially proportional to @xmath122 , with logarithmic corrections depending on the dynamical exponent @xmath13 . for @xmath1 we confirm the results by millis.@xcite for @xmath0 we obtain the same logarithmic corrections as for @xmath1 , in agreement with earlier evidence from a numerical solution of flow equations for the symmetry broken phase.@xcite nevertheless , the size of the ginzburg region @xmath6 , which has been calculated analytically for the first time in this paper , is practically proportional to the distance to the the quantum critical point , @xmath122 . hence , the ginzburg region with its large non - gaussian classical fluctuations covers a substantial part of the phase diagram near a continuous quantum phase transition in two dimensional metals . electronic excitations are strongly scattered by order parameter fluctuations in that region , which can lead to enhanced decay rates , pseudogaps , and other unconventional electronic properties . we are grateful to so takei and hiroyuki yamase for valuable discussions , and to nils hasselmann for useful comments and a critical reading of the manuscript . this work was supported by the german research foundation through the research group for 723 . for a discussion of this point , see , for example , ar . abanov and a. v. chubukov , phys . * 93 * , 255702 ( 2004 ) ; d. belitz , t. r. kirkpatrick , and t. vojta , rev . mod . phys . * 77 * , 579 ( 2005 ) ; h. v. lhneysen , a. rosch , m. vojta , and p. wlfle , rev . phys . * 79 * , 1015 ( 2007 ) ; m. a. metlitski and s. sachdev , phys . b * 82 * , 075127 ( 2010 ) , _ ibid _ * 82 * , 075128 ( 2010 ) .

we compute the transition temperature @xmath0 and the ginzburg temperature @xmath1 above @xmath0 near a quantum critical point at the boundary of an ordered phase with a broken discrete symmetry in a two - dimensional metallic electron system . our calculation is based on a renormalization group analysis of the hertz action with a scalar order parameter . we provide analytic expressions for @xmath0 and @xmath1 as a function of the non - thermal control parameter for the quantum phase transition , including logarithmic corrections . the ginzburg regime between @xmath0 and @xmath1 occupies a sizable part of the phase diagram .

caldern s inverse boundary value problem asks whether the cauchy data at the boundary of an elliptic second order pseudo - differential operator @xmath7 determine the coefficients @xmath8 . it has been solved in a great deal of generality on euclidean domains under the assumption of scalar coefficients @xmath9 @xcite . the problem becomes significantly more difficult when the coefficients @xmath8 are assumed to be anisotropic , that is given by a symmetric positive - definite tensor - field . nonetheless there is a very strong suggestion of general uniqueness provided by the proof in the case of real - analytic conductivities @xcite . the concept of limiting carleman weights was impressively used to prove uniqueness of the caldern problem for metrics in a conformal class , provided the class admitted such a weight @xcite , but the condition is rather limiting on the geometry and topology of the spaces under consideration @xcite,@xcite . in the present paper i show that caldern s problem on a closed manifold with boundary @xmath10 can be reduced to studying a related unique continuation problem on the space @xmath11 $ ] . i introduce the following unique continuation property : a second order differential operator @xmath12)\to c^{\infty}({\partial}{\mathcal{m}}^{2}\times[0,1])$ ] is said to have the _ off diagonal unique continuation property _ ( oducp ) if for any @xmath13)$ ] @xmath14 on @xmath15 $ ] , @xmath16 and @xmath17 implies that @xmath18 equals @xmath19 on @xmath20 $ ] . given a @xmath21-smooth compact manifold with boundary @xmath10 , endowed with a @xmath22 smooth riemannian metric @xmath23 , we can consider the dirichlet neumann map @xmath24 which takes a function @xmath25 and maps it to @xmath26 , where @xmath18 is the solution to the boundary value problem @xmath27 for @xmath28 consider the associated problem for a scalar multiple of @xmath23 , @xmath29 where @xmath30 is a smooth function bounded away from zero . this can be identified with the boundary value problem for a schrdinger equation @xcite . if @xmath31 , then setting @xmath32 the function @xmath33 solves the boundary value problem @xmath34 where @xmath35 . consequently the dirichlet neumann map associated to @xmath29 can be equated with that for the schrdinger equation @xmath36 consequently given two weights @xmath37 and @xmath38 , they define the same dirichlet neumann maps if and only if the dirichlet neumann maps for the associated schrdinger equations are equal , and the weights are equal at the boundary . let @xmath39 and @xmath40 denote the associated potentials , and let @xmath41 and @xmath42 denote the associated dirichlet neumann maps . given a riemannian manifold with boundary and @xmath43 metric tensor , we can define fermi or boundary normal coordinates via the map @xmath44\to { \mathcal{m}}$ ] @xmath45 , where @xmath46 is the inward normal oriented geodesic starting at @xmath47 at time @xmath5 . for @xmath48 sufficiently small this is a diffeomorphism onto its image @xcite . in these coordinates , the metric @xmath23 takes the special form @xmath49 where @xmath50 is a metric tensor on @xmath51 . we let @xmath52 . then we can define a family of operators @xmath4 for @xmath53 $ ] to be the associated dirichlet neumann operators for the operators @xmath54 restricted to the submanifold @xmath55 . now we can define a metric on @xmath11=$ ] by @xmath56 the associated volume form is given by @xmath57 for @xmath58 . with this formalism we are able to introduce our associated operator @xmath59\to c^{\infty}({\partial}{\mathcal{m}}^{2}\times[0,\varepsilon])$ ] for the caldern problem : @xmath60 we call @xmath61 the evolution squared operator , because it arises as the product of an evolution operator and it s adjoint ( _ cf _ [ sec : evo_squared ] ) . we are thus able to condition uniqueness for the caldern problem on the oducp of @xmath61 : [ thm : uniq_cont_loc_cald]let @xmath61 be the operator defined in . if @xmath61 has the oducp , then @xmath62 if and only if @xmath63 on @xmath64 $ ] and @xmath65 . the proof of theorem [ thm : uniq_cont_loc_cald ] follows by constructing a solution to a boundary value problem for @xmath61 on @xmath66 $ ] @xmath67\varphi(\varepsilon)=\lambda^{2}_{\varepsilon}-\lambda^{1}_{\varepsilon } \label{eqn : a_bvp}\end{aligned}\ ] ] where @xmath68 is the measure supported on @xmath69 $ ] given by @xmath70}f(x , x , t)[q^{1}(t , x)-q^{2}(t , x)]\:d{\mathrm{vol}}_{h_{t}}(x)\:dt.\ ] ] this special solution will have @xmath5-derivative at @xmath71 equal to @xmath72 as a distribution on @xmath73 . consequently it follows from the oducp that if @xmath74 then @xmath75 . however , elementary arguments can be used to show that if @xmath76 then @xmath77 on @xmath78 $ ] . subsequently using the existence of an exhaustion ( lemma [ lem : exists_exhaustion ] ) this can be turned into the following global contrapositive of the preceding theorem . [ thm : uniq_cont_impl_cald_uniq ] suppose @xmath62 , then @xmath76 only if there is a collar neighbourhood @xmath79 of @xmath51 such that the operator @xmath61 associated to some fermi coordinates of the complement of @xmath79 does not have the oducp . we work on a @xmath80-smooth compact manifold @xmath10 with boundary , and assume it has a @xmath81-smooth metric @xmath23 . the necessity of such a high degree of smoothness arises from fermi coordinates . let @xmath82 be the mapping taking @xmath83 to the inward normal oriented geodesic starting at @xmath47 at time @xmath5 . @xmath84 is @xmath85 diffeomorphism for a @xmath81 metric . consequently the pullback metric @xmath86 on @xmath64 $ ] is @xmath87 . under the diffeomorphism the metric takes the form @xmath88 where @xmath50 is a metric tensor on @xmath51 @xcite . let @xmath89 denote @xmath90 . we define sobolev spaces on our manifolds via a smooth ( that is as smooth as the manifold allows ) partition of unity @xmath91 , supported on a set @xmath92 with a coordinate chart @xmath93 . a measurable function @xmath18 is in @xmath94 if @xmath95 is in @xmath96 for every @xmath97 . if @xmath10 is @xmath81 then this is valid for @xmath98 $ ] . for @xmath99 $ ] we can define the spaces @xmath100 to be the space of measurable @xmath101-forms @xmath102 for which @xmath103 . we define the spaces @xmath104 and @xmath105 similarly . we define the space @xmath106 to be the dual of @xmath107 for @xmath108 $ ] . this negates the need for a volume form if it is undesirable . if @xmath109 then this definition can be extended to @xmath110 $ ] . a norm @xmath111 is fixed , although it is not particularly important which one . for instant @xmath112 where @xmath113 is some fixed laplace beltrami operator . given a potential @xmath114 , consider the equation @xmath115 where @xmath113 denotes the laplace beltrami operator for @xmath116 . we define the map @xmath117 to be the _ solution operator _ for equation , so @xmath118 . the dirichlet neumann map @xmath119 takes @xmath120 to the the normal derivative @xmath121 . this can be formulated elegantly as @xmath122 where @xmath123 is the hodge - star operator associated to @xmath23 , @xmath124 is the volume form associated to @xmath125 on @xmath51 , and @xmath126 is the volume form associated to @xmath23 . of course an important point is that @xmath127 where @xmath128 is an arbitrary extension operator , because the difference @xmath129 . given two potentials @xmath39 and @xmath40 on @xmath10 , we can consider the difference in the dirichlet neumann maps , here denoted for parsimony s sake by @xmath130 and @xmath131 respectively . by a standard integration by parts technique we can show that @xmath132 by splitting the right hand integral , we can express this as @xmath133 a key observation for the results herein is that the map @xmath134 is the solution to an evolution equation @xmath135 this is the _ tautological evolution equation _ for the boundary value problem . this observation follows from the fact that @xmath136 for the manifold @xmath89 for all @xmath137 $ ] in fermi coordinates . given two different potentials , and functions @xmath18 and @xmath138 satisfying @xmath139 respectively we will be testing the difference in potentials agains their product @xmath140 now the product @xmath141 does not _ a priori _ satisfy an evolution equation like , but the tensor product @xmath142 define as @xmath143 satisfies the evolution equation @xmath144 where @xmath145 . let @xmath142 satisfy @xmath146 then [ lem : evolved_tensor_prod_solves_a ] @xmath147u_{t}\otimes v_{t}=0\ ] ] the operator @xmath148{\partial}_{t}$ ] . when we apply this to @xmath142 we arrive at @xmath149 where we have replaces @xmath150 with @xmath151 and likewise for @xmath152 . [ lem : dn_map_satisfies_ext_props ] suppose @xmath23 and @xmath153 are @xmath81-smooth , @xmath154 , then @xmath155 is weakly @xmath156 i.e. for every @xmath18 and @xmath157 , the map @xmath158 is @xmath156 . furthermore @xmath159 and @xmath160 for @xmath161 $ ] the proof of this follows from layer stripping arguments in @xcite , which were also applied in @xcite . @xmath162\\ & \quad+\int_{\sigma_{t+h}}\bigg[{\mathcal{e}}_{q}^{t+h}(u-{\mathcal{e}}_{q}^{t}(u)|_{\sigma_{t+h}})q{\mathcal{e}}_{q}^{t+h}(v)\\ & \qquad\qquad\qquad+{\mathcal{e}}_{q}^{t}(u)q{\mathcal{e}}_{q}^{t+h}(v-{\mathcal{e}}_{q}^{t}(v)|_{\sigma_{t+h}})\bigg]\:d{\mathrm{vol}}\\ & \quad-\int_{\sigma_{t}\setminus \sigma_{t+h}}d{\mathcal{e}}_{q}^{t}(u)\wedge\star_{g}d{\mathcal{e}}_{q}^{t}(v)i-\int_{\sigma_{t}\setminus\sigma_{t+h}}{\mathcal{e}}^{t}_{q}(u)q{\mathcal{e}}^{t}_{q}(v)\:d{\mathrm{vol}}.\end{aligned}\ ] ] if we divide by @xmath163 and let @xmath163 tend to @xmath19 . we arrive at @xmath164 but @xmath165 . given that the principle symbol of @xmath166 is equal to @xmath167 @xcite @xcite , this yields that the @xmath168 is a bounded operator @xmath169 uniformly in @xmath5 . for the lower bound see @xcite . the second order equation also factorises as the product of two evolution equations in different directions which motivates the name _ evolution squared_. this is in turn justified when we try to make sense of the distributional boundary value problem , which is greatly facilitated compared to heavier machinery , such as the boutet de monvel calculus @xcite . the evolution squared operator @xmath61 is equal to @xmath170 first we note that @xmath171 . here we make use of the fact that @xmath172 from which we deduce that @xmath173 of course @xmath174 and @xmath175={\partial}_{t}(a_{t})$ ] . when we expand @xmath176-(\dot{\mu}(x)+\dot{\mu}(y))a_{t}\ ] ] with these , we arrive at the desired result . now we can see how @xmath61 defines a reasonable elliptic operator . we define a distributional solution @xmath177,{\partial}{\mathcal{m}}^{2}\times\{0,\varepsilon\})$ ] of as one for which @xmath178}\varphi { \mathcal{a}}u\:d{\mathrm{vol}}=\int_{{\partial}{\mathcal{m}}\times[0,\varepsilon]}(q^{1}-q^{2})(t , x)u(t , x , x)\:d{\mathrm{vol}}-(\lambda^{1}_{\varepsilon}-\lambda^{2}_{\varepsilon})(u(\varepsilon,\cdot,\cdot))\ ] ] for every smooth function @xmath79 equal to @xmath19 on @xmath179 and for which @xmath180 for @xmath181 . the space @xmath182 , { \partial}{\mathcal{m}}^{2}\times\{0,\varepsilon\})$ ] is defined to be the topological dual of @xmath183)$ ] . [ lem : evolution_squared_bvp_existence ] assume @xmath23 and @xmath184 are in @xmath185 where @xmath186 . there is a distributional solution to @xmath187\delta_{x , y}\ ] ] @xmath188 we start by solving the inhomogeneous evolution equation @xmath189\psi_{t}=0,\ ] ] subject to @xmath190 as a distribution on @xmath73 . it is an element of @xmath191 we consider a solution @xmath192 , equipped , with an inner product of the form @xmath193 where @xmath194 is an @xmath195 inner product on @xmath196 . then @xmath197 is monotone with respect to this inner product by virtue of @xmath198 . then we have an evolution triple @xmath199 , @xmath200 @xmath201 with this inner product , and so we have unique existence of a solution ( * ? ? ? * theorem 23.a ) in the space @xmath202,h^{-1}({\partial}{\mathcal{m}}),h^{-1/2}({\partial}{\mathcal{m}}),h^{-3/2}({\partial}{\mathcal{m}}))$ ] . we then solve the inhomogeneous initial value problem @xmath203\hat{\varphi}=\psi(t ) , \quad\hat{\varphi}(0)=0.\ ] ] we also solve the inhomogeneous initial value problems @xmath204\tilde{\psi}=\phi_{t},\quad \tilde{\psi}(\varepsilon)=0,\ ] ] and @xmath205\tilde{\varphi}=\tilde{\psi},\quad \tilde{\varphi}(0)=0.\ ] ] however @xmath206 is no longer in @xmath199 , but @xmath207 for @xmath208 for every @xmath5 . so we introduce a new evolution triple @xmath209 , @xmath210 . lastly we set @xmath211 . we apply integration by parts to get the following : [ lem : boundaryvalue ] let @xmath212 be a distributional solution to : @xmath213 then @xmath214 let @xmath215 . we begin with to get @xmath216}({\partial}_{t}^{*}+a_{t})({\partial}_{t}+a_{t})\varphi \:u_{t}\otimes v_{t}\:d(\mu\otimes \mu)\:dt\\ & \quad+(\lambda^{1}_{\varepsilon}-\lambda^{2}_{\varepsilon})(u_{\varepsilon}\otimes v_{\varepsilon})\\ & = -\int_{{\partial}{\mathcal{m}}^{2}}\psi(0)\:u_{0}\otimes v_{0}+\int_{{\partial}{\mathcal{m}}^{2}}\psi(\varepsilon)\:u_{\varepsilon}\otimes v_{\varepsilon}\:d(\mu\otimes \mu)\\ & \quad+(\lambda^{1}_{\varepsilon}-\lambda^{2}_{\varepsilon})(u_{\varepsilon}\otimes v_{\varepsilon})\\ & = -\int_{{\partial}{\mathcal{m}}^{2}}\psi(0)\:u_{0}\otimes v_{0}\:d(\mu\otimes\mu)\end{aligned}\ ] ] lastly we note that @xmath217 because @xmath218 . we now need to show that @xmath76 implies @xmath77 off the diagonal : [ lem : phi_neq0_off_diag ] let @xmath68 be the measure on @xmath66 $ ] given by @xmath219 $ ] . if @xmath220(x_{0},t_{0})\neq 0 $ ] and @xmath221 then @xmath77 on @xmath222 $ ] we will endeavor to show that @xmath223 where @xmath224 , and @xmath225 for @xmath226 , i.e. @xmath227 . first we note that @xmath228 for @xmath229 and @xmath61 is an elliptic second order pseudodifferential operator , so @xmath230 . to see that @xmath231 for @xmath232 , consider a sequence of test functions @xmath233 such that @xmath234 for @xmath235 and @xmath236 , while @xmath237 . first we start with a function @xmath238 supported on @xmath239 such that @xmath240 let @xmath241 . then @xmath242}({\partial}_{t}+a_{t})(\varphi)(d_{t}+a_{t})\xi_{k}\:d{\mathrm{vol}}_{t}\:dt\\ & \leq \|\varphi\|_{1,n/(n-1)}\|\xi_{k}\|_{1,n}\end{aligned}\ ] ] but @xmath243 as @xmath244 , while @xmath245 . armed with most of the necessary tools , we can now prove theorem [ thm : uniq_cont_loc_cald ] . by lemma [ lem : boundaryvalue ] , @xmath246 where @xmath212 is the solution to . by the hypothesized oducp if @xmath247 and @xmath218 then @xmath75 everywhere . but by lemma [ lem : phi_neq0_off_diag ] @xmath212 can not be zero if @xmath248 is non - zero . in order to prove theorem [ thm : uniq_cont_impl_cald_uniq ] we need a lemma for guaranteeing the existence of an exhaustion for smooth manifolds with boundary : [ lem : exists_exhaustion ] let @xmath10 be a @xmath43 smooth manifold with boundary . there exists a map @xmath249\to { \mathcal{m}}$ ] which is a diffeomorphism onto its image when restricted to @xmath250 , such that @xmath251)$ ] is meager . the proof requires the existence of a smooth triangulation @xmath252 of @xmath10 @xcite . let @xmath253 be a smooth @xmath254-simplex , i.e. @xmath43 up to the boundary of each sub - simplex , a diffeomorphism on the interior . by @xmath255 we denote the set @xmath256 we say @xmath257 by the natural inclusion of @xmath258 along the first @xmath259 components . we will assume by induction that we have a diffeomorphism from some @xmath260 simplices @xmath261\to \bigcup_{i=1}^{n}\tau_{i}(\sigma_{n})$ ] . if there is a simplex whose interior is disjoint from the image of @xmath262 , then there is a simplex whose interior is disjoint from the image of @xmath212 and which neighbours some @xmath263 . let @xmath264 denote this simplex , let @xmath265 denote a mutual facet of @xmath263 and @xmath264 . the goal will be to smoothly push @xmath262 from @xmath263 through @xmath265 to @xmath264 . let @xmath266 denote the pre - image of @xmath265 in @xmath263 . define the map @xmath267 by @xmath268 where @xmath269 . let @xmath270 , and let @xmath271 denote the set @xmath272 where @xmath273 is the reflection @xmath274 . @xmath275 can be canonically smoothed . we will construct a map @xmath276 : consider @xmath277 where @xmath278 . this takes @xmath266 to the complement of @xmath266 in @xmath279 . we define @xmath280 inductively by @xmath281 where @xmath282 define @xmath283\\ ( x,0)+(1 - 2t)\min_{t}\{x_{i}\}(-\mathbf{1}/n),1)&t\in[0,1/2 ] , \end{cases}\ ] ] where @xmath284 maps @xmath285 $ ] to @xmath286 $ ] monotonically , and is the identity in a neighbourhood of @xmath287 . lastly we must smooth @xmath288 in a neighbourhood of @xmath266 . finally we define @xmath289 to complete the induction we need an initial step . of course there is no diffeomorphism from @xmath290 for a single simplex , rather we must start with a collar . because @xmath10 is @xmath43 we can apply the collar neighbourhood theorem , to yield a diffeomorphism from @xmath291\to { \mathcal{m}}$ ] onto its image . let @xmath292 $ ] . choose a triangulation of @xmath252 , and extend it to a triangulation of @xmath10 by restricting the triangulation of @xmath252 , to @xmath293 , identifying @xmath51 and @xmath293 , and canonically triangulating @xmath294 $ ] and joining this triangulation to that of @xmath252 to get a triangulation of @xmath10 . now @xmath295 is a diffeomorphism @xmath294 $ ] to some sub - triangulation of @xmath10 . we make use of lemma [ lem : exists_exhaustion ] to give us an exhaustion of @xmath10 via the boundary @xmath296\to { \mathcal{m}}$ ] . if @xmath297 for @xmath298 $ ] then by @xmath299 . consequently by theorem [ thm : uniq_cont_loc_cald ] we can extend it two a boundary normal neighbourhood . but then , because @xmath84 is a homeomorphism , it also true for @xmath300 $ ] for some @xmath48 . but then it is true for every @xmath301 as the set on which it is true is closed and open . finally because @xmath302)$ ] is meager , @xmath63 on all of @xmath10 . although the off - diagonal unique continuation property is a strong assumption for the operator @xmath61 there is some evidence to suggest it might hold . generically , unique continuation properties for pseudodifferential operators are not known and probably false , and it seems unlikely that an appropriate carleman estimate could be derived for @xmath61 because of the non - local behaviour of the dirichlet neumann maps contained therein . however the work of caffarelli and silvestre @xcite shows us that dirichlet neumann maps have very strong unique continuation properties . our operator @xmath61 is the sum of a differential operator and the tensor product of two dirichlet neumann maps , so perhaps clever application of arguments like those in @xcite could be used to derive such a unique continuation . nonetheless , there should be no confusion that the oducp is a stronger condition than the uniqueness of the caldern problem , however , if we restrict ourselves to the study of an operator @xmath61 defined for @xmath303 , then the oducp is equivalent to the uniqueness for the linearised caldern problem for @xmath64 $ ] with the metric @xmath304 . consequently it seems counterintuitive that uniqueness for the caldern problem would be true , while the oducp would be false .

we prove that uniqueness for the caldern problem on a riemannian manifold with boundary follows from a hypothetical unique continuation property for the elliptic operator @xmath0 defined on @xmath1 $ ] where @xmath2 and @xmath3 are potentials and @xmath4 is a dirichlet neumann operator at depth @xmath5 . this is done by showing that the difference of two dirichlet neumann maps is equal to the neumann boundary values of the solution to an inhomogeneous equation for said operator , where the source term is a measure supported on the diagonal of @xmath6 .

the interaction of an intense ultrashort laser pulse with underdense plasmas has attracted much interest for a compact accelerator . using intense laser systems , of which peak powers exceed 10 tw , electrons with energies greater than 100 mev have been observed in various low density range , of which electron densities are from @xmath0 to @xmath1 @xcite . on the other hand , using terawatt class ti : sapphire laser systems , electrons with energies greater than several mega - electron - volts have been observed from moderately underdense plasmas , of which densities are up to near the quarter - critical density@xcite . at the moderately underdense plasmas , the electron energies exceed the maximum energies determined by dephasing length . it is considered recently that the acceleration occurs by the direct laser acceleration@xcite that includes stochastic or chaotic processes . in this paper , we study the propagation of the laser pulses and the generation of high energy electron in the underdense plasmas using two dimensional particle - in - cell ( 2d pic ) simulations . the laser power @xmath2 beyond the critical power @xmath3 is necessary because self - focusing is important in a long - distance propagation@xcite . here , @xmath4 , @xmath5 and @xmath6 are the electron density and critical density , respectively . we assume a terawatt class ti : sapphire laser system as a compact laser system in the simulations parameter , because the plasma electron densities @xmath7 . we use the 2d pic simulation with immobile ions . the peak irradiated intensity , pulse length , and spot size are @xmath8 @xmath9 , @xmath10 fs , and 3.6 @xmath11 m , respectively . @xmath12 tw , namely the energy is @xmath13 mj , when cylindrical symmetry is assumed , although we use two dimensional cartesian coordinates . the rayleigh length @xmath14 m . the plasma electron densities @xmath7 , which correspond to @xmath15 , where @xmath16 for the wavelength @xmath17 . these parameters of the simulations are almost the same as the experiments of compact laser system@xcite . the laser power @xmath12 tw exceeds the critical powers @xmath3 of the relativistic self - focusing for @xmath18 . figures 1(a)-(e ) show the intensities of laser pulses after propagating @xmath19 for electron densities @xmath20 , and @xmath21 , respectively . for @xmath22 and @xmath23 , namely , @xmath24 , the pulses stably propagate without modulation . electrons with energies greater than 2 mev are hardly observed . for @xmath25@xmath26 , the back of pulse is modulated . electrons get energies greater than 20 mev , as shown later . for @xmath27@xmath26 , a pulse separates into the bunches of which size about the plasma wavelength . a pulse breaks up and is not propagate stably any longer , for @xmath28@xmath26 , i.e. @xmath29 . the electron energy spectra for electron densities @xmath30 , and @xmath31 are shown in figs . 2(a)-(c ) , respectively . the maximum energy is greater than @xmath32 mev for @xmath33 . before the pulses propagate about one and two rayleigh length , the maximum electron energies have been saturated at 20 mev and 10 mev for @xmath22 and @xmath34 , respectively . we study the propagation of the intense laser pulses and the generation of high energy electrons from the moderately underdense plasmas using 2d pic simulations . for @xmath35 , the laser pulse of which power and pulse length are @xmath36 tw and @xmath10 fs stably propagates with modulation . as a result , the high energy electrons with energies greater than 20 mev are observed and their energies have not been saturated , namely , electrons can gain higher energies propagating with the intense laser pulse through long size plasmas . for @xmath37 , although the pulses stably propagate , no high energy electron is observed . on the other hand , for @xmath38 , high energy electrons with energies up to 20 mev are observed , although pulses does not propagate stably . the simulation results of the dependence to the plasma density of the maximum electron energy explain the latest experiment well qualitatively@xcite . in the simulations , the maximum propagation distance is @xmath39 is limited by the performance of the computer and simulation code . since the pulse has propagated sufficiently stably to @xmath19 for the plasma densities less than @xmath40@xmath26 , simulations with a longer propagation distance is required . 99 z. najmudin _ et al . _ , phys . plasmas , * 10 * ( 2003 ) 2071 y. kitagawa _ et al . _ , * 92 * ( 2004 ) 205002 . c. gahn _ et al . _ , * 83 * ( 1999 ) 4772 . et al . _ , soc . , * 48 * ( 2003 ) 195 ; k. koyama _ _ , int . j. appl . , * 14 * ( 2001/2002 ) 263 ; m. adachi _ et al . _ , proceedings of the 31st eps conference on plasma phys . london , 28 june - 2 july 2004 eca vol.28 g , p-5.024 ( 2004 ) m. tanimoto _ e , * 68 * ( 2003 ) 026401 ; t. nakamura _ _ , phys . plasmas , * 9 * ( 2002 ) 1801 ; z .- m . sheng _ et al . _ , lett . , * 88 * ( 2002 ) 055004 ; j. meyer - ter - vehn and z. m. sheng , phys . plasmas , * 6 * ( 1999 ) 641 ; a. pukov _ et al . _ , plasmas , * 6 * ( 1999 ) 2847 . g. schmidt and w. horton , comments plasma phys . controlled fusion * 9 * , 85 ( 1985 ) ; p. sprangle _ et al . _ , ieee trans . plasma sci . * ps-15 * ( 1987 ) 1987 ; g .- z . et al . _ , fluids * 309 * ( 1987 ) 526 .

the propagation of intense laser pulses and the generation of high energy electrons from the underdense plasmas are investigated using two dimensional particle - in - cell simulations . when the ratio of the laser power and a critical power of relativistic self - focusing is the optimal value , it propagates stably and electrons have maximum energies .

it is well known experimental fact that identical pions , which are produced in heavy ion collisions , being bosons show bose - einstein correlations . these correlations result from the quantum mechanical interference in the corresponding symmetrized @xmath0-particle wave function ( where @xmath0 denotes the number of produced identical pions ) . they contain large amount of information about the statistical properties of the momentum and configuration space distribution of the system , and thus provide a potentially very useful method to probe the geometry of the hadronizing source ( see , for example , @xcite and references therein ) . according to common understanding we are not able to determine which pion is emitted from which position in the source , so we are required by bose statistics to add amplitudes for all possible alternate histories . in general , symmetrized wave function for @xmath1-pion state can be written in the following way @xcite : @xmath2 , \label{eq : nwave}\ ] ] where @xmath3 denotes the @xmath4 element of a permutation of the sequence @xmath5 and the sum over @xmath6 denotes therefore the sum over all @xmath7 permutations in this sequence ( dependence on positions of points of detection , which will vanish when calculating probabilities , was neglected ) . here s denote the points of production of secondaries . because in the experiment one observes only momenta of produced secondaries these @xmath8 s must be somehow get rid of . this is so far always done by integrating over @xmath9 with some _ assumed _ distribution @xmath10 , which is _ assumed _ to be factorizable and expressed by product of independent single particle distributions @xmath11 @xcite . as result one gets the following probability of the @xmath1-pion state , @xmath12 expressed by _ permanent _ of the matrix @xmath13 , where @xmath14 for usually considered @xmath15 case one recovers well known classical expression for the probability of detecting two pions in the final state @xcite : @xmath16 unfortunately the execution time of direct computation of the permanent , eq.([eq : nwave ] ) , grows exponentially with @xmath1 one has to devise some special methods like _ metropolis procedure _ investigated in @xcite or _ von neumann accepting - rejection method _ proposed in @xcite . the _ metropolis procedure _ proposed in @xcite uses the standard monte carlo technique due to metropolis . this is general method which allows to generate ensemble of @xmath0-body configurations according to some prescribed probability density . that is , the probability of a given configuration in the ensemble is precisely that given by the probability density used to generate `` successive '' configurations . in @xcite this technique was used to modify directions of momentum vectors of number of selected particles from a system of @xmath0 identical particles in order to impose the @xmath0-particle distributions derived from be correlation functions . this procedure is then repeated many times , changing selected particles , until a kind of `` equilibrium '' is achieved . as shown in @xcite one was able in this way to generate typical multipion events which explicitly exhibit all correlations induced by bose statistics . as a result of application of this algorithm a number of objects ( called _ speckles _ ) being clusters of large number of identical pions in the phase - space is being formed . the drawback of this method is that symmetrization of clusters with sizes ( represented by the number of particles inside cluster , @xmath17 ) larger than @xmath18 takes prohibitively long time . in @xcite this method was therefore modified by limiting symmetrization only to particles in clusters . this was possible by using wave packets to describe produced particles instead of plane waves used in @xcite allowing therefore for localization of particles within certain phase - space volume and for providing the suitable criterium for defining a cluster . the _ accepting - rejection method _ investigated in @xcite is based on the well known `` hit - or - miss '' technique of generating a set of random numbers according to a prescribed distribution . the method was designed to model collapse of a multiparticle wave function into a properly symmetrized state , as required by bose quantum statistics . in contrast to the previous one it is sequential because @xmath0 multiparticle event is constructed by first choosing single particle in phase space , then adding to it the second one according to the assumed @xmath19-particle correlation function @xmath20 , then adding @xmath21 particle according to @xmath22 and so on . it is easy to realize that in this way one gets in the allowed phase space a `` cell - like '' structure because regions with some particles inside them already present will have bigger chance to attract new particle . in a sense it looks like follows : first particle forms a seed for a first cell . when second particle is added to event it can , depending on its distance from the first one , either remain in that cell or later on attract new particles and in this way start to form a new cell . this will then continue until all particles are used . unfortunately , this sequential procedure is even more time consuming than the previous one . the above discussion shows that complexity of numerical symmetrization of wave function for all identical pions produced in a given event ( cf . , eq.([eq : nwave ] ) ) can be substantially reduced if only one can justify the idea that such symmetrization should be applied to groups of limited number of particles , as proposed in @xcite . as argued there this can be achieved by dividing , according to some prescribed procedure , the initial set of secondaries into clusters consisting of identical particles with similar momenta . such groups of particles , but this time of ( almost ) equal energies , were introduced in @xcite under the name _ elementary emitting cells _ ( eec s ) . the physical justification was that bosonic particles , because of their statistical properties , tend to occur , as much as possible , in _ the same _ state . one should realize now that such decomposition corresponds to _ factorization _ of permanent given by eq.([eq : permanent ] ) into matrix with a block structure : @xmath23 this idea of eec was exploited by us further in @xcite where instead of symmetrization of multiparticle wave function ( depending on space - time positions , @xmath24 , and energy - momenta , @xmath25 ) we worked in the number of particles basis . the bosonic character of the produced secondaries means in this case the specific bunching of identical particles in the phase space . in fact it is nothing else but modelling the correlations of fluctuations due to quantum statistics present in the system , which for @xmath19-particle case are represented by : @xmath26 @xmath27 is dispersion of the multiplicity distribution @xmath28 and @xmath29 is the correlation coefficient depending on the type of particles produced : @xmath30 for bosons , fermions and boltzmann statistics , respectively ) . the important feature of the eec method is that number of particles in each cell follows by definition the geometrical ( or bose - einstein ) distribution characterized for identical bosonic particles . it should be mentioned at this point that the importance of bunching of particles in modelling quantum statistical effects has been demonstrated already in @xcite . in this paper , following ideas of information theory , authors constructed monte - carlo ( mc ) event generator for multiparticle production processes ( and applied it to @xmath31 reactions ) . the main point was the assumption that particles of the same charge are located in cells ( in their case they were constructed in rapidity space and were of equal size ) . it turned out that such division of phase space into cells was crucial for obtaining the characteristic form of the @xmath19-body bec correlation function @xmath32 ( i.e. , the one peaked and greater than unity at @xmath33 and then decreasing in a characteristic way towards @xmath34 for large values of @xmath35 , out of which one usually deduces the spatio - temporal characteristics of the hadronization source @xcite ) . in our case we argue that the method of production of eec proposed in @xcite can be used to _ define _ the structure of clusters obtained in @xcite ( with , as it turns out , about @xmath36 particles per cluster on average , depending on circumstances @xcite ) . therefore , instead of symmetrizing all particles in a given event , one symmetrizes separately particles in a number of eec s with @xmath37 . in the plane wave approximation used in @xcite , one has therefore for some typical eec the following @xmath38-particle probability function : @xmath39}=1+\frac{2}{n_{cl}!}\sum_{\sigma'>\sigma=1}^{n_{cl } ! } \cos\left\{\sum_{j=1}^{n_{cl}}p_i\left[r_{\sigma(j)}-r_{\sigma'(j)}\right]\right\ } . \label{full}\ ] ] notice that eq . ( [ full ] ) still contains ( unmeasurable ) positions of production of particles , @xmath40 . therefore they will be later eliminated by selecting them from some _ assumed _ distribution in a kind of numerical integration process . it corresponds to analytical integrations encountered before but in our approach we do not limit in any way the form of density function used ( it had to be _ factorizable _ before ) . it should be stressed that when all particles in the cluster are exactly in the same state then one gets , as expected , @xmath39}\left({\rm max}\right)\left|_{p_1=\cdots = p_{n_{cl } } } \right.= n_{cl } ! .\label{max}\ ] ] however , in practice , even now from time to time one encounters eec with @xmath41 , in which case eq.([full ] ) is still very time consuming . for such cases we have to use some approximate schemes . the first one is sequential ( cf . [ fig2 ] ) : one starts with some single particle in eec and adds new particles one by one to all others already present and correlate them by using standard @xmath19-particle probability : @xmath42 . the resulting probability to get @xmath38-particle state is given by : @xmath43}=\mathcal{p}_{1, ... ,n_{cl}-1}\cdot \prod_{j=1}^{n_{cl}-1}\left[1+\cos\left(\delta r_{jn_{cl}}\cdot\delta p_{jn_{cl}}\right)\right ] .\label{sequential}\ ] ] ) .,width=302 ] modelled for single eec with @xmath44 by using @xmath19-particles ( sequential ) correlations as given by eq . ( [ sequential ] ) ( circles ) and full @xmath45-particles correlations as given by eq . ( [ full ] ) ( stars ) . for comparison , the analytical result , @xmath46 , obtained for uniform one dimensional source in space with @xmath47 fm is also shown ( dotted - dashed line).,width=283 ] modelled by using mc event generators with eecs ( circles ) and without eecs but with particles selected directly from the corresponding bose - einstein distribution @xcite . in both cases as reference event the boltzmann distribution was used.,width=283 ] although appealing ( in fact it resembles procedure used in @xcite ) it has drawback that when all particles are in the same state then @xmath43}\left({\rm max } \right)\left|_{p_1=\cdots = p_{n_{cl } } } \right . = 2^{\frac{1}{2}n_{cl}(n_{cl}-1)}\ ] ] i.e. , for @xmath48 the correlations are stronger and the maximal value of probability function gets bigger than allowed limit defined by eq.([max ] ) . both , exact ( [ full ] ) and sequential ( [ sequential ] ) methods are compared in fig . [ fig1 ] . approximation ( [ sequential ] ) keeps the width of @xmath49 the same , however it differs substantially in @xmath50 ( from which one tends to estimate the so called _ chaoticity _ of the hadronizing source @xcite ) . the other possible approximation , which was used in our hitherto applications @xcite , is that all particles in a given eec are correlated only with the first particle defining this cell , not between themselves . it is interesting to notice that the results for @xmath49 obtained this way are almost the same as those obtained by using the full method , eq . ( [ full ] ) , but differ from that of eq . ( [ sequential ] ) . we close this section by noticing recent attempt to imitate the bosonic nature of particles produced by mc event generator , in which they were choosen directly according to bose - einstein distribution @xcite . this method seems to be very natural and occurred to be also very fast . however , as one can see from fig . [ fig3 ] , where it is compared with our approach , it leads to completely different @xmath49 substantially underestimating the correlation function @xmath49 . we claim therefore that such procedure inevitably loses some piece of important information , namely the fact that eec s are formed and that bose - einstein distributions are only for particles in such eec s , not in the whole event . in fact , signal of possible bose - einstein correlations without cells is seen in fig . [ fig3 ] are the trivial correlations , which can be eliminated by the proper choice of the reference event . we would like to summarize by stressing that there is _ no way _ to add to any of the existing mc event generators effects of quantum statistics , in particular bose - einstein one ( be ) . this is because they are all build on basis of classical physics with both the space - time and energy - momentum characteristics of produced secondaries used simultaneously . the only way out advocated here ( albeit , most probably , not very practical one and therefore hardly to be followed ) is to build the multiparticle mc event generator _ ab initio _ , with be properties ( like bunching in phase space ) being one of its basic principles and consisting its first step . all other features of such generator would have to be added only after this . so far there is only one working example of this type @xcite , our efforts @xcite aim for its further generalization and will be continued . ou is grateful for support and for the warm hospitality extended to him by organizers of the ismd2006 . partial support of the ministry of science and higher education ( grants nr 621/e-78/spb / cern / p-03/dwm 52/2004 - 2006 and 1 p03b 022 30 ) is acknowledged ( ou and gw ) . o.v.utyuzh , g.wilk and z.wodarczyk , hep - ph/0503046 , acta phys . hung . * a * - heavy ion phys.*25 * ( 1 ) ( 2006 ) 83 ; hep - ph/0509320 , aip conf . * 828 * ( 2006 ) 75 ; hep - ph/0509342 , to be published in nukleonika * 51 * ( supplement 3 ) ( 2006 ) .

the method of numerical symmetrization of state of identical particles proposed by us before is clarified and discussed .

obtaining accurate results in strongly - correlated systems remains an extremely difficult task in systems where the sign - problem prevents one to use numerically exact quantum monte carlo methods . @xcite in the recent past , a large variety of numerical approaches have been proposed and devised to study simple models of interacting electrons on the lattice . among the most promising ones , we mention post density - matrix renormalization group ( dmrg ) techniques , based upon tensor networks . @xcite as far as insulating phases are concerned , one important issue is to describe mott insulators , in which the insulating character is driven by the strong electron - electron interaction . when lowering the temperature , the general expectation is that some symmetry - breaking phenomena take place , leading for example to magnetic or peierls ( i.e. , dimerized ) phases . in two spatial dimensions , a continuous symmetry can not be spontaneously broken at finite temperatures ; still , the ground state may possess magnetic long - range order . in this regard , the existence of quantum spin liquids , i.e. , states that do not break any symmetry down to zero temperature , in two dimensions represents a fascinating problem . the key feature to impede long - range ordering is given by the presence of frustrating interaction , namely the presence of competing super - exchange couplings that strongly enhance quantum fluctuations . the simplest example that captures the low - energy physics of mott insulators and may give rise to spin - liquid ground states is the frustrated heisenberg model : @xmath0 where both @xmath1 and @xmath2 and @xmath3 are spin-@xmath4 operators at each lattice site @xmath5 ; @xmath6 and @xmath7 denote sum over nearest - neighbor and next - nearest - neighbor sites , respectively . here , we consider both the square and kagome lattices with @xmath8 sites : @xmath9 for the square lattice and @xmath10 for the kagome lattice . all energies are given in units of @xmath1 . the ground state properties of these two models have been widely debated in the past twenty years , with contradicting results . recent dmrg calculations suggest that the ground state on the kagome lattice for @xmath11 is a gapped @xmath12 spin liquid , @xcite while the ground state of the @xmath13 model on the square lattice is a gapless spin liquid for @xmath14 and a ( gapped ) valence - bond solid for @xmath15 . @xcite here , we present a variational monte carlo method to assess ground - state properties of these models . the starting variational wave functions can be substantially improved by the application of few lanczos steps , which can be easily affordable for rather large clusters ( i.e. , few hundreds sites ) . moreover , the calculation of both the energy and its variance makes it possible to perform a zero - variance extrapolation to estimate exact energies , not only for the ground state but also for low - energy excitations . the variational wave functions are defined through the mean - field hamiltonian for the abrikosov - fermion representation of the spin-@xmath4 operators : @xcite @xmath16 where for each bond @xmath17 there are hopping @xmath18 and/or pairing @xmath19 terms ; on - site terms , i.e. , a real chemical potential @xmath20 and/or a complex pairing @xmath21 , may be also considered . given any eigenstate @xmath22 of the mean - field hamiltonian ( [ eq : meanfield ] ) , a physical state for the spin model can be obtained by a projection onto the subspace with one fermion per site : @xmath23 is the gutzwiller projector , @xmath24 being the local density . the flexibility of this approach is given by the fact that _ different _ spin liquids , having for example @xmath25 or @xmath12 gauge structure and gapped or gapless spinon spectrum , may be obtained by changing the pattern of the @xmath18 s and the @xmath19 s . @xcite moreover , dimerized or chiral states can be also easily obtained within this approach . @xcite in the following , we will consider the square lattice with @xmath26 and the kagome lattice with both @xmath11 and @xmath27 , corresponding to cases where the high frustration may stabilize fully symmetric spin liquids . for the square lattice , we will consider a state that is obtained by taking a real pairing @xmath28 ( with @xmath29 symmetry ) on top of the @xmath25 state with nearest - neighbor hopping @xmath30 and real pairing @xmath31 ( with @xmath32 symmetry ) . the @xmath29 term is responsible for breaking the @xmath25 gauge symmetry down to @xmath12 . restricting this coupling along the @xmath33 bonds implies four dirac points in the mean - field spectrum . @xcite for the kagome lattice , we will consider the @xmath25 state that is described by nearest - neighbor real hoppings such to have @xmath34 magnetic flux through triangles and @xmath35 flux through hexagons . @xcite at the mean - field level , there are gapless excitations at two dirac points . a slight improvement of the energy can be achieved by considering a next - nearest - neighbor real hopping . @xcite a gapped wave function , corresponding to a @xmath12 spin liquid ( labelled by @xmath36\beta$ ] ) can be obtained by adding on - site and next - nearest - neighbor pairing terms . @xcite a gapless spin liquid with a large fermi surface ( labeled by uniform rvb ) is obtained when no magnetic fluxes are present . within this formalism , it is straightforward to construct not only an ansatz for the ground state but also for low - energy excitations . in this respect , it is useful to consider a particle - hole transformation for the down electrons on the mean - field hamiltonian ( [ eq : meanfield ] ) , i.e. , @xmath37 , such that the transformed hamiltonian conserves the total number of particles . then , the ground state is obtained by filling the lowest @xmath8 orbitals , with suitable boundary conditions ( either periodic or anti - periodic ) in order to have a unique mean - field state . spin excitations can be obtained by creating the appropriate bogoliubov quasi - particles ( spinons ) and possibly switching boundary condition . here , we will consider only the case of a @xmath38 excitation with momentum @xmath39 , for both the square and the kagome lattices . by computing separately the energies of the @xmath40 and @xmath38 states , the spin gap @xmath41 is studied as a function of the cluster size . the energy per site versus the energy variance per site for the @xmath13 heisenberg model on the square lattice with @xmath26 and @xmath42 . exact lanczos diagonalizations starting from two random initializations ( a ) and quantum monte carlo calculations starting from the best variational state ( [ eq : psivar ] ) ( b ) are reported.,width=528 ] the size scaling of the @xmath38 spin gap for the best variational wave function and the zero - variance extrapolation on the square lattice with @xmath26 . the thermodynamic extrapolation gives @xmath43.,width=268 ] the size scaling of the @xmath38 spin gap for the best variational wave function and the zero - variance extrapolation on the square lattice with @xmath26 . the thermodynamic extrapolation gives @xmath43.,width=268 ] in order to systematically improve the variational wave functions , we can apply a number @xmath44 of lanczos steps to the starting variational wave function : @xmath45 are @xmath44 additional variational parameters . clearly , whenever @xmath46 is not orthogonal to the exact ground state , @xmath47 converges to it for large @xmath44 . unfortunately , on large sizes , only few steps can be efficiently afforded : here , we consider the case with @xmath48 and @xmath49 ( @xmath50 corresponds to the original variational wave function ) . furthermore , an estimate of the exact energy may be obtained by the variance extrapolation . indeed , for accurate variational states @xmath47 with energy @xmath51 and variance @xmath52 , it is easy to prove that @xmath53 where : @xmath54 are the energy and variance per site , respectively . therefore , the exact energy @xmath55 may be extracted by fitting @xmath51 versus @xmath52 and performing a zero - variance extrapolation . it should be emphasized that the lanczos step procedure of eq . ( [ eq : psilan ] ) is not size consistent if @xmath44 is not increased with the number of sites @xmath8 . indeed , both the energy and variance improvements with respect to the original state @xmath46 vanish for @xmath56 and fixed @xmath44 . nevertheless , it is remarkable that a sizable improvement is obtained even for rather large clusters with few hundred sites . by contrast , the zero - variance extrapolation remains size consistent , as shown below . by using the variational monte carlo technique , the properties of the gutzwiller - projected wave function can be easily assessed . indeed , expectation values of any operator @xmath57 , including @xmath58 , are given by : @xmath59 where @xmath60 represents a complete and orthogonal basis . by interpreting @xmath61 as a probability distribution , a markov chain can be constructed to sample eq . ( [ eq : average ] ) . in particular , the gutzwiller projector is automatically implemented by taking @xmath60 written in terms of electron configurations in real space with only singly - occupied sites . within the fermionic representation , an efficient metropolis algorithm can be implemented with local electron moves , e.g. , spin flips , @xmath62 . for @xmath50 , whenever few electrons are moved , the computational cost of @xmath63 only requires @xmath64 operations , while the updating when a new configuration is accepted requires @xmath65 operations . in order to have uncorrelated electron configurations , @xmath8 moves must be done , so that the variational algorithm scales like @xmath66 . when the first lanczos step is implemented , the only extra cost is that @xmath63 scales as @xmath67 , while the second lanczos step gives @xmath65 ; in both cases the updating algorithm remains @xmath65 . therefore , up to @xmath49 , the variational calculations have the same computational effort as the @xmath50 case . in some cases , in order to have a stable simulation for @xmath44 lanczos steps , it is important to adopt a regularization scheme to avoid vanishingly small determinants . here , we consider to sample configurations @xmath68 such that : @xmath69 with @xmath70 ranging from @xmath71 to @xmath72 . we start by showing how the lanczos step procedure and the variance extrapolation work for a case where exact diagonalizations are also available , namely the heisenberg model on the square lattice with @xmath42 and @xmath26 . in fig . [ fig : square36 ] , we report the calculations for the ground state in two different cases , either by starting from random initial states ( i.e. , what is usually done in the lanczos method ) or by initializing with the best variational wave function ( [ eq : psivar ] ) . while in the former cases , many lanczos steps are needed to reach a good energy per site and , therefore , also the linear regime of eq . ( [ eq : extrapol ] ) , the latter one gives excellent results even with two steps , the zero - variance extrapolation being exact within the statistical error . we would like to stress that abrupt reductions of the energy , with almost constant variance , or even non - monotonic behaviors , can appear when the initial wave function has large overlaps with excited states . by using the best gutzwiller projected state , we can extend the lanczos step procedure to larger systems , where exact diagonalizations are not possible . in fig . [ fig : square ] , we report the calculations for both the @xmath40 ground state and the @xmath38 excitation . calculations are performed up to @xmath73 with @xmath50 , @xmath74 , and @xmath75 , and up to @xmath76 with @xmath50 and @xmath74 . even though it is visible that the energy / variance gain reduces when increasing @xmath77 , the slope of the fit remains similar , allowing a size - consistent zero - variance extrapolation . from these results , the @xmath38 gap is shown in fig . [ fig : squarefinal ] . we expect a vanishing gap in the thermodynamic limit for the variational calculations with @xmath50 , since the @xmath12 gauge structure is not expected to alter the mean - field properties . @xcite however , our numerical results up to @xmath78 can not definitively confirm this fact . most importantly , the zero - variance extrapolation for the gap suggests that the frustrated heisenberg model on the square lattice is gapless for @xmath26 . energies per site for the @xmath40 states for the heisenberg model on the kagome lattice on @xmath79 sites ( i.e. , @xmath80 ) for @xmath11 ( a ) and @xmath81 ( b ) . both the @xmath25 dirac spin liquid and the gapped @xmath36\beta$ ] ( for which the next - nearest - neighbor pairing @xmath82 is fixed to 1 ) spin liquid are reported ; for @xmath11 , the uniform rvb state with vanishing magnetic fluxes is also reported.,width=528 ] the size scaling of the @xmath38 spin gap for the @xmath25 dirac state and the zero - variance extrapolation on the kagome lattice with @xmath11 . the thermodynamic extrapolation gives @xmath83.,width=259 ] the size scaling of the @xmath38 spin gap for the @xmath25 dirac state and the zero - variance extrapolation on the kagome lattice with @xmath11 . the thermodynamic extrapolation gives @xmath83.,width=249 ] let us move to the kagome lattice . in fig . [ fig : kagome48 ] , we report the variance extrapolation for a small lattice with @xmath79 sites ( i.e. , @xmath80 ) for both @xmath11 and @xmath81 . for the former case , no matter what is the initial state ( gapless with dirac points or a large fermi surface , or even gapped ) , the extrapolated energy is the same , within the statistical errors . notice , however , that the actual values of the initial energies and variances are quite different for these three cases . by contrast , for the case with @xmath27 , a rather different behavior is found when starting from the @xmath25 dirac state or the @xmath12 gapped one . remarkably , although the latter one gives a slightly better variational energy for @xmath50 , the lanczos extrapolation performs much better for the dirac state , where a smooth fit is possible . instead , for the @xmath36\beta$ ] state , the second lanczos step makes a sensible energy gain , while the variance does not improve . this behavior is reminiscent of what is seen when the initial wave function has some large overlap with excited states ( see fig . [ fig : square36 ] for the square lattice ) . the evidence that the @xmath36\beta$ ] state is not a good approximation even when @xmath11 is reported in fig . [ fig : kagomej20 ] . for @xmath84 , the lanczos extrapolation of the @xmath25 dirac spin liquid gives a better energy than the one obtained from the @xmath12 state . by performing a size - scaling extrapolation of the gap by using the @xmath25 dirac state , we obtain the results shown in fig . [ fig : kagomefinal ] . first of all , it is clear that the gutzwiller projector does not open a spin gap on top of the dirac state : this is a non - trivial outcome , given the fact that a @xmath25 gauge structure may give rise to strong interactions among spinons that may completely change the mean - field picture . most importantly , also after the lanczos step extrapolation , the system is compatible with a gapless spectrum . in summary , we have shown that extremely accurate calculations for the energy per site are possible in frustrated two - dimensional lattice by using variational wave functions that are built from abrikosov fermions . few lanczos steps can be applied to the pure variational state , giving a remarkable energy gain , also for clusters with few hundreds sites . moreover , performing a zero - variance extrapolation , it is possible to obtain accurate values of the exact energies , not only for the ground state but also for excited states . 99 kaul r k , melko r g and sandvik a w 2013 _ annu . con . mat . phys . _ * 4 * 179 schollwoek u 2011 _ annals of physics _ * 326 * 96 orus r 2014 _ annals of physics _ * 349 * 117 yan s , huse d a and white s r 2011 _ science _ * 332 * 1173 gong s s , zhu w , sheng d n , motrunich o i and fisher m p a _ phys . lett . _ * 113 * 027201 hu w j , becca f , parola a and sorella s 2013 _ phys . b _ * 88 * 060402 iqbal y , becca f , sorella s and poilblanc d 2013 _ phys . rev . b _ * 87 * 060405 iqbal y , poilblanc d and becca f 2014 _ phys . rev . b _ * 89 * 020407 baskaran g and anderson p w 1988 _ phys . rev . b _ * 37 * 580 wen x g 2002 _ phys . rev . b _ * 65 * 165113 iqbal y , becca f and poilblanc d 2011 _ phys . rev . b _ * 83 * 100404 bieri s , messio l , bernu b and lhuillier c 2014 _ preprint _ arxiv:1411.1622 ran y , hermele m , lee p a and wen x g 2007 _ phys . lett . _ * 98 * 117205 lu y m , ran y and lee p a 2011 _ phys . b _ * 83 * 224413

gutzwiller - projected fermionic states can be efficiently implemented within quantum monte carlo calculations to define extremely accurate variational wave functions for heisenberg models on frustrated two - dimensional lattices , not only for the ground state but also for low - energy excitations . the application of few lanczos steps on top of these states further improves their accuracy , allowing calculations on large clusters . in addition , by computing both the energy and its variance , it is possible to obtain reliable estimations of exact results . here , we report the cases of the frustrated heisenberg models on square and kagome lattices .

it is well known that adequate relativistic modelling is indispensable for the success of microarcsecond space astrometry . one of the most important relativistic effects for astrometric observations in the solar system is the gravitational light deflection . the largest contribution in the light deflection comes from the spherically symmetric ( schwarzschild ) parts of the gravitational fields of each solar system body . although the planned astrometric satellites gaia , sim , etc . will not observe very close to the sun , they can observe very close to the giant planets also producing significant light deflection . this poses the problem of modelling this light deflection with a numerical accuracy of better than 1 . the exact differential equation of motion for a light ray in the schwarzschild field can be solved numerically as well as analytically . however , the exact analytical solution is given in terms of elliptic integrals , implying numerical efforts comparable with direct numerical integration , so that approximate analytical solutions are usually used . in fact , the standard parametrized post - newtonian ( ppn ) solution is sufficient in many cases and has been widely applied . so far , there was no doubt that the post - newtonian order of approximation is sufficient for astrometric missions even up to microarcsecond level of accuracy , besides astrometric observations close to the edge of the sun . however , a direct comparison reveals a deviation between the standard post - newtonian approach and the exact numerical solution of the geodetic equations . in particular , we have found a difference of up to 16 in light deflection for solar system objects observed close to giant planets . this error has triggered detailed numerical and analytical investigations of the problem . usually , in the framework of general relativity or the ppn formalism analytical orders of smallness of various terms are considered . here the role of small parameter is played by @xmath1 where @xmath2 is the light velocity . standard post - newtonian and post - post - newtonian solutions are derived by retaining terms of relevant analytical orders of magnitude . on the other hand , for practical calculations only numerical magnitudes of various terms are relevant . in this note we attempt to close this gap and combine the analytical post - post - newtonian solution derived in @xcite with estimates of numerical magnitudes of various terms . in this way we will derive a compact analytical solution for the boundary problem for light propagation where all terms are indeed relevant at the level of 1 . the derived analytical solution is then verified using high - accuracy numerical integration of the differential equations of light propagation and found to be correct at the level well below 1 . we use fairly standard notations : * @xmath3 is the newtonian constant of gravitation ; * @xmath2 is the velocity of light ; * @xmath4 and @xmath5 are the parameters of the parametrized post - newtonian ( ppn ) formalism which characterize possible deviation of the physical reality from general relativity theory ( @xmath6 in general relativity ) ; * lower case latin indices @xmath7 , @xmath8 , take values 1 , 2 , 3 ; * lower case greek indices @xmath9 , @xmath10 , take values 0 , 1 , 2 , 3 ; * repeated indices imply the einstein s summation irrespective of their positions ( e.g. @xmath11 and @xmath12 ) ; * a dot over any quantity designates the total derivative with respect to the coordinate time of the corresponding reference system : e.g. @xmath13 ; * the 3-dimensional coordinate quantities ( `` 3-vectors '' ) referred to the spatial axes of the corresponding reference system are set in boldface : @xmath14 ; * the absolute value ( euclidean norm ) of a `` 3-vector '' @xmath15 is denoted as @xmath16 or , simply , @xmath17 and can be computed as @xmath18 ; * the scalar product of any two `` 3-vectors '' @xmath15 and @xmath19 with respect to the euclidean metric @xmath20 is denoted by @xmath21 and can be computed as @xmath22 ; * the vector product of any two `` 3-vectors '' @xmath15 and @xmath19 is designated by @xmath23 and can be computed as @xmath24 , where @xmath25 is the fully antisymmetric levi - civita symbol ; * for any two vectors @xmath15 and @xmath19 the angle between them is designated as @xmath26 . the paper is organized as follows . in section [ section - schwarzschild ] we present the exact differential equations . high - accuracy numerical integration of these equations is discussion in section [ section : numerical_integration ] . in section [ section - standard - pn ] we discuss the standard post - newtonian approximation and demonstrate the problem with the standard post - newtonian solution by direct comparison between numerical results and the ppn solution . in section [ section - ppn - solution ] , the formulas for the boundary problem in post - post - newtonian approximation are considered . a detailed estimation of all relevant terms is given , and simplified expressions are derived . we demonstrate by explicit numerical examples the applicability of this analytical approach for the gaia astrometric mission . in section [ section - stars ] we consider the important case of objects situated infinitely far from the observer as a limit of the boundary problem . the results are summarized in section [ section - conclusion ] . in the appendices detailed derivations for a number of analytical formulas are given . for the reasons given above we need a tool to calculate the real numerical accuracy of some analytical formulas for the light propagation . to this end , we consider the exact schwarzschild metric and its null geodesics in harmonic gauge . those exact differential equations for the null geodesics will be solved numerically with high accuracy ( see below ) and that numerical solution provides the required reference . as it has been already discussed in @xcite in harmonic gauge @xmath27 the components of the covariant metric tensor of the schwarzschild solution are given by @xmath28 where @xmath29 @xmath30 is the schwarzschild radius of a body with mass @xmath31 . the contravariant components read @xmath32 considering that the determinant of the metric can be computed as @xmath33 one can easily check that this metric satisfies the harmonic conditions ( [ harmonic - conditions ] ) . the christoffel symbols of second kind are defined as @xmath34 using ( [ exact_5 ] ) and ( [ exact_10 ] ) one gets @xmath35 and all other christoffel symbols vanish . as it has been pointed out in section ii.c of @xcite the condition of isotropy @xmath36 leads to the following integral of the equations of light propagation @xmath37 where @xmath38 is the coordinate direction of propagation ( @xmath39 ) , @xmath40 is the position of the photon and @xmath41 is the absolute value of the coordinate light velocity normalized by @xmath2 : @xmath42 . reparametrizing the geodetic equations @xmath43 by coordinate time @xmath44 ( see e.g. section ii.d of @xcite ) and using the christoffel symbols computed above one gets the differential equations for the light propagation in metric ( [ exact_5 ] ) : @xmath45 { \mbox{\boldmath$x$ } } + 2 \frac{a}{x^2 } \;\frac{2 - a}{1 - a^2 } ( { \mbox{\boldmath$x$ } } \cdot \dot{{\mbox{\boldmath$x$ } } } ) \ , \dot{{\mbox{\boldmath$x$ } } } \ , . \label{exact_25}\end{aligned}\ ] ] eq . ( [ isotropic_15 ] ) for the isotropic condition together with @xmath46 could be used to avoid the term containing @xmath47 , but it does not simplify the equations . our goal is to integrate eq . ( [ exact_25 ] ) numerically to get a solution for the trajectory of a light ray with an accuracy much higher than the goal accuracy of @xmath48 . for this numerical integration a simple fortran 95 code using quadrupole ( 128 bit ) arithmetic has been written . numerical integrator odex @xcite has been adapted to the quadrupole precision . odex is an extrapolation algorithm based on the explicit midpoint rule . it has automatic order selection , local accuracy control and dense output . using forth and back integration to estimate the accuracy , each numerical integration is automatically checked to achieve a numerical accuracy of at least @xmath49 in the components of both position and velocity of the photon at each moment of time . the numerical integration is first used to solve the initial value problem for differential equations ( [ exact_25 ] ) . ( [ isotropic_15 ] ) should be used to choose the initial conditions . the problem of light propagation has thus only 5 degrees of freedom : 3 degrees of freedom correspond to the position of the photon and two other degrees of freedom correspond to the unit direction of light propagation . the absolute value of the coordinate light velocity can be computed from ( [ isotropic_15 ] ) . fixing initial position of the photon @xmath50 and initial direction of propagation @xmath51 one gets the initial velocity of the photon as function of @xmath51 and @xmath41 computed for given @xmath51 and @xmath40 : @xmath52 the numerical integration yields the position @xmath40 and velocity @xmath53 of a photon as function of time @xmath54 . the dense output of odex allows one to obtain the position and velocity of the photon on a selected grid of moments of time . ( [ isotropic_15 ] ) holds for any moment of time as soon as it is satisfied by the initial conditions . therefore , ( [ isotropic_15 ] ) can be also used to estimate the accuracy of numerical integration at each moment of integration . for the purposes of this work we need to have an accurate solution of two - value boundary problem . that is , a solution of eq . ( [ exact_25 ] ) with boundary conditions @xmath55 where @xmath56 and @xmath40 are two given constants , @xmath57 is assumed to be fixed and @xmath54 is unknown and should be determined by solving ( [ exact_25 ] ) . instead of using some numerical methods to solve this boundary problem directly , we generate solutions of a family of boundary problems from our solution of initial value problem ( [ num_5 ] ) . each intermediate result computed by odex during the integration with initial conditions ( [ num_5 ] ) gives us a high - accuracy solution of the corresponding two - value boundary problem ( [ num_10 ] ) : @xmath54 and @xmath40 are just taken from the intermediate steps of our numerical integration . in the following discussion we will compare predictions of various analytical models for the unit direction of light propagation @xmath58 for a given moment of time @xmath54 . the reference value for these comparisons can be derived directly from the numerical integration as @xmath59 the accuracy of this numerically computed @xmath60 in our numerical integrations is guaranteed to be of the order of @xmath49 radiant and can be considered as exact for our purposes . in this section we will recall the standard post - newtonian approach and will compare the results for the light deflection with the accurate numerical solution of the geodetic equations described in the previous section . the well - known equations of light propagation in first post - newtonian approximation with ppn parameters have been discussed by many authors . the differential equations for the light rays are given by the post - newtonian terms of eq . ( 22 ) of @xcite : @xmath61 the analytical solution of ( [ pn_15 ] ) can be written in the form @xmath62 where @xmath63 solution ( [ pn_20])([pn_25 ] ) satisfies the following initial conditions : @xmath64 from eqs . ( [ pn_20])([pn_25 ] ) it is easy to derive the following expression for the unit tangent vector at observer s position ( note , in boundary problem we consider @xmath65 as the exact position @xmath40 , according to eq . ( [ pn_20 ] ) ) : @xmath66 where @xmath67 , @xmath68 . by means of eq . ( [ omega_5 ] ) given below we obtain that for the angle @xmath69 between @xmath70 and @xmath71 one has ( for @xmath72 ) @xmath73 where @xmath74 in the limit of a source at infinity one gets @xmath75 in order to determine the accuracy of the standard post - newtonian approach we have to compare the post - newtonian predictions of the light deflection with the results of the numerical solution of geodetic equations . here , we compare the difference between the unit tangent vector @xmath70 defined by ( [ pn_30 ] ) and the vector @xmath60 calculated from the numerical integration using ( [ eq : n - numerical ] ) . having performed extensive tests , we have found that , in the real solar system , the error of @xmath70 for observations made by an observer situated in the vicinity of the earth attains 16 . these results are illustrated by table [ table0 ] and fig . [ fig : numeric1 ] . table [ table0 ] contains the parameters we have used in our numerical simulations as well as the maximal deviation between @xmath70 and @xmath60 in each set of simulations . we have performed simulations with different bodies of the solar systems , assuming that the minimal impact distance @xmath76 is equal to the radius of the corresponding body , and the maximal distance @xmath77 between the gravitation body and the observer is given by the maximal distance between the gravitational body and the earth . the simulation shows that the error of @xmath70 is generally increasing for larger @xmath77 and decreasing for larger @xmath76 . the dependence of the error of @xmath70 for fixed @xmath76 and @xmath77 and increasing distance between the gravitating body and the source at @xmath78 is given on fig . [ fig : numeric1 ] for the case of jupiter , @xmath76 being taken to be minimal and @xmath77 to be maximal as given in table [ table0 ] . moreover , the error of @xmath70 is found to be proportional to @xmath79 which leads us to the necessity to deal with the post - post - newtonian approximation for the light propagation . .numerical parameters of the sun and giant planets are taken from @xcite . @xmath80 is the minimal value of the impact parameter @xmath76 that was used in the simulations . for each body @xmath80 are equal its radius . for the sun at @xmath81 the impact parameter is computed as @xmath82 . @xmath83 is the maximal absolute value of the position of observer @xmath77 that was used in the simulations . @xmath84 is the maximal angle between @xmath70 and @xmath60 found in the numerical tests . [ cols="<,^,^,^,^,^,^ " , ] as soon as we accept the equality of @xmath85 and @xmath71 for our case the only relevant step is the transformation between @xmath85 and @xmath60 . this transformation in the post - post - newtonian approximation is given by eqs . ( 53)(54 ) of @xcite . introducing impact vector computed using @xmath85 and the position of the observer @xmath40 @xmath86 we can re - write eqs . ( 53)(54 ) of @xcite as @xmath87 where @xmath88 . now we need to estimate the effect of the individual terms in eq . ( [ sigma - n - stars ] ) on the angle @xmath89 between @xmath85 and @xmath60 . this angle can be computed from vector product @xmath90 . the term in ( [ sigma - n - stars ] ) proportional to @xmath85 obviously plays no role and can be ignored . for the other terms taking into account that @xmath91 and considering the general - relativistic values @xmath92 we get @xmath93 where @xmath94 is the sum of all terms of order @xmath95 in ( [ sigma - n - stars ] ) . estimate ( [ psi - estimate ] ) obviously agrees with estimate ( [ n_20 ] ) for @xmath96 . numerical values of this estimate can be found in table [ table2 ] . the estimates show that these terms can be neglected at the level of 1 except for the observations within 5 angular radii from the sun . omitting these terms one gets an expression valid at the level of 1 in all other cases : @xmath97 note that for @xmath98 this coincides with ( [ n_85-better])([p - sso ] ) and with ( [ n - sigma - better])([n - sigma - t ] ) . this formula together with @xmath99 can be applied for sources at distances larger than 1 pc to attain the accuracy of 1 . alternatively eqs . ( [ n_85-better])([p - sso ] ) can be used for the same purpose giving slightly better accuracy for very close stars . however , distance information ( parallax ) is necessary to use ( [ n_85-better])([p - sso ] ) . in this report the numerical accuracy of the post - newtonian and post - post - newtonian formulas for light propagation in the parametrized schwarzschild field has been investigated . analytical formulas have been compared with high - accuracy numerical integrations of the geodetic equations . in this way we demonstrate that the error of the standard post - newtonian formulas for the boundary problem ( light propagation between two given points ) can not be used at the accuracy level of 1 for observations performed by an observer situated within the solar system . the error of the standard formula may attain @xmath100 16 . detailed analysis has shown that the error is of post - post - newtonian order @xmath101 . on the other hand , the post - post - newtonian terms are often thought to be of order @xmath102 and can be estimated to be much smaller than 1 in this case . to clarify this contradiction we have investigated the post - post - newtonian solution for the light propagation derived in @xcite . for each individual term in relevant formulas upper estimates have been found . it turns out that in each case one post - post - newtonian term may become much larger than the other ones and can not be estimates as @xmath103 . these terms depend only on @xmath5 and do not come from the post - post - newtonian terms of the corresponding metric . the formulas for transformations between directions @xmath85 , @xmath60 and @xmath71 containing both post - newtonian terms and post - post - newtonian ones that can be relevant at the level of 10 cm for the shapiro delay and 1 for the directions have been derived . the formulas are given by eqs . ( [ tau_30 ] ) , ( [ sigma - k - better])([sigma - k - s ] ) , ( [ n - sigma - better])([n - sigma - t ] ) , ( [ n_85-better])([p - sso ] ) , and ( [ sigma - n - stars - simplified])([q - stars ] ) . these formulas should be considered as formulas that guarantee this numerical accuracy . the derived analytical solution shows that no `` native '' post - post - newtonian terms are relevant for the accuracy of 1 in the conditions of this note ( no observations closer than five angular radii of the sun ) . `` native '' refers here to the terms coming from the post - post - newtonian terms in the metric tensor . it is , therefore , not the post - newtonian solution itself , but the standard analytical way to convert the solution of the initial value problem into the solution for the boundary problem that is responsible for the numerical error of 16 mentioned above . let us finally note that the post - post - newtonian term in ( [ n_85-better])([p - sso ] ) is closely related to the standard gravitation lens formula . here we only note that all the formulas given in @xcite and in this paper are not valid for @xmath104 ( @xmath76 always appear in the denominators of the relevant formulas ) . on the other hand , the standard post - newtonian lens equation successfully treats this case , known as the einstein ring solution . the relation between the lens approximation and the standard post - newtonian expansion is a different topic which will be considered in a subsequent paper . this work was partially supported by the bmwi grant 50qg0601 awarded by the deutsche zentrum fr luft- und raumfahrt e.v . ( dlr ) . 999 s.a . klioner , s. zschocke , gaia - ca - tn - lo - sk-002 - 1 e. hairer , s. p. norsett , g. wanner , _ solving ordinary differential equations 1 . nonstiff problems _ , springer , berlin , 1993 . moyer , t.d . ( 2000 ) formulation for observed and computed values of deep space network data types for navigation , deep space communications and navigation series , jpl publication 00 - 7 . weissman , l .- a . mcfadden , t.v . johnson , encyclopedia of the solar system , ( san diego : academic ) eds . iers conventions ( 2003 ) . dennis d. mccarthy and grard petit . ( iers technical note 32 ) frankfurt am main : verlag des bundesamts fr kartographie und geodsie , 2004 , 127 pp . in order to get ( [ tau_20 ] ) we write the corresponding term as @xmath105 where @xmath106 is the angle between @xmath40 and @xmath56 , and @xmath107 . it is easy to see that for @xmath108 and @xmath109 @xmath110 this immediately gives ( [ tau_20 ] ) . here and below we always give estimates that can not be improved in the sense that they are reachable for certain values of the parameters . for ( [ tau_25 ] ) we write @xmath111 here and below @xmath106 is the angle between @xmath40 and @xmath56 , and @xmath107 . one can show that for @xmath108 and @xmath109 @xmath112 and this immediately gives ( [ tau_25 ] ) . ( [ estimate - rho-0 ] ) we note that @xmath113 again @xmath106 is the angle between @xmath40 and @xmath56 , and @xmath107 . one can show that for @xmath108 and @xmath109 @xmath114 \displaystyle{\frac{2}{1 + z } } , & z>1 \end{array } \right.\ , \le 2 \label{for - rho-0 - 2}\end{aligned}\ ] ] and this leads to ( [ estimate - rho-0 ] ) . the discontinuity of @xmath115 and its estimate at @xmath116 are discussed in the main text after eq . ( [ rho-4 ] ) . the term @xmath117 can be written as @xmath118}{r^3 } \,\right| \nonumber\\ & = & 4\,\frac{m^2}{d^2}\,{x\over d}\ , z\,(1+z)\,{1-\cos\phi\over 1+z^2 - 2z\,\cos\phi}\,\left(1+{1-z\over\sqrt{1+z^2 - 2z\,\cos\phi}}\right ) . \label{for - rho-4 - 1}\end{aligned}\ ] ] for @xmath108 and @xmath109 one has @xmath119 \displaystyle{4\,\frac{z}{(1+z)^2 } } , & z<{1\over 2}\ { \rm or}\ z > 1\ , . \end{array } \right . \nonumber\\ \label{for - rho-4 - 2}\end{aligned}\ ] ] this gives eq . ( [ rho-4 ] ) . the function itself and its estimate ( [ for - rho-4 - 2 ] ) are again not continuous for @xmath120 ( implying @xmath121 ) . this is discussed after eq . ( [ rho-4 ] ) . in order to get ( [ sigma_40 ] ) we write @xmath122 one can show that for @xmath108 and @xmath109 @xmath123 this immediately leads to ( [ sigma_40 ] ) . estimate ( [ estim_5 ] ) for @xmath124 is trivial . for estimate ( [ estim_10 ] ) of @xmath125 we write @xmath126 \ , \biggr|\ , \left(1 + \frac{{\mbox{\boldmath$k$ } } \cdot { \mbox{\boldmath$x$}}}{x}\right)\ , \frac{r^2}{|\,{\mbox{\boldmath$x$ } } \times { \mbox{\boldmath$x$}}_0\,|^4}\,\frac{r^2 - ( x - x_0)^2}{2 } \nonumber\\ & = & 4\,\frac{m^2}{d^2}\,{r\over d}\ , \left(1 + \frac{{\mbox{\boldmath$k$ } } \cdot { \mbox{\boldmath$x$}}}{x}\right)\ , \frac{r^2 - ( x - x_0)^2}{2\,r^2 } \nonumber\\ & = & 4\,\frac{m^2}{d^2}\,{r\over d}\ , \left({1-z\,\cos\phi\over \sqrt{1+z^2 - 2z\,\cos\phi}}+1\right)\ , { z\,(1-\cos\phi)\over 1+z^2 - 2z\,\cos\phi},\end{aligned}\ ] ] where again @xmath106 is the angle between @xmath40 and @xmath56 , and @xmath107 . it is easy to see that for @xmath108 and @xmath109 @xmath127 this immediately leads to ( [ estim_10 ] ) . for eq . ( [ estim_30 ] ) we write @xmath128 where @xmath129 is the angle between vectors @xmath71 and @xmath40 . here we used that @xmath130 and @xmath131 . for @xmath132 we have @xmath133 and this proves eq . ( [ estim_30 ] ) . in order to get ( [ omega_5 ] ) we write @xmath134 here again @xmath106 is the angle between @xmath40 and @xmath56 , and @xmath107 . one can show that for @xmath108 and @xmath109 @xmath135 that immediately gives ( [ omega_5 ] ) . to derive ( [ omega-1 ] ) and ( [ omega-1-alternative ] ) we write @xmath136 for @xmath108 and @xmath109 one gets @xmath137 this gives the first estimate in ( [ omega-1 ] ) . trivial inequalities @xmath138 , @xmath139 and @xmath140 give the second and third estimates in ( [ omega-1 ] ) and estimate ( [ omega-1-alternative ] ) , respectively . for ( [ omega_22 ] ) we write @xmath141 for @xmath108 and @xmath109 one can demonstrate that @xmath142 & & \quad\le 15\,\pi\end{aligned}\ ] ] and this leads to ( [ omega_22 ] ) . estimates ( [ psi-0])([psi-3 ] ) are trivial . for ( [ psi - estimate ] ) we write @xmath143 where @xmath144 is the angle between vectors @xmath85 and @xmath40 . here we use @xmath145 and @xmath146 . therefore , for @xmath147 one can use estimate ( [ f7 ] ) for @xmath148 to prove ( [ psi - estimate ] ) .

* gaia - ca - tn - lo - sz-002 - 2 * issue 2 , numerical integration of the differential equations of light propagation in the schwarzschild metric shows that in some extreme situations relevant for practical observations ( e.g. for gaia ) the well - known standard post - newtonian formula for the boundary problem has an error up to 16 . the aim of this note is to identify the reason for this error and to derive an extended formula accurate at the level of 1 as needed e.g. for gaia . the analytical parametrized post - post - newtonian solution for light propagation derived by @xcite gives the solution for the boundary problem with all analytical terms of order @xmath0 taken into account . giving an analytical upper estimates of each term we investigate which post - post - newtonian terms may play a role for an observer in the solar system at the level of 1 . we conclude that only one post - post - newtonian term remains important for this numerical accuracy and derive a simplified analytical solution for the boundary problem for light propagation containing all the terms that are indeed relevant at the level of 1 . the derived analytical solution has been verified using the results of a high - accuracy numerical integration of differential equations of light propagation and found to be correct at the level well below 1 for arbitrary observer situated within the solar system .

_ how do characters of states change with variation of a parameter which specifies the property of the system or of the environment where the system is placed ? _ this is a general issue discussed in various phenomena of physics ; deformed nuclei depending on the deformation parameter of nuclear mean - field potential @xcite , electronic wave function configurations of diatomic molecules depending on the internuclear distance @xcite , and conversion of solar neutrinos depending on the distance from the center of the sun @xcite . in quantum mechanics , one starts with a hermite model hamiltonian @xmath5 with a real parameter @xmath6 . here one can assume that the eigenstates @xmath7 ( @xmath8 ) of @xmath5 at @xmath9 can be an appropriate basis with clear characters to classify the properties of the eigenstates @xmath10 ( @xmath8 ) for finite @xmath6 . now , if the energy expectation values @xmath11 ( @xmath8 ) cross with each other at a certain value @xmath12 , the energy eigenvalues @xmath13 of @xmath10 with finite mixing have level repulsion , i.e. , anticrossing at @xmath14 ( see fig . [ fig_1d_2d](a ) ) due to the neumann - wigner non - crossing rule @xcite . at this point , the overlap between @xmath10 and @xmath7 is exceeded by that between @xmath10 and @xmath15 ( @xmath16 ) as @xmath17 . therefore , due to orthogonality , @xmath10 and @xmath18 exchange their characters in terms of the appropriate basis @xmath7 and @xmath15 at the anticrossing point @xmath19 , which we call `` nature transition '' in this paper . in fact , the critical value @xmath14 is very useful to know the internal structure of the quantum states with variation of certain parameter @xmath6 . in this paper , we consider the quantum systems with dissipation into decay channels outside of the model space . such systems are often called _ open quantum systems _ with resonance states , which are effectively described by a non - hermite model hamiltonian @xmath5 with complex energy eigenvalues @xcite . the real and imaginary parts of the eigenvalues correspond to the mass and decay width of the resonance states , respectively . it can be shown in the feshbach reduction formalism that the hamiltonian with a reduced model space becomes non - hermitian @xcite . in such open quantum systems , @xmath20 of @xmath7 can move on the complex energy plane ( see fig . [ fig_1d_2d](b ) ) without having degeneracy at a certain value of @xmath6 except for an accidental case @xcite . therefore , a simple criterion should be newly found to judge the existence of the nature transition between the resonances , and its critical value @xmath14 can be used to know the internal structure of the resonance states depending on the parameter . in this paper , we construct complex two - dimensional ( 2d ) matrix model to discuss the nature transition between two resonance states . ( two dimensions represent two levels of resonances . ) this 2d model will give an elementary understanding for higher dimensional problems because the latter can often be reduced locally to the 2d problems . we show that , by extending @xmath6 to a _ complex variable _ , the geometry on the complex-@xmath6 plane gives a criterion of the nature transition within the real parameter subspace @xmath21 . + after establishing the general framework , we apply it to the hadron physics with strong interaction , which is governed by _ quantum chromodynamics _ ( qcd ) as the @xmath22 gauge theory with color number @xmath23 @xcite . by extending @xmath0 to an arbitrary number , @xmath3-expansion provides a systematic perturbative treatment . the leading order of `` large-@xmath0 qcd '' reproduces lots of qcd phenomenologies @xcite . in fact , in large-@xmath0 qcd , the internal structure of mesons becomes clear : mesons as quark - antiquark ( @xmath24 ) pairs appear with masses of @xmath25 and zero widths , while `` mesonic molecules '' can also appear as resonances with masses and widths increasing along with @xmath0 because the meson - meson interactions are suppressed with @xmath26 @xcite . from such considerations in large-@xmath0 , one often expects that exotics can also be suppressed in the real world @xcite . however the internal structures of hadrons can be easily changed due to the development of hadron dynamics scaled by @xmath3 . here , a basic but essential question arises : _ what is the internal structure of hadrons with continuous variation of @xmath0 from @xmath1 to @xmath2 ? _ to find a typical feature for such @xmath0-dependence of the internal structure of hadrons , we adopt the complex 2d matrix model . by regarding @xmath24 and mesonic molecule states in large-@xmath0 as the appropriate basis @xmath7 ( @xmath27 ) with clear characters , and by identifying @xmath3 to @xmath6 in the complex 2d matrix model , we will calculate a critical color number of nature transitions in terms of appropriate basis from the geometry on the complex-@xmath0 plane . as an example , we investigate the internal structure of @xmath28 meson with admixed nature of @xmath24 and @xmath29-molecule components . for the large-@xmath0 effective theory , we employ the chiral lagrangian induced by holographic qcd with d4/d8/@xmath4 multi - d brane system in the type iia superstring theory @xcite . in sec . [ c2d ] , we formulate complex 2d matrix model . in sec . [ cnc ] , we discuss the application of the model to @xmath0-dependence of internal structure of hadrons . sec . [ sum ] is devoted to summary and outlook . in appendix [ a1 ] , we calculate the attaching number @xmath30 which characterizes the geometry near the origin on the complex - parameter space . in appendix [ a2 ] , we show a simple prescription of writing geometrical maps for arbitrary matrix elements of the model . first we formulate complex 2d matrix model to treat a two - level problem in a quantum system with resonances . we describe resonance states by using the bi - orthogonal representation as @xmath31@xmath32 : its bra - state is defined by the complex conjugate of the dirac bra - state @xmath33 , which was firstly introduced for the unstable nuclei in nuclear physics @xcite . only by taking such bi - orthogonal representation , resonance states with different eigenvalues become orthogonal to each other as @xmath34 , which is needed to employ the matrix representation of operators in such basis . as anticipated , we assume that @xmath31 , the eigenstates of @xmath5 at @xmath9 , are the appropriate basis with clear characters and are useful to classify the quantum states . hence we consider the hamilton matrix @xmath35 $ ] in this basis : @xmath36 where @xmath37 is the energy of @xmath31 and @xmath38 are the interaction satisfying @xmath39 . @xmath40 is a parameter , controlling the development of the two eigenstates @xmath41 which can be obtained in terms of the basis @xmath31 as @xmath42 the coefficients @xmath43 carry the information for the internal structure of the eigenstates @xmath41 in terms of @xmath31 . there is a subtlety for the interpretation of component weights from @xmath43 , since the norms @xmath44 can be complex numbers due to the bi - orthogonality . several attempts have been made to interpret such complex probability of resonances ( for example , see ref . @xcite ) , while a consensus has not been achieved yet . recently ref . @xcite has considered a probabilistic interpretation of resonance states by taking the integral of the modulus square of resonance wave function over a limited spatial domain expanding with the speed of leaking particles . normalization of the resonance wave function over such domain makes the modulus square finite and could be suitable for the probabilistic interpretation . however , as for the expansion coefficients in eq . ( [ de1 ] ) , their probabilistic interpretation still remains unsolved . in this work we simply presume the module , @xmath45 , to be interpreted as the component weights , as it is suitable for narrow resonances . at @xmath9 , @xmath41 coincides with @xmath31 due to @xmath39 , so that @xmath46 for @xmath47 . now , if @xmath5 is hermite with real eigenvalues , the level crossing of @xmath31 is known to give the level anticrossing of @xmath41 as shown in fig . [ fig_1d_2d](a ) @xcite . at this anticrossing point @xmath19 , @xmath41 exchange their characters as `` nature transition '' with the transition condition @xmath48 , where the two basis components @xmath49 and @xmath50 are equally mixed as a character exchanging point . in this paper , we newly consider the case that @xmath5 is non - hermite with complex eigenvalues for resonance states . as we will show below , @xmath51 can be satisfied _ at least _ by the energy coincidence @xmath52 , which can be realized if one extends @xmath6 to a _ complex variable _ @xcite . therefore , to get a geometrical insight for the existence of nature transition , here we introduce the _ complex-@xmath6 plane_. by solving the schrdinger equation : @xmath53 , we find the two eigenvalues @xmath13 ( @xmath54 ) as @xmath55 and the coefficient ratios @xmath56 of the eigenstates @xmath57 ( @xmath54 ) in eq . ( [ de1 ] ) as @xmath58 the upper ( lower ) sign in eqs . ( [ eab_1 ] ) and ( [ rab_1 ] ) corresponds to @xmath59 ( @xmath60 ) . the ratios ( [ rab_1 ] ) are sufficient to discuss the nature transition between two levels as below . now we consider the transition condition @xmath61 on the complex-@xmath6 plane . due to the bi - orthogonality @xmath62 , i.e. , @xmath63 , the transition condition can be written only by the ratios ( [ rab_1 ] ) as @xmath64 , which is equivalent from eq . ( [ rab_1 ] ) to @xmath65=0.\label{cond3}\end{aligned}\ ] ] due to the square root in ( [ eab_2 ] ) , eq . ( [ cond3 ] ) becomes equivalent to the two conditions : @xmath66=0,\label{condf1}\\ & & |a(\lambda)|^4-\{{\rm im}[a(\lambda)^*\overline{v}(\lambda)]\}^2 \leq 0 . \label{cond4}\end{aligned}\ ] ] from eq . ( [ condf1 ] ) , @xmath67|=|a(\lambda)||\overline{v}(\lambda)|$ ] , so that the condition ( [ cond4 ] ) becomes @xmath68 since @xmath6 s for @xmath69 trivially satisfy the conditions ( [ condf1 ] ) and ( [ condf2 ] ) . now the `` transition line '' is defined as the region satisfying @xmath61 , i.e. , both conditions ( [ condf1 ] ) and ( [ condf2 ] ) on the complex-@xmath6 plane . therefore , the line ( [ condf1 ] ) , named `` line 1 '' , can be the candidate of the transition line , and the region ( [ condf2 ] ) with the boundary @xmath70 , named `` line 2 '' , selects the proper part for the transition line . the region ( [ condf2 ] ) always excludes the origin @xmath9 for the case @xmath71 , because @xmath72 and @xmath73 . then , if the transition line crosses the real-@xmath6 axis , the nature transition occurs at the crossing point @xmath74 ( see schematic fig . [ fig_1](a ) ) . now , from ( [ condf1 ] ) and ( [ condf2 ] ) , the crossing points @xmath75 ( @xmath76 ) of line 1 and line 2 satisfy the condition @xmath77 for @xmath78 , which is equivalent to @xmath79 therefore , at @xmath80 , the mass difference in eq . ( [ eab_1 ] ) becomes zero and two eigenvalues coincide as @xmath81 . @xmath82 are called the `` exceptional points '' on the complex-@xmath6 plane @xcite . in fact , the importance of the exceptional points has been intensively studied both theoretically @xcite and experimentally @xcite in the area of quantum chaos , where the dense exceptional points on the complex-@xmath6 plane correspond to the development of quantum chaos in the energy - level statistics @xcite . now , in this paper , we can show that line 1 and line 2 cross each other at all exceptional points , so that these points can always be the _ end points _ of the transition lines . therefore , the location of the exceptional points is very important to geometrically judge the existence of @xmath83 . + + one simple example is the `` linear-@xmath6 model '' with hamilton matrix : @xmath84 where @xmath85 and @xmath86 ( @xmath87 ) are @xmath6-independent quantities . two exceptional points and one transition line appear ( see fig . [ fig_1](b ) ) , which are simply checked from the power counting about @xmath6 in ( [ excond1 ] ) . in this model , eq . ( [ condf1 ] ) can be equally written with @xmath88 , @xmath89 , and @xmath90 as @xmath91\}^{-1 } ) \label{circle1}\end{aligned}\ ] ] so that line 1 is a circle crossing at @xmath9 , @xmath92 and @xmath93 , and the transition line has an arc shape . ref . @xcite shows that , in the linear-@xmath6 model , the eigenvalue behaviors for @xmath94 depend on the location of the two exceptional points ; if the two locate on the opposite sides striding over the real-@xmath6 axis ( see in fig . [ fig_two_excep](a ) ) , level anticrossing / width crossing occurs , while , if not ( see fig . [ fig_two_excep](b ) ) , level crossing / width anticrossing occurs as in fig . [ fig_two_excep](c ) . therefore , by comparing fig . [ fig_1](b ) and fig . [ fig_two_excep ] , we can newly suggest that the nature transition occurs only in the level anticrossing / width crossing case . in this way , as for the linear-@xmath6 model with two exceptional points , we can relate the behaviors of poles on the complex - energy plane and their internal structures through the geometry on the complex-@xmath6 plane . the linear-@xmath6 model also suggests that , if @xmath95 , the radius of the circle of line 1 in eq . ( [ circle1 ] ) diverges : @xmath96 , so that there is no nature transition for finite @xmath6 . there only occurs the mixing of the basis components up to @xmath97 at most . therefore the @xmath6-dependence in the diagonal components of the matrix form ( [ h_1 ] ) is needed to have the nature transition between resonance states . + we can show more general cases of the geometrical map as the `` @xmath98 model '' by using the following matrix : @xmath99 where @xmath100 and @xmath101 are the powers of @xmath6 in the matrix elements . as examples , geometrical maps of @xmath102 and @xmath103 with @xmath85 and @xmath86 fixed are presented in fig . [ pqr_fig ] . by changing the values of @xmath100 and @xmath101 , various types of geometry can be observed on the complex-@xmath6 plane . [ pqr_fig ] and fig . [ fig_1](b ) classified as @xmath104 also show that the geometry around @xmath105 is rather independent of the values of @xmath106 and @xmath101 with the same values of @xmath85 and @xmath86 . on the other hand , @xmath107 effectively corresponds to the linear-@xmath6 model with @xmath95 , so that there is no nature transition for finite @xmath6 as discussed below eq . ( [ circle1 ] ) . ( if @xmath85 and @xmath86 ( @xmath108 ) are specially chosen for line 1 to exactly cross the point @xmath105 , @xmath107 still corresponds to the linear-@xmath6 model with @xmath95 where line 1 as a circle of infinite radius coincides with the real-@xmath6 axis itself . ) these features can be roughly understood as follows ; first , let us consider a situation that line 1 of eq . ( [ condf1 ] ) appears near @xmath105 , i.e. , eq . ( [ condf1 ] ) is satisfied at @xmath109 ( @xmath110 ) as @xmath111 } = 0,\label{formxi1}\end{aligned}\ ] ] with @xmath112 and @xmath113 . lhs of eq . ( [ formxi1 ] ) can be expanded up to @xmath114 as @xmath115 } + { \rm re}{\left [ } \xi^\ast v_{11}^\ast \bar{v } { \right ] } p -{\rm re}{\left [ } \xi^\ast v_{22}^\ast \bar{v } { \right ] } q + { \rm re}{\left [ } ( \delta\varepsilon + \delta v)^\ast \xi \bar{v } { \right ] } r = 0 , \label{formxi2}\end{aligned}\ ] ] with @xmath116 . ( [ formxi2 ] ) implies that , with increase of @xmath117 , @xmath118 and @xmath119 , @xmath120 tends to decrease to maintain eq . ( [ formxi2 ] ) . that is , a part of line 1 approaches asymptotically to @xmath105 as its `` fixed point '' . actually , @xmath117 , @xmath118 and @xmath119 determine , via eq . ( [ formxi2 ] ) , the order of @xmath120 . furthermore , line 1 always crosses the point @xmath9 for @xmath121 ( see appendix [ a1 ] ) , so that @xmath120 should be _ at least @xmath122 or less _ for any powers with @xmath121 . therefore , @xmath117 , @xmath118 and @xmath119 which are _ sufficiently larger than unity _ tend to develop a power - independent geometry of line 1 in the vicinity of @xmath105 in comparison to a length scale @xmath122 on the complex-@xmath6 plane . so far , we have formulated the model described by the hamiltonian ( [ h_1 ] ) with arbitrary complex functions : @xmath20 and @xmath123 , and studied two - level problems on the complex - energy plane . we have shown that the geometrical map on the complex-@xmath6 plane provides the geometrical insight for the existence of nature transition within the real parameter subspace @xmath94 . for convenience , we supply in appendix [ a2 ] a prescription of writing geometrical maps for arbitrary @xmath20 and @xmath123 , instead of numerically solving the high - powered algebraic equations ( [ condf1 ] ) and ( [ condf2 ] ) . let us now utilize the complex 2d matrix model to find the typical @xmath0-dependence of the internal structure of hadrons . as a demonstration , we consider the @xmath28 meson which has admixed nature of @xmath24 and @xmath29-molecule components . first , we prepare the appropriate basis for the @xmath24 and the @xmath29-molecule states in large-@xmath0 . for the large-@xmath0 effective theory , we make use of the chiral lagrangian induced by holographic qcd with d4/d8/@xmath124 multi - d brane system in the type iia superstring theory @xcite . due to the large-@xmath0 condition of the duality with `` classical '' supergravity , the @xmath125 meson appearing as a gauge field in holographic qcd should correspond to the @xmath24 state . on the other hand , the holographic action also induces the energy - dependent @xmath126-@xmath127 interaction as the weinberg - tomozawa ( wt ) interaction of order @xmath26 . due to its attractive interaction , the non - perturbative @xmath126-@xmath127 dynamics gives a resonance pole as the `` @xmath29-molecule state '' . the @xmath125 meson as the @xmath29-molecule is also studied in the chiral unitary model @xcite . thus , by preparing the @xmath24 and @xmath29-molecule states as the appropriate basis @xmath7 ( @xmath27 ) and identifying @xmath3 to @xmath6 in the complex 2d matrix model , we will calculate the critical color number of the nature transition from the geometry on the complex-@xmath0 plane . here we investigate the scattering equation for the @xmath126-@xmath127 propagator in the @xmath128 channel . by reducing the relativistic eigenvalue equation to the schrdinger equation of the model ( [ h_1 ] ) with a non - relativistic approximation as discussed below , we will derive the geometrical map on the complex-@xmath0 plane for the @xmath125 meson . from the lagrangian in holographic qcd @xcite , we obtain the three - point interaction @xmath129 and the wt interaction @xmath130 in fig . [ fig_diagram ] after proper @xmath131-wave projection @xcite in the form , @xmath132 by taking the two experimental inputs , e.g. , @xmath133mev and @xmath134mev , all the masses and coupling constants of hadrons can be uniquely determined in the holographic approach as @xmath135mev and @xmath136 @xcite . ( in the d4/d8/@xmath124 model , pion is massless , whereas we use an isospin - averaged mass value : @xmath137mev . ) + now we introduce a two - dimensional g - function with @xmath29 and @xmath24 channels , having @xmath128 as the @xmath125 meson : @xmath138 where @xmath139 is a propagator for the @xmath24 state as the @xmath125 meson and @xmath140 is @xmath29 loop function @xcite as @xmath141 with @xmath142 a total incident momentum as @xmath143 . we use a dimensional regularization with the natural condition @xcite to avoid the effect of cdd pole in eq . ( [ gpirho ] ) . in fact , the loop integral of eq . ( [ gpirho ] ) appears in the scattering equation of the t - matrix with the separable approximation for the interactions @xcite . then one can sum up the diagonal component of the potential in eq . ( [ gform1 ] ) as @xmath144 with @xmath145 . we numerically find that @xmath146 has single resonance pole above the @xmath29 threshold as @xmath147 this pole appears due to non - perturbative dynamics between @xmath126 and @xmath127 through the 4-point coupling @xmath130 , so that we interpret @xmath148 in eq . ( [ g_molecule1 ] ) as the propagator of `` @xmath29-molecule state '' with a wave function renormalization factor @xmath149 . to renormalize ( [ g_molecule1 ] ) , @xmath149 can be attached to the interaction sector by @xmath150 as @xmath151 where the first term is the inverse of the `` free '' propagator for the @xmath29-molecule state and the @xmath24 state as the @xmath125 meson . now , by solving the relativistic eigenvalue equation for @xmath152 as @xmath153 we have arrived at two - level model for the @xmath125 meson with the @xmath29-molecule and the @xmath24 components having proper mixing . now , to get the geometrical map on the complex-@xmath0 plane for the @xmath125 meson , we reduce eq . ( [ kg1 ] ) to the schrdinger equation for eq . ( [ h_1 ] ) , with a non - relativistic approximation . we approximate the molecule propagator and the renormalization factor in eq . ( [ g_molecule1 ] ) as @xmath154 and @xmath155 estimated at @xmath156 in mev unit . we also approximate the @xmath24 propagator and the coupling constant as @xmath157 and @xmath158 at @xmath159 in mev unit . such energy fixing has been traditionally employed in nuclear - physics shell - model study , where the absorptive effects into decay channels outside of the model space are represented by the non - hermite matrix elements @xcite . then , eq . ( [ kg1 ] ) can be written as @xmath160 with @xmath161 . from the schrdinger equation ( [ nonrela1 ] ) , we can construct the two dimensional hamilton matrix as @xmath162 now we evaluate @xmath0-counting for the matrix elements in eq . ( [ hnonrela ] ) . according to large-@xmath0 qcd @xcite , @xmath163 , @xmath129 and @xmath140 @xcite have @xmath0-dependence as @xmath164 for energy region far from the threshold ; @xmath165 , the weinberg - tomozawa interaction ( [ v_wt ] ) can be simplified as @xmath166 as the mesonic four - point interaction @xcite . therefore , eq . ( [ g_molecule1 ] ) can be rewritten as @xmath167 by comparing eqs.([g_molecule1 ] ) and ( [ wtmod ] ) , we can also estimate the @xmath0 dependence of @xmath168 and @xmath169 as @xmath170 where energy dependence of the loop function @xmath140 is approximately ignored . such increasing behavior of @xmath168 with @xmath0 as in eq . ( [ count2 ] ) can also be observed in the second reference of @xcite . by using eq . ( [ count2 ] ) , we can also estimate the @xmath0-dependence of energy scale @xmath171 introduced in eq . ( [ nonrela1 ] ) as @xmath172{n_c } ) . \label{count3}\end{aligned}\ ] ] by using eqs . ( [ count1 ] ) , ( [ count2 ] ) and ( [ count3 ] ) for the matrix elements in eq . ( [ hnonrela ] ) , we eventually get the complex 2d matrix model for @xmath125 meson with @xmath0 dependence factored out by @xmath6 as @xmath173{3/n_c},\hspace{15mm}\end{aligned}\ ] ] where @xmath168 , @xmath169 , @xmath129 and @xmath171 in ( [ h_c2da1 ] ) are the constants estimated at @xmath23 as shown above eq . ( [ nonrela1 ] ) . @xmath174 and @xmath163 in eq . ( [ h_c2da1 ] ) are the energies of the @xmath29-molecule state and the @xmath24 state , and they form the appropriate basis . the ( 1,1 ) element with negative power of @xmath6 reflects that a resonance state appears due to highly nonperturbative hadron dynamics . then , by applying the conditions ( [ condf1 ] ) and ( [ condf2 ] ) to the hamiltonian ( [ h_c2da1 ] ) , we can get the geometrical map on the complex-@xmath0 plane for the @xmath125 meson in fig . [ fig_2 ] . ( a prescription of writing geometrical maps is shown in appendix [ a2 ] . ) six exceptional points and four transition lines , two of which are half - lines , appear on this map . these numbers can be derived from the power counting about @xmath6 in eq . ( [ excond1 ] ) . the transition line shown by the solid curve can cross the real @xmath6 axis between @xmath9 ( @xmath175 ) and @xmath105 ( @xmath23 ) . the crossing point shows a critical color number for transition as @xmath176{3/n_c}\sim 0.93 $ ] , i.e. , @xmath177 . this result indicates that , with continuous change of @xmath0 from @xmath1 to @xmath2 , the internal structures of two hadronic states can be exchanged in terms of appropriate basis @xmath24 and @xmath29-molecule at the critical color number @xmath177 . such a critical color number with character exchange for the @xmath125 meson is also reported from the analysis of the pole residues in ref . @xcite . in this way , by looking into the existence of nature transition from the geometry on the complex-@xmath0 plane , we can discuss the typical @xmath0-dependence of the internal structure of hadrons from @xmath175 to @xmath2 . finally , we check the stability of the present results , especially that of the appearance of the critical color number near @xmath23 , when corrections are made in the @xmath0-counting of eq . ( [ h_c2da1 ] ) . below eq . ( [ hnonrela ] ) , we have adopted two approximations : i ) @xmath178 for @xmath165 , and ii ) neglecting the energy dependence of @xmath140 . these two are introduced to perform a simple @xmath0-counting of the matrix elements in eq . ( [ h_c2da1 ] ) as a first study . by including effects of their energy dependence , the counting in eq . ( [ h_c2da1 ] ) can be moderately changed . first , eq . ( [ h_c2da1 ] ) can be classified as @xmath179 of the @xmath98 model ( [ pqrmodel ] ) . now , there are three physical constraints ; i ) @xmath180 : mass and decay width of molecule resonance should increases at large-@xmath0 , ii ) @xmath181 : mass of @xmath24 is independent of @xmath0 , and iii ) @xmath121 : qcd should become a free meson theory at large-@xmath0 @xcite . with these constraints , we have checked various types of geometrical maps on the complex-@xmath0 plane for physical sets of @xmath142 , @xmath182 and @xmath101 ( for example , see fig . [ nc_maps ] for @xmath183 , @xmath184 and @xmath185 as test cases ) . we find that the appearance of the critical color number near @xmath23 in this paper is not so much affected by the corrections of the @xmath0-counting in eq . ( [ h_c2da1 ] ) . this is reasonably expected because @xmath23 corresponds to the fixed point on the complex-@xmath0 plane as discussed in the @xmath98 model ( [ pqrmodel ] ) . we have formulated the complex 2d matrix model to get typical features about the parameter dependence of the internal structure of resonances . we suggest that the geometry on the complex - parameter space will give a simple criterion for the nature transition between resonance states within the real - parameter subspace . by applying the model to hadron physics , we have discussed the @xmath0-dependence of the internal structure of hadrons from the geometry on the complex-@xmath0 plane . we show that , with continuous change of @xmath0 from @xmath1 to @xmath2 , the internal structures of hadrons can be exchanged in terms of appropriate basis at the critical color number . we hope that the new concept of geometry on the complex-@xmath0 plane and its possible topological classification will shed light on the exotic physics in qcd for the future . our model can be employed to general multi - level problems of resonances to analyze their internal structures with variation of a parameter in each system . wide applications of our model to resonance physics are expected as a future prospect . the authors thank hiroki nakamura , koichi yazaki , and tetsuo hyodo for their fruitful communications . the authors also thank masuo suzuki for his valuable suggestions during the international workshop `` resonances and non - hermitian systems in quantum mechanics ( 2012 ) '' in the yukawa institute for theoretical physics at kyoto university . k. n. thanks naomichi hatano for his meaningful discussions about probabilistic interpretation of resonances . this work is supported by grant - in - aid for scientific research on innovative areas `` elucidation of new hadrons with a variety of flavors '' ( nos . 22105509 ( k. n. ) , 22105510 ( h. n. ) , 24105706 ( d. j. ) and e01:21105006 ( a. h. ) ) from the ministry of education , culture , sports , science , and technology(mext ) of japan . k. n. is supported by the special postdoctoral research program of riken . in this appendix , we show line 1 always crosses the point @xmath9 for @xmath121 in the @xmath98 model . first , we calculate @xmath30 showing how many times line 1 attaches to @xmath9 . ( [ condf1 ] ) can be written in the @xmath98 model ( [ pqrmodel ] ) as @xmath186= \frac{1}{2}{\rm re}[(\delta\varepsilon+ \lambda^p v_{11}-\lambda^q v_{22})^\ast \lambda^r \bar{v}]=0,\label{condf1_pqr}\end{aligned}\ ] ] with @xmath112 and @xmath113 . we define @xmath187 to find a largest contribution within @xmath188 at @xmath189 . then , at @xmath189 , eq . ( [ condf1_pqr ] ) becomes @xmath190= ar^{r+n_a}{\rm re}{\left [ } e^{i\{(r - n_a)\theta+\phi\ } } { \right ] } = 0,\label{condf1_n_a}\end{aligned}\ ] ] with @xmath191 and @xmath192 . now there are two cases as follows . 1)@xmath193 : by definition , @xmath194 so that @xmath195 and @xmath196 becomes unity or divergent at @xmath197 . therefore , eq . ( [ condf1_n_a ] ) is not satisfied at @xmath9 , i.e. , @xmath198 . here we implicitly exclude the accidental case @xmath199 @xmath200 which makes the equation indefinite . 2)@xmath201 : if the argument _ strongly _ approaches to certain angles at @xmath197 as @xmath202 eq . ( [ condf1_n_a ] ) is satisfied instead of possible divergence of @xmath196 for @xmath203 . ( [ angle1 ] ) is equal to @xmath204 therefore @xmath205 is an angle between nearest - neighboring two segments of line 1 near @xmath9 . @xmath30 is given by @xmath206 in table i , we summarize @xmath30 for various sets of @xmath207 discussed in figs . [ fig_1 ] , [ pqr_fig ] , [ fig_2 ] and [ nc_maps ] . @xmath208 @xmath30 in table i is consistent with the resultant geometry of line 1 around @xmath9 in figs . [ fig_1 ] , [ pqr_fig ] , [ fig_2 ] and [ nc_maps ] . now , if @xmath121 consistently with @xmath39 in eq . ( [ h_1 ] ) , @xmath201 because @xmath194 . therefore @xmath209 and line 1 always crosses @xmath9 . + in this appendix , we show a prescription of writing geometrical maps . ( [ condf1 ] ) and a boundary of eq . ( [ condf2 ] ) can be rewritten as algebraic equations : @xmath210 where @xmath211 is a real function of @xmath212 and @xmath59 ( @xmath60 ) corresponds to eq . ( [ condf1 ] ) ( boundary of eq . ( [ condf2 ] ) ) . by drawing a three - dimensional plot of @xmath213 , _ contours _ at @xmath214 correspond to line 1 ( for @xmath59 ) and line 2 ( for @xmath60 ) ( see , e.g. , fig . [ 3dim_plot ] ) . in this way , by using a contour method , one can get geometrical maps without numerically solving the high - powered algebraic equations ( [ condf1 ] ) and ( [ condf2 ] ) . if @xmath0-dependent cut - off @xcite is considered , @xmath140 would have complicated @xmath0-dependence ( even in a non - analytic way ) . in our present study , to simply perform the analytic continuation into the complex-@xmath0 plane , we alternatively use @xmath0-independent cut - off corresponding to the mass scale of @xmath24 meson , which is also suggested as another candidate in ref . @xcite .

we study the parameter dependence of the internal structure of resonance states by formulating complex two - dimensional ( 2d ) matrix model , where the two dimensions represent two - levels of resonances . we calculate a critical value of the parameter at which `` nature transition '' with character exchange occurs between two resonance states , from the viewpoint of geometry on complex - parameter space . such critical value is useful to know the internal structure of resonance states with variation of the parameter in the system . we apply the model to analyze the internal structure of hadrons with variation of the color number @xmath0 from @xmath1 to a realistic value @xmath2 . by regarding @xmath3 as the variable parameter in our model , we calculate a critical color number of nature transition between hadronic states in terms of quark - antiquark pair and mesonic molecule as exotics from the geometry on complex-@xmath0 plane . for the large-@xmath0 effective theory , we employ the chiral lagrangian induced by holographic qcd with d4/d8/@xmath4 multi - d brane system in the type iia superstring theory .

the effects of substitutions of magnetic impurities on the antiferromagnetic spin chain have attracted great interests in the past decade . it has been shown theoretically that the ground state properties vary with different dilution cases . for the random substitutions , the most interested case is of the s=1/2 impurities in haldane chain @xcite . for example , inelastic neutron scattering experiment on the compound y@xmath4banio@xmath5 substituting ca@xmath6 for ni@xmath6 @xcite show a substantial increase of the spectral function below the haldane gap to indicate the creation of states below the energy of the spin gap . this effects are also studied by numerical works by s. wessel @xcite . for regular substitutions , these systems are the mixed - spin chains which have been extensively studied by many authors in the past a few years . analytical methods of non - linear sigma model , mean field theory and spin - wave method @xcite as well as numerical works by density matrix renormalization group @xcite and quantum monte carlo @xcite have been applied extensively for such systems . so far , it is well known that the topology of spin arrangements in the mixed chains plays an essential role on the ground state properties and thermodynamics in the mixed - spin systems . experimently , many quasi-1d mixed - spin materials have been synthesized in the past two decades , such as acu(pba)(h@xmath4 o)@xmath7 @xmath8 n(h@xmath4 o ) and acu(pbaoh)(h@xmath4 o)@xmath7 @xmath8 nh@xmath4o ( where pba=1,3-propylenebis(oxamato),pbaoh=2-hydroxo-1,3-propylenebis and a= ni , fe , co , mn , zn ) . these materials contains two different transition metal ions per unit cell , and their properties were studied as ferrimagnetic chains @xcite . the experiment results imply that the magnetic properties of the mixed - spin compounds can all be described by a heisenberg model with nearest - neighbor antiferromagnetic coupling as @xmath9 where @xmath10 denotes a spin-@xmath11 moment at site @xmath12 , @xmath13 is system size and @xmath14 . t. fukui and n. kawakami @xcite have studied spin chain composed by a periodic array of impurities @xmath1 embedded in the host @xmath15 spin chain with the period @xmath16 , i.e. , @xmath17 the dilutions of the model denoted by the impurity concentration @xmath18 has two limits : ( _ i _ ) @xmath19 , the undoped pure antiferromagnetic @xmath20 chain , it has a non - magnetic ground state ; ( _ ii _ ) @xmath21 , the alternating spin chain of @xmath1 and @xmath20 . according to marshall theorem and lieb - schultz - mattis ( lsm ) theorem @xcite , ground state of the doped cases are specified by the spin quantum number @xmath22 for @xmath23 or @xmath24 , it is either a spin singlet or ferrimagnetic . if the effective spin in a unit composed of @xmath16 spins @xmath25 is half - integer , so the system has a gapless energy spectrum . but when @xmath25 is integer , lsm theorem fails to predict the energy spectrum to be gaped or gapless . by applying non - linear @xmath26 model , it is found that the system has an energy gap when the @xmath25 is integer@xcite . but details of ground state properties and thermodynamics can not be given by non - linear @xmath26 model analyses . the authors of present paper have recently studied the model ( 1 ) with the case of @xmath27 and @xmath28 by applying quantum monte carlo simulations @xcite , where the numerical results reveal different non - trivial magnetic properties happened between two kinds of diluting cases , i.e. for @xmath29 @xmath28 spins in a unit , system has magnetic ground state and it shows ferrimagnetic features ; while for @xmath24 @xmath28 spins in a unit , systems behave non - magnetic ground states with antiferromagnetic - like features . for both the @xmath30 cases , the ground states are _ gapless _ steadily . and the system gradually transits from the ferrimagnetic ground state of the alternating @xmath1-@xmath20 chain to the disordered ground state of pure @xmath31 chain in two different tendencies . in this letter , we study an opposite case with @xmath0 and @xmath32 . previous analytical work predicted that if @xmath29 @xmath32 spins in a unit , the effective spin @xmath25 is half - integer , the ground state is ferrimagnetic with a gapless energy spectrum ; while if @xmath24 @xmath32 in a unit , @xmath25 is integer , the ground state is non - magnetic and the system has an energy gap . our numerical study will focus on how the ground state properties depend on the concentration and the finite temperature magnetic properties evolute with decreasing of the @xmath0 concentration @xmath18 . we use the efficient continuous imaginary time version of loop cluster algorithm to perform the quantum monte carlo simulation @xcite , which has been successfully applied for the other mixed - spin chains @xcite . we confine our calculation to isotropic antiferromagnetic coupling cases , i.e. @xmath33 in equation ( [ hamiltonian ] ) , and the positions of spin @xmath34 and @xmath35 are arranged as represented in equation ( [ spin_arrange ] ) with @xmath16 taking the values from 2 to 11 . we carry out @xmath36 monte carlo steps for measuring physical quantities after @xmath37 monte carlo steps for the thermalization . in order to clearly explore the ground state properties , the simulations are performed at the very low temperature @xmath38 for system sizes @xmath39 in condition of even number of unit . the physical quantities we measure are the ground state energy @xmath40 , the uniform magnetic susceptibility @xmath41 and staggered susceptibility @xmath42 by using the improved estimators in the loop cluster algorithm , e.g. , @xmath43 @xmath44 where @xmath45 is winding number of cluster @xmath46 , and @xmath47 is the cluster size . the magnetization and staggered magnetization are estimated by @xmath48 and @xmath49 the energy gap @xmath50 is also estimated in the way given by todo , @xcite @xmath51 where @xmath52 is the correlation length in the imaginary time direction . the results for magnetizations and uniform susceptibility are plotted in fig . [ mz ] and fig . [ sus_u ] . we find that the magnetic properties are apparently different for two cases of @xmath23 and @xmath53 . when @xmath53 , the magnetization is finite and approaches zero linearly with decreasing of @xmath18 . while @xmath23 , the magnetization remains almost at zero value . on the other hand , it can be observed from our results that the uniform susceptibilities @xmath41 is finite for @xmath53 , but it vanishes when @xmath23 . thus there is magnetic long - range order ( lro ) in the ground state when @xmath53 , but the order is absent when @xmath53 . we further estimate the staggered magnetization and its susceptibility as a function of concentration shown in fig . [ smz ] and fig . [ sus_s ] respectively . the two observables are both finite for @xmath23 and @xmath53 cases , but the data for the cases of @xmath53 have much stronger values than the cases of @xmath23 . in order to confirm the results observed above , we begin to investigate the finite temperature uniform magnetic susceptibility . as displayed in fig . [ sus_t ] , one can easily find that @xmath41 diverges when the temperature @xmath54 goes to zero in the cases @xmath53 . this is the typical behavior of a system with magnetic lro . in the cases @xmath23 , all the @xmath41 approach zero when @xmath55 , a remarkable evidence to reveal the existence of the energy gap . up to now , our results verify numerically that there are magnetic lro and antiferromagnetic lro in the ground states when @xmath53 . they clearly show that the ground states are ferrimagnetic in such cases . while for @xmath23 , there should exist of spin liquid phases denoted by the vanish of the magnetizations . consequently we believe our numerical results consist correctly with the previous analytical predictions . more important , one can easily see that the magnetism decreases with decreasing of impurity concentration in the case of @xmath53 . but there is not notable change of the magnetic properties when the @xmath0 concentration decreases as @xmath23 . next , we consider the feature of the energy gap @xmath50 on different regular dilutions . not surperised for us , the energy gap is closed when @xmath16 is @xmath24 and it opens again while @xmath16 is @xmath29 as shown in fig . these results is consist with the prediction by non - linear @xmath26 model and lsm theorem @xcite . it is interesting that the energy gap @xmath50 tends to be narrow as decreasing of @xmath0 concentration when @xmath23 . we confirm such behavior by fitting @xmath50 to the curve of @xmath56 as one can see in fig . [ gap ] . moreover , we show the finite - size effect of @xmath50 results for several cases with @xmath57 increasing in fig . [ size ] . in our estimations , although the gaps are not exact closed for @xmath53 due to the finite - size simulations , we find the data of the gaps decrease fast than @xmath58 , so it is obvious that the gaps will trend to zero as @xmath59 . for the cases @xmath23 , where the gap opens all the time , there is almost no finite - size effect . in order to identify the ground state phases , we calculate the valence - bond - solid ( vbs ) @xcite order parameter @xmath60\rangle,\ ] ] according to the lsm theorem , @xmath61 vanishes in the gapless phase as system size @xmath62 . on the other hand , one expects that @xmath61 varies in between @xmath63 but @xmath64 in a given gaped phase . in exact vbs states , @xmath65 @xcite . our calculations are plotted in fig . it is clear that @xmath66 for all cases of @xmath23 to present the system located in a vbs phase ; while @xmath67 , it reveals the gapless energy spectrum for all @xmath53 cases . especially , all these ground state phases can be understood under the scenario of vbs picture . in vbs picture , each impurity @xmath0 can be regarded as two spin-1/2 in a triplet state , these two spin-1/2 can form singlet with their nearest neighbor s=1/2 spin due to the antiferromagnetic coupling . when @xmath23 , each unit have _ even _ number of s=1/2 host spins , so they can fall into singlets with their nearest neighbors including the two spin-1/2 of @xmath31 to induce the vbs order as seen in fig . [ illus_1 ] ( a ) . as a result , the system now shows a gaped energy spectrum . = 3.5 cm but for @xmath53 , _ odd _ number of spin-1/2 exist in a unit and there will be an active spin which is not used to form singlet as shown in fig . [ illus_1](b ) , thus there is no vbs order and the system emerges no spin gap . our results of the vbs order parameter @xmath61 clearly verify this picture , @xmath68 when @xmath23 and @xmath67 when @xmath53 as shown in fig . [ vbs ] . at last , we note that the vbs phase is stable with the variation of @xmath0 concentration @xmath18 when @xmath23 . our monte carlo study verifies that two brunches of different magnetic behaviors emerge in cases of regular s=1 diluted s=1/2 host chains . according marshall theorem , the cases with @xmath53 have the ferrimagnetic ground states which can be specified by quantum number @xmath69 , so the magnetization per site is finite and it decreases linearly as a function of @xmath18 to @xmath70 , the case of the pure s=1/2 antiferromagnetic heisenberg chain . this feature can be easily observed from our results in fig . [ mz ] and fig . when @xmath16 is @xmath29 , the ground state is singlet with @xmath71 , thus the magnetization per site keeps zero and this is a non - magnetic state . as observed in our simulations there is no notable variations of the ground state magnetic properties in the cases of @xmath23 . to compare our ground state results of the model in this letter for @xmath0 and @xmath32 ( _ system i _ ) with the one we studied previously @xcite when @xmath27 and @xmath28 ( _ system ii _ ) , we collect the main points of the numerical calculations in table i. .comparison of ground state properties of two model , where @xmath25 is effective spin in a unit , @xmath72m@xmath73 is magnetization , @xmath72m@xmath74@xmath73 is staggered magnetization , @xmath41 is uniform susceptibility , @xmath42 is staggered susceptibility , @xmath50 is energy gap and @xmath61 is vbs order parameter . [ cols="^,^,^,^,^ " , ] one can easily see that both systems behave with two kinds of different ground state phases , magnetic or non - magnetic , respectively . if @xmath53 , the ground states are ferrimagnetic for both _ system i _ and _ system ii _ , and their magnetizations and staggered magnetizations are all finite and decrease linearly with decreasing of impurity concentration . however , for the cases of @xmath23 , there appears vbs order in _ system i _ which is gaped , but the order is absent in _ system ii _ where the spin arrangements can not induce such order , so the gap is constantly closed . this feature reveals that this topological order plays an important role to the behavior of the energy gap in the mixed - spin system . we believe that the fitted relation of @xmath75 , to denote the energy gap as function of @xmath0 concentration , provides a good stuff to study how the topological order affects the energy gap in the mixed - spin systems . in conclusion , we have studied the ground state and finite temperature magnetic properties of the regular @xmath0 diluted in @xmath32 antiferromagnetic chain . our calculations show that there exist different phases in the ground state as a function of @xmath1 concentration . when there is one @xmath1 impurity and @xmath29 number of host @xmath20 spins in a unit cell , the ground states are ferrimagnetic and the system has a gapless energy spectrum . the ferrimagnetism becomes weaker as the impurity concentration reduced . while for one @xmath1 and @xmath24 number of @xmath20 in one unit cell , the ground state is a vbs phase where there is a gaped energy spectrum and the energy gap gradually approaches to zero with decreasing the concentration @xmath18 . an interesting observation is that the behavior of the energy gap can be numerically well fitted by @xmath76 . further analytical work , for exmaple using the mean - field theory @xcite , is required to explain why such dependence of the energy gap exist in vbs phases . the authors would like to thank prof . jianhui dai for stimulating discussions and comments . this work was supported in part by the nnsf and srfdp of china , and by the nsf of zhejiang province . 99 e. s. sorensen and i. affleck , phys . rev . b * 51 * , 16115(1995 ) j.f . ditusa , s .- w . cheong , j .- h . park , g. aeppli , c. broholm and c.t . chen , phys . lett . * 73*,1857(1994 ) s. wessel and s. haas , phys . b * 65 * , 132402(2002 ) t. fukui and n. kawakami , phys . rev . b * 55 * , r14709(1997 ) ; _ ibid . _ * 56 * , 8799(1997 ) k. takano , phys b * 61 * , 8863(2000 ) . c .- wu , b. chen , x. dai , y. yu , and z .- b . su , phys . b * 60 * , 1057(1999 ) . s. yamamoto , t.fukui , k. maisinger and u. schollwock , j. phys.:condens . matter * 10 * , 11033(1997 ) s.k . pati , s. ramasesha and d. sen , phys . b * 55 * , 8894(1997 ) ; s.k . pati , s. ramasesha and d. sen , j. phys . matter * 9 * , 8707(1997 ) . t. tonegawa , t. hikihara , m. kaburagi , t. nishino , s. miyashita , and h .- j . mikeska , j. phys . * 76 * , 1000(1998 ) . s. yamamoto , j. phys . . jpn . * 64 * , 4051(1995 ) . o. kahn , y. pei and y. journaux , _ inorganic materials _ , john wiley @xmath77 sons ltd . , new york , 59 - 114 ( 1992 ) . o. kahn , _ molecular magnetism _ , vch , new york , 1993 . m. verdaguer , a. gleizes , j. p. renard and j. seiden , phys . b * 29*,5144 - 5155(1984 ) . a. gleizes and m. verdaguer , j. am . * 106 * , 3727(1984 ) . y. pei , m. verdaguer , o. kahn , j. sletten and j.p . renard , inorg . chem . * 26*,138(1987 ) . m. hagiwara , k. minami , y. narumi , k. tatani and k. kindo , j. phys . . jpn . * 67*,2209 - 2211(1998 ) . lieb , t. schultz and d.j . mattis , ann . 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the effects of regular s=1 dilution of s=1/2 isotropic antiferromagnetic chain are investigated by the quantum monte carlo loop / cluster algorithm . our numerical results show that there are two kinds of ground - state phases which alternate with the variation of @xmath0 concentration . when the effective spin of a unit cell is half - integer , the ground state is ferrimagnetic with gapless energy spectrum and the magnetism becomes weaker with decreasing of the @xmath1 concentration @xmath2 . while it is integer , a non - magnetic ground state with gaped spectrum emerges and the gap gradually becomes narrowed as fitted by a relation of @xmath3 .

the electroweak standard model ( sm ) has so far been very successful and consistent with experiments . however , it is likely that there exist other models , of which the sm is the effective low energy limit . the future colliding facilities , which will operate at tev scale , are likely to reveal new physics beyond sm . the symmetry - breaking and the non - abelian gauge - boson sectors are the most peculiar natures of sm . a lot have been discussed on the possibilities of the future hadronic and @xmath10 colliders to probe the gauge - boson and the symmetry - breaking sectors . with the recent discussion of the physics possibilities at @xmath9 and @xmath11 colliders @xcite , they might be as important as the hadronic and @xmath10 colliders . they have backgrounds much cleaner than the hadronic colliders and should be as clean as the @xmath10 colliders , and also photon has anomalous gluon and quark contents @xcite that enable one to study qcd directly . the @xmath12 collisions at @xmath10 machines can be realized by directing a low energy ( a few @xmath13 ) laser beam almost head - to - head to the incident positron beam . by compton scattering , there are abundant , hard back - scattered photons in the same direction as the incident positron beam , and carry a substantial fraction of the energy of the incident positrons . therefore , we have the @xmath12 collisions . for details please see refs . other possibilities include the bremsstrahlung and beamstrahlung effects @xcite but these methods produce photons mainly in the soft region @xcite , and beamstrahlung depends critically on the beam structure @xcite . therefore we shall limit all the calculations to @xmath9 collisions produced by the laser back - scattering method . in recent studies of the higgs production in @xmath9 collisions @xcite through @xmath14 the cross section is just a factor of 2 or 3 smaller than that of @xmath15 for @xmath16 tev , and so this production might be a possible channel in searching for the higgs boson . however , the backgrounds have not been fully analysed , therefore we can not draw any decisive conclusions . for the higgs in the intermediate mass range ( imh ) the signature , due to the dominate decay of @xmath17 and hadronic decay of @xmath18 , will be @xmath19 where there are 4 jets plus missing energy in the final state . two of the four jets are reconstructed to the @xmath18 mass and the other two can be reconstructed as a resonance peak at the higgs mass . for this signature the backgrounds are the @xmath0 , @xmath20 and @xmath3 productions when the @xmath18 and @xmath21 bosons decay hadronically into four jets . if the higgs mass falls very close to the @xmath18 or @xmath21 masses , the signal is much more difficult to identify and precise calculation of the backgrounds under the @xmath18 and @xmath21 peaks is necessary . calculations of the @xmath0 , @xmath20 and @xmath3 pair productions are desirable as important potential backgrounds to the imh search through @xmath22 production . in addition , it also suffers backgrounds from the @xmath23 production @xcite with @xmath24 , and @xmath25 . also attention has been focused on the single @xmath18 production in the channel @xmath26 @xcite to probe the @xmath27 coupling and search for any anomalous gauge - boson interactions . to probe the @xmath28 coupling , however , we must go for the boson - pair productions of @xmath29 . another interesting point is that the quartic @xmath30 and @xmath31 couplings first come in in the @xmath32 and @xmath33 productions . the calculation of these processes involves delicate cancellation among the contributions from triple gauge - boson , quartic gauge - boson and the other feynman diagrams , which consist of well - tested vertices . therefore , any anomalous interactions of the triple or quartic gauge - boson vertices will result in deviations from sm predictions . it is then favorable to quantify precisely the production of these gauge - boson pairs , @xmath0 , @xmath1 and @xmath20 , within sm so that any deviations from these predictions will indicate some new physics in the gauge - boson sector . one advantage of these gauge - boson pair productions in @xmath9 collisions over hadronic collisions is that they do not have large qcd background as they do in hadronic collisions . also these processes as probes to test the triple and quartic gauge couplings should be as important as the three gauge - boson productions in @xmath10 colliders@xcite . in this paper we calculate the following processes of boson - pair productions in @xmath12 collisions , @xmath34 the processes in eqs . ( [ wz])([zz ] ) are important because they are the major potential backgrounds to the imh search in the channels of eqs . ( [ wh ] ) and ( [ zh ] ) @xcite . besides , the processes in eqs . ( [ wz])([ww ] ) are important tests for sm because they involve non - abelian gauge couplings . these processes must be quantified precisely within sm before any anomalous triple or quartic gauge - boson interactions can be realized in these channels . the organization of the paper is as follows : we briefly describe the calculation methods including the photon luminosity function in sec . [ ii ] , following which we present the results in sec . [ iii ] , and then summarize in sec . we will also present detail formulas for the matrix elements of the processes involved in the appendix [ amp ] . we use the energy spectrum of the back - scattered photon given by @xcite @xmath35 \ , , % \ ] ] where @xmath36 @xmath37 , @xmath38 is the energy of the incident laser photon , @xmath39 is the fraction of the incident positron s energy carried by the back - scattered photon , and the maximum value @xmath40 is given by @xmath41 it is seen from eq . ( [ lum ] ) and ( [ d_xi ] ) that the portion of photons with maximum energy grows with @xmath42 and @xmath38 . a large @xmath38 , however , should be avoided so that the back - scattered photon does not interact with the incident photon and create unwanted @xmath10 pairs . the threshold for @xmath10 pair creation is @xmath43 , so we require @xmath44 . solving @xmath45 , we find @xmath46 for the choice @xmath47 one finds @xmath48 , @xmath49 , and @xmath50 ev for a 0.5(1 ) tev @xmath10 collider . here we have assumed that the electron , positron and back - scattered photon beams are unpolarized . we also assume that , on average , the number of back - scattered photons produced per positron is 1 ( i.e. , the conversion coefficient @xmath51 equals 1 ) . the @xmath18 and @xmath21 bosons are detected through their leptonic or hadronic decays . so we are not going to impose any acceptance cuts on the @xmath18 and @xmath21 bosons for their detections , instead , we assume some detection efficiencies for their decay products to estimate the number of observed events . on the other hand , @xmath52 can be observed directly in the final state by imposing a typical experimental acceptance , say , @xmath53 we use the helicity amplitude method of ref . @xcite to evaluate the feynman amplitudes , and keep the electron mass @xmath54 finite in all the calculations . there are totally 11 contributing feynman diagrams in the process @xmath55 , 9 in @xmath56 , 18 in @xmath57 , and 6 in @xmath58 , in the general @xmath59 gauge . the helicity amplitudes for the processes of eqs . ( [ wz])([zz ] ) are given in appendix [ amp ] . the processes of eqs . ( [ wh ] ) and ( [ zh ] ) have been calculated in detail in refs . the total cross - section @xmath60 is obtained by folding the subprocess cross - section @xmath61 in with the photon luminosity function of eqs . ( [ lum ] ) and ( [ d_xi ] ) ; shown in appendix [ amp ] . we show the dependence of the cross sections for all the processes of eqs . ( [ wz])([zh ] ) , together with @xmath62 , in fig . [ cross ] . we typically choose @xmath63 = 100 gev in the intermediate mass range , and impose the acceptance cuts of eq . ( [ photoncut ] ) on the photon that occurs in the final state . the cross section of @xmath20 is overwhelming due to a hugh contribution from the feynman diagrams with an almost on - shell @xmath64-channel @xmath52 propagator . this hugh cross section , of order 10 pb in the energy range of 0.52 tev , is an advantage to probe the triple or quartic gauge - boson couplings . the @xmath6 and @xmath7 cross section is of order 0.5 and 1 pb in the same energy range , respectively . for a yearly luminosity of 10 @xmath65 , we have about 5000 @xmath6 and 10000 @xmath7 events . the hadronic branching fraction of both @xmath18 and @xmath21 is @xmath66 and assuming 50% hadronic detection efficiency , we still have 625 observed events for @xmath6 and 3500 events for @xmath7 , which are numerous enough to observe any anomalous gauge - boson interactions . therefore , even with 1% anomalous gauge coupling it could result in about 6 and 35 events in @xmath6 and @xmath7 productions respectively , and in the order of hundreds of events for @xmath8 production . therefore , as mentioned above these gauge - boson pair productions as probes to test the triple and quartic gauge couplings are as important as the three gauge - boson productions in @xmath10 collisions , which are of order 0.1 pb for @xmath672 tev @xcite . @xmath3 production is insignificant at all for the energy range that we are considering . the @xmath68 production is of order 0.10.2 pb for @xmath16 tev and @xmath69 gev , and @xmath70 production is so much smaller that it will never be discovered . in comparison we also show the cross section of @xmath71 , which is dominant over the @xmath72 production for @xmath73 tev . we can see that at @xmath74 tev the @xmath75 cross section is only about a factor of 2.5 ( 2 ) smaller than that of @xmath15 . for the imh search in @xmath68 production total background from @xmath8 , @xmath6 and @xmath3 is about two order of magnitudes larger ( see fig . [ cross ] ) . but from fig . [ ptvv ] we can see that the hugh cross section of @xmath8 can go down sharply by requiring a moderate transverse momentum @xmath76 cut , say @xmath77 gev at @xmath78 tev , on the @xmath8 system to keep the @xmath52-propagator far from being on - shell . further reduction of the @xmath8 cross section can be achieved by central electron vetoing method , i.e. , rejecting events with electrons detected in the central rapidity region ( @xmath79 ) . then the total background from @xmath6 , @xmath3 and @xmath8 is only a few times larger than the imh signal in @xmath22 production . furthermore , if @xmath80-tagging has a high efficiency and the invariant mass reconstruction has a good resolution these backgrounds can be substantially reduced , so @xmath68 production remains a possible channel to search for the imh . however , a more detail analysis taking into account the other backgrounds from @xmath81 and detector resolutions is necessary to establish the higgs - boson signal . in fig . [ mvv ] , we show the dependence of the differential cross section @xmath82 on the invariant mass @xmath83 of the boson pair at @xmath84 and 2 tev . as expected , these curves rise a little bit above their corresponding @xmath83 threshold and then fall gradually as @xmath83 increases further . however , the presence of any anomalous triple or quartic gauge - boson interactions can alter the @xmath6 , @xmath7 , @xmath8 and @xmath68 curves to some extent . these spectra can therefore serve as sm predictions to probe the anomalous gauge - boson sector . we have quantified the productions of @xmath85 , @xmath86 , @xmath33 , @xmath87 within sm , and presented the helicity amplitudes for these processes . these processes can probe the non - abelian gauge sector of the sm , and should be as good as the three gauge - boson pair productions in @xmath10 collisions and better than those in hadronic collisions . the production rate of @xmath20 pair is hugh , and that of @xmath0 and @xmath1 are large enough that a percent - level anomalous gauge - boson interactions can be detected if they exist . on the other hand , the imh search in the @xmath5 channel seems impossible due to hugh background from @xmath8 and @xmath6 . however , we have shown in fig . [ ptvv ] that a @xmath76 cut can substantially reduce the @xmath8 background , together with central electron vetoing method and @xmath80-tagging the total background from boson - pair productions is only a few times larger than the imh signal . nevertheless , a more detail signal - background analysis is needed . in this appendix we present the matrix elements for processes @xmath88 , from which explicit helicity amplitudes can be directly computed . to start with , we introduce some general notation : @xmath89 d^v(k ) \ , , \\ % \gamma^\alpha ( k_1,k_2;\epsilon_1,\epsilon_2 ) & = & ( k_1-k_2)^\alpha \epsilon_1 \cdot \epsilon_2 + ( 2k_2+k_1 ) \cdot \epsilon_1 \epsilon_2^\alpha - ( 2k_1+k_2 ) \cdot \epsilon_2 \epsilon_1^\alpha\ , , \\ % g_{vww } & = & \left \ { \begin{array}{ll } e \cot \theta_{\rm w } \quad & { \rm for\ } v = z \\ e & { \rm for\ } v=\gamma \ , . \end{array } \right . \\ \nonumber % \end{aligned}\ ] ] here @xmath90 and @xmath91 are the electric charge ( in units of the positron charge ) and the third component of weak isospin of the fermion @xmath92 , @xmath93 is the su(2 ) gauge coupling , and @xmath94 , @xmath95 , with @xmath96 being the weak mixing angle in the standard model . dots between 4-vectors denote scalar products and @xmath97 is the minkowskian metric tensor with @xmath98 ; @xmath99 is a gauge - fixing parameter . in figs . [ fey - wz ] and [ fey - ww ] , the momentum - labels @xmath100 denote the momenta flowing along the corresponding fermion lines in the direction of the arrows . we shall always denote the associated spinors by the symbols @xmath101 and @xmath102 for the ingoing and outgoing arrows , which is usual for the annihilation and creation of fermions , respectively . the contributing feynman diagrams for @xmath104 are given in fig . [ fey - wz ] . we define a shorthand notation @xmath105 then the helicity amplitudes are given by @xmath106 \ , , \\ % { \cal m}^{(d , e ) } & = & g_{\gamma ww } \gamma_\alpha(-k_1,\,p_2;\,\epsilon(k_1),\ , \epsilon(p_2 ) ) p^{\alpha\beta}_w(p_2- k_1 ) \nonumber \\ & & \times \left [ \bar u(q_1 ) \gamma_\beta g^w ( e ) \frac{\overlay{/}{p}_1 - \overlay{/}{k}_2 + m_e } { ( p_1-k_2)^2 - m^2_e}\ , \overlay{/}{\epsilon}(k_2 ) g^z(e ) u(p_1 ) \right . \nonumber \\ & & \qquad \qquad \left . + \ ; \bar u(q_1 ) \overlay{/}{\epsilon}(k_2 ) g^z(\nu ) \frac{\overlay{/}{q}_1 + \overlay{/}{k}_2}{(q_1+k_2)^2}\ , \gamma_\beta g^w(e ) u(p_1 ) \right ] \ , , \\ % % { \cal m}^{(f ) } & = & g_{zww } \gamma_\alpha ( k_2,\,k_1;\,\epsilon(k_2),\ , \epsilon(k_1 ) ) p^{\alpha\beta}_w(k_1+k_2 ) \nonumber \\ & & \times\ ; \bar u(q_1 ) \gamma_\beta g^w(e ) \frac{\overlay{/}{p}_1 + \overlay{/}{p}_2 + m_e}{(p_1+p_2)^2 - m_e^2}\ , \overlay{/}{\epsilon}(p_2 ) g^\gamma(e ) u(p_1)\ , , \\ % % { \cal m}^{(g ) } & = & - \bar u(q_1 ) \overlay{/}{\epsilon}(k_1 ) g^w(e ) \frac { \overlay{/}{q}_1 + \overlay{/}{k}_1 + m_e}{(q_1+k_1)^2 - m_e^2 } \overlay{/}{\epsilon}(k_2 ) g^z(e ) \frac { \overlay{/}{p}_1 + \overlay{/}{p}_2 + m_e}{(p_1+p_2)^2 -m_e^2 } \nonumber \\ & & \qquad \overlay{/}{\epsilon}(p_2 ) g^\gamma(e ) u(p_1 ) \ % \noalign{\break } % { \cal m}^{(h ) } & = & - \bar u(q_1 ) \overlay{/}{\epsilon}(k_1 ) g^w(e ) \frac { \overlay{/}{q}_1 + \overlay{/}{k}_1 + m_e}{(q_1+k_1)^2 - m_e^2 } \overlay{/}{\epsilon}(p_2 ) g^\gamma(e ) \frac { \overlay{/}{p}_1 - \overlay{/}{k}_2 + m_e}{(p_1-k_2)^2 -m_e^2 } \nonumber \\ & & \qquad \overlay{/}{\epsilon}(k_2 ) g^z(e ) u(p_1 ) \ \\ % % { \cal m}^{(i ) } & = & - \bar u(q_1 ) \overlay{/}{\epsilon}(k_2 ) g^z(\nu ) \frac { \overlay{/}{q}_1 + \overlay{/}{k}_2}{(q_1+k_2)^2 } \overlay{/}{\epsilon}(k_1 ) g^w(e ) \frac { \overlay{/}{p}_1 + \overlay{/}{p}_2 + m_e}{(p_1+p_2)^2 -m_e^2 } \nonumber \\ & & \qquad \overlay{/}{\epsilon}(p_2 ) g^\gamma(e ) u(p_1 ) \ , , \\ % % { \cal m}^{(j ) } & = & - g^2 m_w^2 x_{\rm w } \tan\theta_{\rm w } \frac{\xi}{\xi(p_2-k_1)^2 -m_w^2}\ , \epsilon(k_1 ) \cdot \epsilon(p_2 ) \ , \epsilon(k_2 ) \cdot j_1 \,\\ % % { \cal m}^{(k ) } & = & - g^2 m_w^2 x_{\rm w } \tan\theta_{\rm w } \frac{\xi}{\xi(k_1+k_2)^2 -m_w^2 } \ , \epsilon(k_1 ) \cdot \epsilon(k_2 ) \ , \epsilon(p_2 ) \cdot j_1 \ , . % \end{aligned}\ ] ] the contributing feynman diagrams for @xmath107 are obtained from those in fig . [ fey - wz ] by replacing the @xmath21 by @xmath52 . the helicity amplitudes for @xmath108 can be obtained from the above expressions by replacing the corresponding @xmath109 and @xmath110 by @xmath111 and @xmath112 respectively , and substituting the @xmath113 in @xmath114 and @xmath115 by -1 . since @xmath116 , diagrams [ fey - wz](e ) and ( i ) do not contribute to the @xmath1 production . the contributing feynman diagrams for the process @xmath118 are shown in fig . [ fey - ww ] . we can also define a shorthand notation @xmath119 then the helicity amplitudes are given by @xmath120 \ , , \\ % % { \cal m}^{(d ) } & = & - \bar u(q_1 ) \overlay{/}{\epsilon}(k_2 ) g^w(e ) \frac{\overlay{/}{q}_1 + \overlay{/}{k}_2}{(q_1+k_2)^2 } \overlay{/}{\epsilon}(k_1 ) g^w(e ) \frac{\overlay{/}{p}_1 + \overlay{/}{p}_2 + m_e}{(p_1+p_2)^2 - m_e^2 } \nonumber \\ & & \qquad \overlay{/}{\epsilon}(p_2 ) g^\gamma(e ) u(p_1 ) \ , , \\ % % { \cal m}^{(e ) } & = & - \bar u(q_1 ) \overlay{/}{\epsilon}(p_2 ) g^\gamma(e ) \frac{\overlay{/}{q}_1 - \overlay{/}{p}_2 + m_e}{(q_1-p_2)^2 - m_e^2 } \overlay{/}{\epsilon}(k_2 ) g^w(e ) \frac{\overlay{/}{p}_1 - \overlay{/}{k}_1 } { ( p_1 - k_1)^2 } \nonumber \\ & & \qquad \overlay{/}{\epsilon}(k_1 ) g^w(e ) u(p_1 ) \ \\ % % { \cal m}^{(f ) } & = & \sum_{v=\gamma , z } g_{vww } d^v(k_1+k_2 ) \ , \gamma_\alpha ( k_1,\,k_2;\ , \epsilon(k_1),\ , \epsilon(k_2 ) ) \nonumber \\ & & \quad \times \,\bar u(q_1 ) \gamma^\alpha g^v(e ) \frac{\overlay{/}{p}_1 + \overlay{/}{p}_2 + m_e } { ( p_1 + p_2)^2 - m_e^2 } \ , \overlay{/}{\epsilon}(p_2 ) g^\gamma(e ) u(p_1 ) \ , , \\ % % { \cal m}^{(g ) } & = & \sum_{v=\gamma , z } g_{vww } d^v(k_1+k_2 ) \ , \gamma_\alpha ( k_1,\,k_2;\ , \epsilon(k_1),\ , \epsilon(k_2 ) ) \nonumber \\ & & \quad \times \ , \bar u(q_1 ) \overlay{/}{\epsilon}(p_2 ) g^\gamma(e ) \frac{\overlay{/}{q}_1 - \overlay{/}{p}_2 + m_e } { ( q_1 - p_2)^2 - m_e^2 } \ , \gamma^\alpha g^v(e ) u(p_1 ) \ , , \\ % % { \cal m}^{(h ) } & = & g_{\gamma ww } p^{\alpha\beta}_w(p_2-k_2 ) \gamma_\alpha ( p_2,\ , -k_2;\ , \epsilon(p_2),\ , \epsilon(k_2 ) ) \nonumber \\ & & \quad \times \ , \bar u(q_1 ) \gamma_\beta g^w(e ) \frac{\overlay{/}{p}_1 - \overlay{/}{k}_1}{(p_1 - k_1 ) ^2 } \ , \overlay{/}{\epsilon}(k_1 ) g^w(e ) u(p_1 ) \ , , \\ % % { \cal m}^{(i ) } & = & g_{\gamma ww } p^{\alpha\beta}_w(p_2-k_1 ) \ , \gamma_\alpha ( -k_1,\ , p_2;\ , \epsilon(k_1),\ , \epsilon(p_2 ) ) \nonumber \\ & & \quad \times \ , \bar u(q_1 ) \overlay{/}{\epsilon}(k_2 ) g^w(e ) \frac{\overlay{/}{q}_1 + \overlay{/}{k}_2}{(q_1 + k_2 ) ^2 } \ , \gamma_\beta g^w(e ) u(p_1 ) \ , , \\ % % { \cal m}^{(j ) } & = & \sum_{v=\gamma , z } g^2 m_w^2 x_{\rm w } \ , \frac{\xi}{\xi(p_2-k_2)^2 - m_w^2}\ , \epsilon(p_2 ) \cdot \epsilon(k_2 ) \ , \epsilon(k_1 ) \cdot j_v \nonumber \\ & & \qquad \times \ , \left \ { \begin{array}{ll } -\tan\theta_{\rm w } & \quad { \rm for}\ ; v = z \\ 1 & \quad { \rm for}\ ; v=\gamma \end{array } \right . \ , % % { \cal m}^{(k ) } & = & \sum_{v=\gamma , z } g^2 m_w^2 x_{\rm w } \ , \frac{\xi}{\xi(p_2-k_1)^2 - m_w^2}\ , \epsilon(p_2 ) \cdot \epsilon(k_1 ) \ , \epsilon(k_2 ) \cdot j_v \nonumber \\ & & \qquad \times \ , \left \ { \begin{array}{ll } -\tan\theta_{\rm w } & \quad { \rm for}\ ; v = z \\ 1 & \quad { \rm for}\ ; v=\gamma \end{array } \right . \ , . % \end{aligned}\ ] ] the contributing feynman diagrams for the process @xmath122 are the same as in fig . [ fey - ww](d ) with the @xmath18-bosons replaced by @xmath21-bosons plus all possible permutations . totally it has six contributing feynman diagrams . they are given by @xmath123 plus those terms with @xmath124 . these matrix elements are to be summed over polarizations and spins of the final state gauge - bosons and fermions respectively , and average over the polarizations of the incoming photon and spins of the initial state electron . then the cross section @xmath60 is obtained by folding the subprocess cross - section @xmath61 in with the photon luminosity function as @xmath125 where @xmath126 and @xmath127 is the sum of the masses of the final state particles . d. borden , d. bauer and d. caldwell , slac preprint , slac - pub-5715 , ( 1992 ) . e. witten , nucl . phys . * b120*,189 ( 1977 ) . v. telnov , nucl . & methods * a294 * , 72 ( 1990 ) ; i. ginzburg , g. kotkin , v. serbo and v. telnov , nucl . instr . & methods * 205 * , 47 ( 1983 ) ; _ idem _ * 219 * , 5 ( 1984 ) . v. n. baier and v. m. katkov , phys . lett . * 25a * , 492 ( 1967 ) ; r. j. noble , nucl . . & meth . * a256 * , 427 ( 1987 ) ; r. b. palmer , ann . sci . * 40 * , 529 ( 1990 ) . e. boos , _ et al . . lett . * b273 * , 173 ( 1991 ) . k. hagiwara , i. watanabe , and p. zerwas , phys . * b278 * , 187 ( 1992 ) . g. jikia , nucl . phys . * b374 * , 83 ( 1992 ) . i. ginzburg , _ et al . _ , nucl . phys . * b228 * , 258 ( 1983 ) ; k. mikaelian , phys . rev . * d30 * , 1115 ( 1984 ) ; j. robinson and t. rizzo , phys . rev . * d33 * , 2608 ( 1986 ) ; g. couture , s. godfrey and p. kalyniak , phys . lett . * b218 * , 361 ( 1989 ) ; e. yehudai , phys . * d41 * , 33 ( 1990 ) ; s. choi and f. schrempp , desy report 91 - 155 , to appear in _ proc . of ee500 european working groups _ , @xmath10 _ collision at 500 gev : the physics potential _ , ed p. zerwas , ( desy , hamburg , 1992 ) ; o. philipsen , desy report 92 - 004 ; a. denner and s. dittmaier , cern report cern - th.6585/92 . v. barger , t. han and r. n. j. phillips , phys . rev . * d39 * , 146 ( 1989 ) ; a. tofighi - niaki and j. f. gunion , phys . rev . * d39 * , 720 ( 1989 ) . v. barger , a. stange , and r. j. n. phillips , phys . rev . * d44 * , 1987 ( 1991 ) .

we calculate the gauge - boson pairs @xmath0 , @xmath1 , @xmath2 , @xmath3 productions in the @xmath4 collisions , where the photon beam is realized by the laser back - scattering method . these processes are important tests for the non - abelian gauge sector of the standard model ( sm ) . precise calculations of these processes can therefore probe the anomalous gauge - boson interactions . besides , these processes are important potential backgrounds for the intermediate mass higgs ( imh ) search in the @xmath5 production . = 10000 * @xmath6 , @xmath7 , @xmath8 and @xmath3 pair productions at tev @xmath9 colliders * dept . of physics & astronomy , northwestern university , evanston , illinois 60208 , usa +

consider a continuous path @xmath0\to\mathbb{r}$ ] . the @xmath1-variation of @xmath2 is defined for @xmath3 by @xmath4 for subdivisions @xmath5 of @xmath6 $ ] . it is well known that the finiteness of @xmath7 is closely related to the possibility of constructing integrals @xmath8 for some functions @xmath9 . the simplest case is when @xmath10 is finite ( @xmath2 has finite variation ) ; then a signed measure @xmath11 ( the lebesgue stieltjes measure ) is defined from @xmath2 , and the integral is well defined for any bounded borel function @xmath9 ; if moreover @xmath9 has left and right limits , then the integral is also a riemann stieltjes integral ( it is the limit of riemann sums ) . if now @xmath2 has infinite variation ( @xmath12 ) but @xmath7 is finite for a larger value of @xmath1 , it was proved by young @xcite that a riemann stieltjes integral can still be constructed as soon as @xmath13 is finite for @xmath14 such that @xmath15 ; as an application , one can consider and solve stochastic differential equations driven by a multidimensional path with finite @xmath1-variation if @xmath16 ( in particular a typical fractional brownian path with hurst parameter @xmath17 ) . if now @xmath1 is greater than 2 , lyons s theory of rough paths @xcite provides a richer framework which is still suitable to consider and solve these equations . on the other hand , one can associate to @xmath2 a metric space @xmath18 which is a compact real tree and which can be used to describe the excursions of @xmath2 above any level ; see @xcite or chapter 3 of @xcite . the tree @xmath19 can be endowed with its length measure @xmath20 , and our aim is to relate the properties of @xmath21 to the questions of @xmath1-variation of @xmath2 and of integration with respect to @xmath2 . these questions are also considered for cdlg paths @xmath2 ( paths which are right - continuous and have left limits ) , since these paths can be considered as time - changed continuous paths . as an application , we consider the case where @xmath22 is a path of a stochastic process such as a lvy process or a fractional brownian motion ( the case of a standard brownian path has been considered in @xcite ) . in section [ trees ] , we introduce the tree @xmath19 and study its basic properties . in particular , in the finite variation case , we work out the interpretation of its length measure @xmath20 by means of the lebesgue stieltjes measure of @xmath2 , extending a result of @xcite ; this result is fundamental for the construction of integrals in section [ integrals ] ( see below ) . we also explain how the tree can be defined in the cdlg case . in section [ pvariation ] , we see in theorem [ dimtree ] ( theorem [ vpomp ] for the cdlg case ) that the finiteness of @xmath7 is related to some metric properties of @xmath19 , particularly its upper box dimension @xmath23 ; more precisely , @xmath24 we give applications of these results to martingales , fractional brownian motions and lvy processes . we prove in particular that upper box and hausdorff dimensions of @xmath19 coincide for fractional brownian motions ( with hurst parameter @xmath25 ) and stable lvy processes ( with index @xmath26 ) ; we also construct an estimator of @xmath25 based on @xmath19 , which can be computed by means of a sequence of stopping times ( proposition [ estimator ] ) . the aim of section [ integrals ] is to construct integrals with respect to @xmath2 by means of the tree . let us assume that @xmath2 is continuous and @xmath27 ( considering the general case adds some notational complication ) . the construction of the integral is based on the following remark ( propositions [ lebesgue ] and [ finitevar ] ) : when @xmath2 has finite variation , the positive and negative parts @xmath28 and @xmath29 of @xmath30 can be viewed as the images of the length measure @xmath20 by two maps @xmath31 and @xmath32 from @xmath19 to @xmath6 $ ] ; thus @xmath33 when @xmath2 has infinite variation , this procedure can still be applied to construct @xmath28 and @xmath29 ; these measures are @xmath34-finite but no more finite . however , can be viewed as a definition of @xmath35 provided the term in the right - hand side is integrable ; this means that the tree can provide a mechanism by means of which @xmath28 and @xmath29 compensate each other . for instance , if @xmath15 , @xmath36 moreover , in this case , the integral defined by coincides with the young integral ( theorems [ young ] and [ cadlag ] ) . consequently , differential equations driven by multidimensional paths with finite @xmath1-variation with @xmath16 enter our framework . actually , we may take @xmath37 for one of the components ( theorem [ edo ] ) ; this is due to the fact that the condition @xmath38 can be replaced by some weaker condition @xmath39 . we also prove that the tree approach can be used to consider multidimensional fractional brownian motions with parameter @xmath40 ( theorem [ intfbm ] ) ; in this case , the right - hand side of should be understood as a generalized integral on @xmath19 ( a limit of integrals on subtrees @xmath41 obtained by trimming @xmath19 ) , and we recover the integrals of the rough paths theory . the is devoted to two results which are needed in the article , and which may also be of independent interest . in appendix [ mixing ] , we prove that increments of fractional brownian motions are asymptotically independent from the past . in appendix [ roughpaths ] , we study the time discretization of integrals in the rough paths calculus , in a spirit similar to @xcite . a lot of work has been devoted to the links between random trees and excursions of some stochastic processes ; these links are an extension of the classical harris correspondence between random walks and random finite trees . historically , they have first been investigated in the context of brownian excursions in @xcite ( see also the courses @xcite ) with the aim of studying branching processes . in order to consider more general branching mechanisms , lvy trees , defined by means of lvy processes @xmath42 without negative jumps , have been introduced and studied in @xcite ; they have been related to the notion of real tree in @xcite . however , we will not focus here on properties of lvy trees ; a lvy tree is indeed a tree which is associated to some continuous process related to @xmath42 ( the height process ) , whereas we will rather consider in our applications the tree which is associated directly to the lvy process @xmath42 . we work out here a nonlinear approach to integration with respect to one - dimensional paths ; consequently , the integral with respect to @xmath43 is not simply related to integrals with respect to @xmath44 and @xmath45 ; moreover , integration with respect to a multidimensional path can be worked out by summing integrals with respect to each component , but this depends on the choice of a frame . in the proofs of this article , the letter @xmath46 will denote constant numbers which may change from line to line . for quantities depending on the path @xmath2 of a stochastic process , we will rather use the notation @xmath47 . in this section , we first define the tree associated to a continuous path , describe its length measure , and extend these objects to cdlg paths . consider a continuous function @xmath48 . the function @xmath49}\omega\ ] ] is a semi - distance on @xmath6 $ ] , where @xmath50}\omega.\ ] ] the quotient metric space @xmath51/\delta,\delta)$ ] is a real tree ; this means that between any two points @xmath52 and @xmath53 in @xmath19 , there is a unique arc denoted by @xmath54 $ ] ( @xmath19 is a topological tree ) , and that @xmath54 $ ] is isometric to the interval @xmath55 $ ] of @xmath56 ; see @xcite . actually , real trees can also be characterized as connected metric spaces satisfying the so - called four - point condition , and one can use this condition to prove that @xmath19 is a real tree ; see @xcite . we will denote by @xmath57 the projection of @xmath6 $ ] onto @xmath19 ; notice that if @xmath22 is constant on some interval @xmath58 $ ] , then all the points of this interval are projected on the same point of @xmath19 . the continuity of @xmath57 follows from the continuity of @xmath2 ; in particular , @xmath19 is compact . in this article we implicitly assume that @xmath2 is not constant , so that @xmath19 is not reduced to a singleton . with its tree @xmath19 represented by dashed lines ( the vertical lines represent points of the skeleton , and each branching point is represented by a horizontal line ) ; maps @xmath59 , @xmath32 and @xmath60 are also depicted . ] we now suppose @xmath61 , or equivalently @xmath62}\omega.\ ] ] an example is given in figure [ fig1 ] . we explain at the end of the subsection how general paths can be reduced to this case . under this condition , @xmath19 becomes a rooted tree by considering as the root of the tree , and we can say that a point @xmath52 is above @xmath53 if @xmath63 $ ] . we consider on @xmath19 the level function @xmath64 defined by @xmath65 then @xmath66 . for @xmath67 in @xmath19 , define @xmath68 so that @xmath69}\omega = \ell(\tau).\ ] ] in particular @xmath70 and @xmath71 . the set @xmath72)$ ] is exactly the set of points above @xmath67 . if now we consider the set @xmath72)\setminus\{\tau\}$ ] of points which are strictly above @xmath67 , it is made of connected components which are subtrees , and which are called the branches above @xmath67 ; each of these branches is the projection of a connected component of @xmath73\setminus\pi^{-1}(\tau)$ ] , and corresponds to an excursion of @xmath2 above level @xmath74 . if there is more than one branch above @xmath67 , then @xmath67 is said to be a branching point ; this means that there is more than one excursion , and the times between these excursions are local minima of @xmath2 ( a local minimum may be a constancy interval ) . on the other hand , if there is no branch above @xmath67 , then @xmath67 is said to be a leaf ; this means that @xmath75 $ ] , so this holds when @xmath76 or when @xmath77 $ ] is a constancy interval of @xmath2 . local maxima of @xmath2 are projected on leaves of @xmath19 , but there may be leaves which are not associated to local maxima . points which are not leaves constitute the skeleton @xmath78 of the tree . we say that @xmath2 is piecewise monotone if there exists a finite subdivision @xmath5 of @xmath6 $ ] such that @xmath2 is monotone on each @xmath79 $ ] . we also say that @xmath19 is finite if it has finitely many leaves . if @xmath19 is not finite , then it has infinitely many branching points , or it has at least a branching point with infinitely many branches above it ; in both of these cases , @xmath2 has infinitely many local minima and is therefore not piecewise monotone . conversely , if @xmath2 is not piecewise monotone , then it has infinitely many local maxima , and each of them is projected on a different leaf of @xmath19 , so @xmath80 is not finite . thus @xmath81 we shall also need an operation called trimming , or leaf erasure , due to @xcite ( see also @xcite ) ; to this end , we introduce the function @xmath82 this is the height of the ( or of the highest ) branch above @xmath67 . in particular , @xmath83 if and only if @xmath67 is a leaf . is represented by dashed lines , and its leaves by double arrows ; the flattened path @xmath84 is represented by dots when it differs from @xmath2 ; times @xmath85 , @xmath86 and @xmath87 , @xmath88 , are respectively represented on the curve by bullets , circles and triangles . ] now consider the trimmed tree @xmath89 then @xmath41 is nonempty if and only if @xmath90 , and in this case , it is a rooted subtree of @xmath19 ( it contains the root @xmath91 ) . an example is drawn in figure [ fig2 ] . as @xmath92 , the tree @xmath41 increases to the skeleton of @xmath19 ; each branch grows at unit speed , and a new branch appears at @xmath67 if @xmath67 is a branching point of @xmath19 such that one of the branches above @xmath67 has height exactly @xmath93 , and another one has height at least @xmath93 . this subtree has been introduced in @xcite and is related to @xmath93-minima and @xmath93-maxima of the path . more precisely , starting with @xmath94 , define @xmath95 ; \omega(t)-\sup_{[s_i^a , t]}\omega<-a\biggr\},\cr \displaystyle s_{i+1}^a:=\inf\biggl\{t\in [ t_{i+1}^a,1 ] ; \omega(t)-\inf_{[t_{i+1}^a , t]}\omega > a\biggr\},\cr \displaystyle n^a:=\inf\{i ; t_i^a\mbox { or } s_i^a=\inf\varnothing\}. } % \ ] ] actually , in the case @xmath61 , @xmath96 is still well defined , but not @xmath97 ( notice in particular that if @xmath2 is a path of an adapted stochastic process , then @xmath86 and @xmath85 are stopping times ) . then @xmath98 is the number of leaves of @xmath41 ; the set of leaves @xmath99 and the set of times @xmath100 are in bijection by means of @xmath57 and its inverse map @xmath101 . moreover @xmath102}\omega=\inf_{[t_i^a , s_i^a]}\omega = \omega(s_i^a)-a \qquad\mbox{for $ 1\le i < n^a$.}\ ] ] the approximation of @xmath19 by @xmath41 can also be interpreted as an approximation of the path @xmath2 ; trimming the tree is equivalent to flattening some excursions of the path . more precisely , let @xmath103 be the projection of @xmath104 on @xmath41 ( assuming @xmath105 ) , and let @xmath106 for the level function @xmath64 defined in . then @xmath41 is the associated tree of @xmath84 . the path @xmath84 is continuous , is obtained from @xmath2 by means of the change of time @xmath107 and satisfies @xmath108 . since @xmath41 is finite , it follows from that @xmath84 is piecewise monotone . actually , if @xmath87 is a time of @xmath109 $ ] at which @xmath2 is minimal ( for @xmath110 ) and if @xmath111 , @xmath112 , then @xmath113 and @xmath114$,}\cr \mbox{$\omega^a$ is nonincreasing on $ [ t_i^a , u_i^a]$. } } % \ ] ] consider now a general continuous map @xmath2 which does not satisfy @xmath61 . then we can again associate the tree @xmath19 by means of @xmath115 defined by , but some of the above properties differ . however , it is still possible to apply the above discussion to an extended path @xmath116 defined on a greater interval , say @xmath117 $ ] , coinciding with @xmath2 on @xmath6 $ ] , and satisfying @xmath118}\omega'$ ] . then the associated tree @xmath119 contains @xmath19 as a subtree , and the projection @xmath120\to\mathbb{t}$ ] is the restriction of @xmath121\to\mathbb{t}'$ ] to @xmath6 $ ] . among these paths , we will only consider the minimal extensions ; they are those such that @xmath122 . this means that @xmath123}\omega,\cr \mbox{$\omega'$ is nondecreasing on $ [ -1,0]$ , nonincreasing on $ [ 1,2]$. } } % \ ] ] let @xmath124 be a time of @xmath6 $ ] at which @xmath2 is minimal and consider @xmath125 ( these points are drawn in figure [ fig3 ] below , in the more general case of paths with jumps ) . we choose @xmath91 as the root of @xmath19 . then @xmath91 belongs to @xmath126 $ ] , the points of @xmath127 $ ] are those such that @xmath128 , and the points of @xmath129 $ ] are those such that @xmath130 ; for the points of @xmath131 $ ] , one has @xmath132 . is the curve augmented by the jumps ( dotted lines ) . are also depicted the map @xmath57 from @xmath133 to @xmath19 , the maps @xmath31 , @xmath32 from @xmath19 to @xmath6 $ ] ; in particular , @xmath134 and @xmath135 . ] in particular , if we trim the tree @xmath19 and if @xmath105 , then the flattened path @xmath84 of is the restriction of @xmath136 to @xmath6 $ ] . moreover , the quantities @xmath98 , @xmath85 and @xmath86 defined in and the similar quantities for @xmath116 satisfy @xmath137 at @xmath138 , the time @xmath139 may be after time 1 , and in this case @xmath96 is not defined . the length measure on @xmath19 is the unique measure @xmath20 which is supported by the skeleton ( the set of leaves have zero measure ) and such that the measure of an arc is equal to its length ; in particular , this measure is @xmath34-finite and atomless . the existence and uniqueness of @xmath20 is elementary for the finite subtrees @xmath41 , and it is not difficult to deduce the result for @xmath19 by letting @xmath140 . it can be identified to either of the two following measures . [ lambda12 ] define @xmath141 where @xmath142 denotes the dirac mass at @xmath67 . then @xmath143 . notice that the number of terms in the sum is at most countable for any @xmath144 in the definition of @xmath145 , whereas it is finite for any @xmath146 in the definition of @xmath147 . the integrals are supported by the interval @xmath148 $ ] for the first one , and @xmath149 $ ] for the second one . proof of proposition [ lambda12 ] the two measures are supported by the skeleton of the tree ; in order to check that they are equal to @xmath20 , it is sufficient to verify that they coincide with it on arcs @xmath150 $ ] for any @xmath67 in the skeleton @xmath151 . the maps @xmath64 and @xmath152 are injective on @xmath153 $ ] , so , if @xmath154 denotes the lebesgue measure on @xmath56 , @xmath155)=\lambda_\mathbb{r}(\ell([o,\tau ] ) ) , \qquad \lambda_2([o,\tau])=\lambda_\mathbb{r}(h([o,\tau])).\ ] ] moreover , @xmath64 induces a bijection between @xmath150 $ ] and @xmath156 $ ] , so @xmath155)=\ell(\tau)-\ell(o)=\delta(o,\tau)=\lambda ( [ o,\tau]).\ ] ] thus @xmath157 . for the study of @xmath147 , notice that @xmath158 is the distance between @xmath159 and any of the highest points above it . when @xmath159 goes from @xmath91 to @xmath67 , then @xmath158 is decreasing ; more precisely , it jumps at @xmath159 , when @xmath159 is a branching point so that no highest point above it is in the direction of @xmath67 ; thus @xmath152 has a finite number of negative jumps , and between these jumps , it is affine with slope @xmath160 . consequently , @xmath152 induces a bijection from @xmath150 $ ] onto its image , and this image has lebesgue measure @xmath161 . we deduce that @xmath162 . the measure @xmath20 is closely related to the two following measures on @xmath6 $ ] . say that an excursion begins at time @xmath163 above level @xmath164 if for some @xmath165 , @xmath166 for @xmath167 . let @xmath168 be the set of beginnings of excursions above any level ; we can define similarly the set @xmath169 of ends of excursions . these two sets are in bijection with each other ; to each beginning @xmath163 of an excursion we can associate its end @xmath170 . if we restrict ourselves to a fixed level @xmath144 , the sets of beginnings and ends of excursions above @xmath144 are at most countable , and we can define @xmath171 [ lebesgue ] assume . the measures @xmath172 and @xmath173 are @xmath34-finite and are respectively the images of @xmath20 by the maps @xmath31 and @xmath32 , and @xmath174 is the image of @xmath172 and @xmath175 by the projection @xmath57 . if does not hold , then , with the notation , the maps @xmath31 and @xmath32 are respectively defined on @xmath176 $ ] and @xmath177 $ ] ; the relation between @xmath178 and @xmath20 ( or between @xmath175 and @xmath20 ) again holds by restricting @xmath20 to @xmath179 $ ] ( or @xmath180 $ ] ) . we only work out the proof under ; the general case is easily deduced by considering an extension of @xmath2 satisfying . we want to compare the measure @xmath178 carried by the set @xmath168 of beginnings of excursions , with the measure @xmath20 carried by the skeleton @xmath151 . if @xmath181 is in @xmath168 , then @xmath182 is in @xmath151 and @xmath183 except if @xmath181 is at a local minimum , or the end of a constancy interval of @xmath2 ; on the other hand , if @xmath67 is in @xmath151 , then @xmath184 and @xmath185 is in @xmath168 except if it is the beginning of a constancy interval of @xmath2 . since there are at most countably many local minima and constancy intervals , we deduce that there exists @xmath186 and @xmath187 such that @xmath188 and @xmath189 are at most countable , and the maps @xmath31 and @xmath57 are inverse bijections between @xmath190 and @xmath191 . @xmath20 and @xmath172 are atomless , so they are supported respectively by @xmath191 and @xmath190 . thus the relation between @xmath20 and @xmath172 claimed in the proposition follows from this one - to - one property , the definition of @xmath178 and the property @xmath192 of proposition [ lambda12 ] . the case of @xmath175 is similar , and the @xmath34-finiteness follows from the @xmath34-finiteness of @xmath20 . we now give a condition on @xmath19 with which one can decide whether @xmath2 has finite or infinite variation ( this characterization is also given in @xcite ) . [ finitevar ] the measures @xmath20 , @xmath172 and @xmath175 are finite if and only if @xmath2 has finite variation . in this case , @xmath172 and @xmath175 are respectively the positive and negative parts of the lebesgue stieltjes measure of @xmath2 . moreover , @xmath193 we first work out the proof under the condition , so that @xmath194 . suppose also that @xmath19 is finite , so that @xmath20 is finite and @xmath2 is piecewise monotone [ as explained in ] . if , for instance , @xmath2 is nondecreasing on @xmath195 $ ] , then it is easily checked from the definitions that @xmath196)=\omega(t_2)-\omega(t_1),\qquad \omega^\nwarrow([t_1,t_2])=0.\ ] ] a similar result holds for intervals on which @xmath2 is nonincreasing , so we deduce that the proposition holds true in this case . if @xmath80 is not finite , consider the tree @xmath41 of and its path @xmath84 of . notice that @xmath197 as @xmath92 . for @xmath198 , one has @xmath199 , and the path @xmath84 is obtained from @xmath200 by a change of time , so the variation of @xmath84 increases as @xmath93 decreases , and is bounded by the variation of @xmath2 ; since the variation is a lower semicontinuous function of the path , it follows that the variation of @xmath84 converges to the variation of @xmath2 as @xmath140 , so @xmath201 ( we have applied the first part of the proof to @xmath84 and @xmath41 ) . thus @xmath2 has finite variation if and only if @xmath20 is finite . moreover , if @xmath2 has finite variation , one checks similarly that the positive part @xmath28 of the lebesgue stieltjes measure of @xmath2 satisfies @xmath202)=\lim(d\omega^a)^+([s , t ] ) = \lim\omega^\nearrow\bigl([s , t]\cap\pi^{-1}(\mathbb{t}^a)\bigr ) = \omega^\nearrow([s , t])\ ] ] where we have used the fact that @xmath203 is the restriction of @xmath172 to @xmath204 . if does not hold , we can consider an extension of @xmath2 satisfying and then restrict to @xmath6 $ ] . in this case , with the notation , the points @xmath67 of @xmath126 $ ] are such that @xmath205 or @xmath206 and should not be counted twice in the total variation of @xmath2 in . the correction which has to be made is @xmath207)=\delta(0,1)$ ] , so we obtain . let us explain how our construction of @xmath19 can be extended to cdlg paths @xmath2 ( paths which are right - continuous and have left limits ) , see figure [ fig3 ] ; we apply the classical idea of embedding these paths into continuous paths by opening temporal windows at times of jumps and considering interpolated continuous paths ( this idea has been used for the rough paths theory in @xcite ) . let @xmath133 be the set of points @xmath208 such that @xmath209 and @xmath144 is between @xmath210 and @xmath211 . this is the graph of @xmath2 augmented by the segments joining @xmath212 and @xmath213 . then define @xmath214 and @xmath215 if @xmath216 . if @xmath2 is continuous , then @xmath133 and @xmath6 $ ] are naturally identified , in such a way that @xmath115 coincides with the previous definition . let us say that two points of @xmath133 satisfy @xmath217 if either @xmath218 , or @xmath219 and @xmath144 is between @xmath210 and @xmath220 . this is a total order , and @xmath133 can be endowed with the topology generated by open intervals for this order ; actually , this topology coincides with the topology of @xmath133 considered as a subset of @xmath221 . [ graphtree ] the map @xmath115 is a semi - distance on @xmath133 , and @xmath222 is a compact real tree . actually , there exists a continuous map @xmath116 such that @xmath2 is obtained from @xmath116 by an increasing ( not necessarily surjective ) time change , and @xmath19 is the tree associated to @xmath116 . suppose that @xmath2 is not continuous ( the result is evident otherwise ) . let @xmath223 be the set of times where @xmath2 jumps , and let @xmath224 be a family of ( strictly ) positive numbers such that @xmath225 . let @xmath226 then @xmath227 is an increasing bijection from @xmath133 onto @xmath6 $ ] , so @xmath133 and @xmath6 $ ] can be identified , and previous results on the tree representation for continuous functions defined on @xmath6 $ ] can also be applied to continuous functions on @xmath133 . thus , in order to prove the proposition , it is sufficient to find a map @xmath116 defined on @xmath133 . put @xmath228 . it induces the semi - distance @xmath229}\omega ' = \delta((s , x),(t , x')),\ ] ] so its tree is @xmath19 . moreover , @xmath230 for the increasing time change @xmath231 . in this setting , let @xmath57 be the projection of @xmath133 on @xmath19 . we extend the notation by @xmath232 where @xmath124 is a time at which @xmath233 . let @xmath168 be the set of @xmath208 in @xmath133 such that @xmath234 for any @xmath167 and some @xmath165 , define @xmath235 similarly , and let @xmath236 notice also that all the points of @xmath237 are at the same level ; we let @xmath185 and @xmath238 be the infimum and supremum of the time component of this set . the measures @xmath172 and @xmath175 are the images of @xmath20 by @xmath31 and @xmath101 [ after restricting @xmath20 as in proposition [ lebesgue ] if does not hold ] . the statements of proposition [ finitevar ] about the finite variation case again hold true . let us use the notation of the proof of proposition [ graphtree ] . the set @xmath168 is the set of beginnings of excursions of @xmath116 , so @xmath172 is the projection on the time component of @xmath239 ; we deduce the first statement . moreover , @xmath240 is monotone on the intervals corresponding to the jumps of @xmath241 , so the total variations of @xmath2 and @xmath116 coincide ( a more general result will be proved in theorem [ vpomp ] ) , and the lebesgue stieltjes measure of @xmath2 is again deduced from its analogue for @xmath116 by projection on the time component . let us now assume that @xmath2 has finite @xmath1-variation for some @xmath3 , so that latexmath:[\[\label{pvarfinite } v_p(\omega):=\sup_{(t_i)}v_p(\omega,(t_i ) ) : = \sup_{(t_i)}\sum_i|\omega(t_{i+1})-\omega(t_i ) subdivisions of @xmath6 $ ] ( notice that a nonconstant continuous map can not have finite @xmath1-variation for @xmath243 ) . let us first assume that @xmath2 is continuous ( the cdlg case will be dealt with in section [ jumps ] ) . we first want to describe the property by means of the geometry of @xmath19 . in particular , @xmath244 implies @xmath245 for @xmath246 , and we are interested in the variation index @xmath247 let us recall that we have defined in approximations @xmath41 of @xmath19 obtained by trimming the tree , that @xmath98 , defined by , is the number of leaves of @xmath41 , and that the flattened path @xmath84 of is associated to @xmath41 ; let @xmath248 be its total length . as @xmath140 , each branch of @xmath41 grows at unit speed at its leaves , so @xmath249 if @xmath61 , we deduce from proposition [ finitevar ] that @xmath250 is the mass of the positive part of @xmath251 , so , by applying and , @xmath252 if @xmath253 , then this equation has to be corrected as in ; notice , however , that the correction is bounded , so if @xmath2 has infinite variation , then @xmath254 thus @xmath250 is easily estimated from the path @xmath2 , the times @xmath86 and @xmath85 , and the number @xmath98 of . we consider two other metric characteristics of @xmath19 , namely its upper box ( or minkowski ) dimension ( see , for instance , @xcite ) defined by @xmath255 where @xmath256 is the minimal number of balls of radius @xmath93 which are needed to cover @xmath19 , and the index @xmath257 where @xmath258 is the height of the highest branch above @xmath67 . the aim of this subsection is to prove that all these quantities are related to the variation index @xmath259 defined in , and in particular prove the result announced in . [ dimtree ] let @xmath2 be a ( nonconstant ) continuous function . then @xmath260 denoting by @xmath261 the successive terms of the theorem , we prove that @xmath262 these five inequalities are proved in the five following steps . proof of @xmath263 . let @xmath216 be two times , and let @xmath159 be the most recent common ancestor of @xmath182 and @xmath264 . then @xmath265}\omega\ ] ] so @xmath266 and @xmath267 on the other hand , @xmath268}\bigl(\omega(t)-\ell(\tau ) \bigr)^{p-1}\lambda ( d\tau)\\ & \le & p\int_{[\tau_0,\pi(t)]}\bigl(h(\tau)-h(\pi(t ) ) \bigr)^{p-1}\lambda ( d\tau)\\ & \le & p\int_{[\tau_0,\pi(t)]}(h(\tau))^{p-1}\lambda ( d\tau)\end{aligned}\ ] ] where we have used in the second line the property @xmath269}\ell -\ell(\tau ) -\max_{[\pi(t)^\nearrow,\pi(t)^\nwarrow]}\ell+\omega(t)\\ & \ge&\omega(t)-\ell(\tau)\end{aligned}\ ] ] valid for @xmath104 above @xmath67 . the same property holds at time @xmath181 , so by addition , @xmath270 } ( h(\tau))^{p-1}\lambda(d\tau).\ ] ] if @xmath5 is a subdivision of @xmath6 $ ] , we can sum up these estimates for @xmath271 and @xmath272 . since almost any @xmath67 appears at most twice in the right - hand sides ( at times @xmath185 and @xmath273 ) , we deduce @xmath274 in particular @xmath263 . proof of @xmath275 . it follows from @xmath276 ( proposition [ lambda12 ] ) and from that for @xmath277 , @xmath278 we deduce that if @xmath279 for some @xmath280 , then the integral is finite , so . proof of @xmath281 . this inequality follows from . proof of @xmath282 . above each @xmath283 there is a @xmath284 such that @xmath285 , and the @xmath98 balls with centers @xmath284 and radius @xmath93 are disjoint ; this implies that the number of balls of radius @xmath286 which is needed to cover @xmath19 is at least @xmath98 ; we also have @xmath287 , so we deduce that @xmath282 . proof of @xmath288 . for @xmath146 , let @xmath289 and @xmath290 let @xmath291 be the most recent common ancestor of @xmath292 and @xmath293 , so that @xmath294}\omega$ ] . consider the closed ball @xmath295 of @xmath19 with center @xmath291 and with radius @xmath296 , so that @xmath297)$ ] is included in this ball . then the union of @xmath295 is a covering of @xmath19 . moreover , the number of these balls is dominated by @xmath298 , so the upper box dimension of @xmath80 is dominated by @xmath1 as soon as @xmath299 . we deduce that @xmath288 . if @xmath61 , we have @xmath300}\omega-\omega ( \tau ^\nearrow)\biggr)^p + \biggl(\sup_{[\tau^\nearrow,\tau^\nwarrow]}\omega-\omega(\tau ^\nwarrow ) \biggr)^p\biggr ) = 2a^pn^a.\ ] ] if @xmath253 , we have to omit the first term for the first leaf of @xmath41 ( @xmath185 may be before time 0 ) , and the second term for the last leaf of @xmath41 ( @xmath238 may be after time 1 ) . thus @xmath301 and the right - hand side can be doubled if @xmath61 . other related estimates of @xmath7 using numbers of upcrossings were previously known ; see @xcite . the link between the dimension of @xmath19 and the behavior of @xmath98 is similar to the link between the dimension of the boundary of discrete trees and their growth ( see page 201 of @xcite ) . a more classical fractal dimension related to a path @xmath2 is the dimension of its graph as a subset of @xmath221 . this dimension ( which is bounded by 2 ) is of course generally different from the dimension of @xmath19 . other well - known notions of dimensions ( @xcite ) are the packing dimension @xmath302 and the hausdorff dimension @xmath303 , and we always have @xmath304 some of these inequalities may be strict . for instance , consider the path @xmath2 which is affine on each interval @xmath305 $ ] , and such that @xmath306 then @xmath307 for @xmath3 , so @xmath308 . on the other hand , the tree is a star with a countable number of branches , and its hausdorff and packing dimensions are 1 . on the other hand , if @xmath2 has the same variation index @xmath259 on any interval @xmath58 $ ] with @xmath216 , then any open subset of @xmath19 has the same upper box dimension , so in this case ( @xcite ) @xmath309 we will now see an example where the hausdorff dimension is also equal to @xmath259 . we now consider the case where @xmath2 is a typical path of a fractional brownian motion @xmath310 . this is a centered gaussian process @xmath311 with covariance function @xmath312 for the hurst parameter @xmath313 and the coefficient @xmath314 . it satisfies the scaling property @xmath315 for @xmath316 . in this subsection , we let @xmath2 be a path of @xmath310 restricted to @xmath6 $ ] and extended to @xmath117 $ ] by the technique of ; we compute the hausdorff dimension of the tree @xmath19 , and describe an estimator of @xmath25 based on @xmath19 . the property @xmath317 for @xmath318 is well known ; it is classically obtained from the @xmath319-hlder continuity of the paths , which itself is obtained by means of the kolmogorov criterion and the estimation @xmath320 on the @xmath321 norm of the increments for any @xmath322 . it actually follows from this estimation that the moments of @xmath323 are finite . [ dimfbm ] for almost any path @xmath2 of @xmath310 , one has @xmath324 the property @xmath325 follows from the discussion preceding the proposition . from and theorem [ dimtree ] , it is therefore sufficient to prove that @xmath326 . the constants involved in this proof depend on @xmath25 and @xmath34 . it is known from @xcite that @xmath327}w>-u\biggr]=o(u^\gamma)\ ] ] as @xmath328 , for any @xmath329 . moreover , if @xmath330 is the filtration of @xmath310 , the conditional law of @xmath331 given @xmath332 is a gaussian law with deterministic positive variance , so @xmath333\le c u.\ ] ] the event @xmath334 is included in the intersection of the two events of and , so @xmath335=o(u^{\gamma+1}).\ ] ] we deduce that @xmath336 is integrable for @xmath337 . from the scaling property , @xmath338 is also integrable , and @xmath339 so @xmath340 for the projection @xmath341 of the lebesgue measure of @xmath6 $ ] on @xmath342 . the double integral of the left - hand side is the @xmath1-energy of the measure @xmath341 on the metric space @xmath18 . its almost sure finiteness implies that @xmath343 for any @xmath337 ( see , for instance , @xcite ) , so @xmath326 . dimensions of lvy trees have been computed in @xcite . this includes our tree @xmath19 for @xmath344 , and for this tree , the exact hausdorff measure has been obtained in @xcite . here , we do not look for a so precise result , but verify that the normalization of the length measure @xmath20 on @xmath41 converges to the measure @xmath341 of the previous proof ; the same property is verified for the uniform measure on leaves of @xmath41 . in this sense , @xmath341 can be viewed as a uniform measure on the leaves of the tree . this will be a corollary of the following result ( proposition [ nu12 ] ) . [ nala ] for almost any path @xmath2 of @xmath310 , we have @xmath345 as @xmath140 , for some @xmath346 . since @xmath98 and @xmath250 are related to each other by means of , it is sufficient to study @xmath98 . moreover , @xmath34 acts as a multiplicative coefficient on the path , so @xmath98 for the process with parameter @xmath34 has the same law as @xmath347 for the process with parameter 1 ; thus it is sufficient to consider the case @xmath348 . if @xmath318 , it follows from the finiteness of the moments of @xmath323 and from that @xmath349 for any @xmath322 and some @xmath350 . in the two following steps , we study successively the expectation and the variance of @xmath98 . study of @xmath351 $ ] . consider in this proof the whole path @xmath352 of @xmath310 , and its associated ( noncompact ) tree @xmath353 . for @xmath216 , let @xmath354 , respectively @xmath355 , be the numbers of leaves of the trimmed tree @xmath356 such that @xmath357 , respectively @xmath358 . then @xmath359 ( actually one may have @xmath360 if @xmath181 is some @xmath185 , but this happens with zero probability for any fixed @xmath181 ) . on the other hand , it follows from the scaling property of @xmath310 that @xmath361=\mathbb{e}[\widetilde n_{0,a^{-1/h}}^1].\ ] ] the law of @xmath310 is shift invariant and @xmath362 is additive , so @xmath363 $ ] is proportional to @xmath364 , and @xmath361=a^{-1/h}\mathbb{e}[\widetilde n_{0,1}^1].\ ] ] thus the result of the proposition holds in expectation for @xmath365 $ ] . study of @xmath366 . it follows from and the additivity of @xmath367 that @xmath368 for @xmath369 . thus , by considering a regular subdivision of @xmath6 $ ] with mesh @xmath370 , we have @xmath371 moreover , @xmath372 \le\mathbb{e}[(n^{a \delta t^{-h}})^2 ] \le c a^{-2p}(\delta t)^{2ph}\end{aligned}\ ] ] for @xmath318 , where we have used the scaling property in the second equality , and in the last inequality . since @xmath373 depends only on the increments of @xmath2 on @xmath79 $ ] , we deduce from the result of appendix [ mixing ] that @xmath374\\[-8pt ] & \le & c ' a^{-2p}(\delta t)^{2ph - h-1},\nonumber\end{aligned}\ ] ] so , by joining and , @xmath375 we choose @xmath376 for @xmath377 , so @xmath378 by choosing @xmath1 and @xmath26 close enough to @xmath379 , we have @xmath380 for some @xmath165 . conclusion of the proof . the two previous steps show that @xmath381 converges in @xmath382 to a constant , and that the rate of convergence is at most of order @xmath383 . from the borel cantelli lemma , the convergence is almost sure on a sequence @xmath384 for @xmath385 large enough . since @xmath386 is monotone , we deduce from @xmath387 for @xmath388 , that the convergence is actually almost sure as @xmath140 . [ nu12 ] for almost any path @xmath2 of @xmath310 , the measures @xmath389 converge weakly to the projection @xmath341 on @xmath19 of the lebesgue measure of @xmath6 $ ] . let @xmath390 and @xmath391 be the images of @xmath392 and @xmath393 by @xmath31 . one has @xmath394 , so @xmath392 and @xmath395 are the images of @xmath390 and @xmath391 by @xmath57 . since @xmath57 is continuous , it is sufficient to prove that @xmath390 and @xmath391 converge weakly to the lebesgue measure of @xmath6 $ ] , and therefore that @xmath396)$ ] and @xmath397)$ ] converge to @xmath364 . but @xmath396)$ ] counts the proportion of leaves of @xmath41 which satisfy @xmath398 ; the number of such leaves is close to the number @xmath355 of the proof of proposition [ nala ] ; it can be estimated from proposition [ nala ] and the scaling property , and we can conclude . the study of @xmath391 is similar . we can deduce estimators for @xmath25 from proposition [ nala ] . our result is an alternative to the generalized quadratic variation approach @xcite . for instance , we can consider @xmath399 or @xmath400 , so that the unknown coefficient @xmath34 is eliminated . however , we can also use @xmath401 roughly speaking , the estimator @xmath402 counts the normalized number of changes in the sense of variation of @xmath84 . the smaller @xmath25 is , the more often the sense of variation of @xmath84 changes . from , we deduce the following result . [ estimator ] the hurst parameter @xmath25 of the fractional brownian motion @xmath403 can be estimated from the relation @xmath404 which holds almost surely , where @xmath98 , @xmath86 , @xmath85 were defined in . we now consider a cdlg path @xmath2 . we have seen in proposition [ graphtree ] how it can be written as a time - changed path @xmath230 for a continuous @xmath116 defined on @xmath133 , and the trees of @xmath2 and @xmath116 coincide . actually , the variations also coincide , so the tree @xmath19 can again be used to study the variations of @xmath2 . [ vpomp ] let @xmath2 be a cdlg path and @xmath116 the associated continuous path . one has @xmath405 for any @xmath3 . in particular , @xmath406 and theorem [ dimtree ] again holds . the relation @xmath230 immediately implies @xmath407 . in order to verify the reverse inequality , we notice that when computing @xmath408 , it is sufficient to consider subdivisions @xmath5 consisting of local extrema of @xmath116 ; thus these times are in the closure of the image of @xmath241 ; consequently , from the continuity of @xmath116 , it is sufficient to consider times in the image of @xmath241 , so that we can conclude . we now give applications of the tree representation to martingales and lvy processes . in the following result , we recover with our method a result of @xcite ( which was given in discrete time ) . notice , however , that our results are only for the real - valued case , whereas @xcite considers the banach space - valued case . [ super ] consider a purely discontinuous martingale @xmath409 for a filtration @xmath410 . let @xmath411 ; then @xmath412\le c_p \mathbb{e}\sum|\delta x_t|^p.\ ] ] the proof is divided into two steps ; in the first step , we reduce the problem to a particular case . [ step1 ] let @xmath413 and @xmath414 be the times of jumps of an independent standard poisson process , and consider @xmath415 ( @xmath42 is supposed to be constant after time 1 ) . then @xmath416 is a martingale in its filtration ; if the proposition were proved for @xmath416 , we would have @xmath417 where we have used in the second line the classical burkholder davis gundy inequalities ; it is then sufficient to let @xmath418 tend to 0 . thus it is sufficient to prove the result for martingales varying only on a sequence of totally inaccessible stopping times . by separating the positive and negative parts of the jumps , such a martingale is the difference of two martingales with finite variation and with no negative jump , so we only have to prove the result for these martingales . [ step2 ] we suppose therefore that @xmath42 has finite variation with positive jumps at a sequence of stopping times @xmath419 . thus the positive part @xmath420 of the lebesgue stieltjes measure of @xmath42 is purely atomic ; it is carried by the times of jumps of @xmath42 . let @xmath421 be in @xmath19 ; it is the projection of some @xmath422 of @xmath133 , and @xmath423 with @xmath424 then implies that @xmath425 where @xmath426 . the first term corresponds to the integral on the arc @xmath427 $ ] of @xmath19 , on which @xmath205 ; its expectation is dominated by the expectation of @xmath428 ( doob s inequality ) which can be estimated by the right - hand side of with the technique of . the second term corresponds to the integral on the remaining part of the tree , for which @xmath429 . in order to estimate it , consider some jump @xmath430 and notice that since @xmath42 is a martingale with no negative jump , @xmath431 \le\frac{x_s - x}{a}\ ] ] for @xmath432 and @xmath433 . we deduce that @xmath434 \le(x_s - x)\int_{x_s - x}^m a^{p-3}\,da,\ ] ] so @xmath435 \le(\delta x_s)^p/\bigl(p(2-p)\bigr)\ ] ] and we can conclude by summing on the times of jumps @xmath430 . we now give for lvy processes the analogue of proposition [ dimfbm ] . let @xmath42 be an @xmath26-stable lvy process . then , for almost any path @xmath2 of @xmath42 , @xmath436 for @xmath437 , the process has finite variation , so @xmath438 and the dimension is 1 . for @xmath439 , the fact that @xmath440 is classical and can be deduced from proposition [ super ] ; thus @xmath441 our result is therefore proved for @xmath442 . suppose now @xmath443 . we will use the notation @xmath444}x.\ ] ] it is known ( proposition viii.2 of @xcite ) that @xmath445}x>-u\biggr]\le c u^{\alpha\beta } = o(u^{\alpha-1})\ ] ] as @xmath328 , for @xmath446\ge(\alpha -1)/\alpha $ ] . we also have @xmath447 & \le&\sup_x\mathbb{p}[x - u < x_1-x_{1/2}<x+u]\\ & = & \sup_x\mathbb{p}[x - u < x_{1/2}<x+u]=o(u)\end{aligned}\ ] ] because @xmath448 has a bounded density , so by taking the intersection of these two events , @xmath335=o(u^\alpha).\ ] ] we deduce that @xmath336 is integrable for any @xmath449 . the variables @xmath450 satisfy the same property , and by scaling , @xmath451 this can be used to prove for any @xmath452 , so we deduce as in proposition [ dimfbm ] that the hausdorff dimension is bounded below by @xmath26 . another real tree , called the lvy tree , has been associated to @xmath42 in @xcite when @xmath42 has only positive jumps . this tree is different from @xmath19 but is related to it ; times which project on the same point of @xmath19 also project on the same point of the lvy tree , but an arc of @xmath19 associated to a jump of @xmath42 is concentrated in the lvy tree into a single point . let us now give an analogue of proposition [ nala ] for lvy processes . [ proposition314 ] let @xmath42 be a lvy process . suppose that almost surely , @xmath42 has no interval on which it is monotone , and define @xmath453\ ] ] for @xmath454}x - a\biggr\},\qquad s^a:=\inf\biggl\{t;\>x_t>\inf_{[0,t]}x+a\biggr\}.\ ] ] then @xmath455 , and @xmath456 ( for the process @xmath42 on the time interval @xmath6 $ ] ) converges in probability to 1 as @xmath140 . if @xmath457 for some @xmath458 , then the convergence is almost sure . when the assumption about @xmath42 is not satisfied , then @xmath42 or @xmath459 is the sum of a subordinator and a compound poisson process . in this case , @xmath342 is finite , so @xmath98 is bounded . proof of proposition [ proposition314 ] consider the times @xmath460 and @xmath461 defined by . on the other hand , notice that our assumption implies that @xmath462 and @xmath463 tend almost surely to 0 as @xmath140 . since @xmath42 is a lvy process , times @xmath464 and @xmath465 are independent , and have the same law as @xmath463 and @xmath462 . thus @xmath466}x\biggr)\ge\sup_{1\le j\le k } \biggl(\sup_{(j-1)\mu\le t\le j\mu } \biggl(x_t-\inf_{[(j-1)\mu , t]}x\biggr)\biggr)\ ] ] and the right - hand side is the supremum of @xmath467 independent identically distributed variables , so @xmath468 = \mathbb{p}\biggl[\sup_{0\le t\le k\mu}\biggl(x_t-\inf_{[0,t]}x\biggr)<a\biggr ] \le(\mathbb{p}[s^a\ge\mu])^k\ ] ] for @xmath469 . this probability is smaller than 1 from our assumption on @xmath42 . we deduce that the moments of @xmath462 ( and @xmath463 ) are finite , so @xmath470 and @xmath471\le1/2^k.\ ] ] thus @xmath472)$ ] , and similarly @xmath473)$ ] , are dominated by a geometric variable , so the variances of @xmath462 and @xmath463 are dominated by @xmath474)^2 $ ] and @xmath475)^2 $ ] . thus @xmath476=n \xi(a),\qquad \operatorname{var}(s_n^a)=n \bigl(\operatorname{var}(s^a)+\operatorname{var}(t^a)\bigr)\le c n \xi(a)^2.\ ] ] if @xmath477 as @xmath140 , then @xmath478 has expectation 1 and has a variance dominated by @xmath479 ; in particular it converges in probability to 1 . by taking @xmath480 , we see from and the definition of @xmath98 in that @xmath98 is between @xmath481 and @xmath482 with a high probability , so the convergence in probability of the proposition is proved . moreover , for the second statement , it follows from the borel cantelli lemma that @xmath483 converges almost surely to 1 as soon as @xmath484 . we can apply this result to the above @xmath485 for @xmath486 and @xmath385 large enough , and we deduce that @xmath487 converges almost surely to 1 . we conclude as in proposition [ nala ] from the monotonicity of @xmath98 . the almost sure convergence holds in particular for @xmath26-stable processes such that @xmath488 is not a subordinator . in this case indeed , @xmath489 is proportional to @xmath490 from the scaling property . for the standard brownian motion , @xmath491 and @xmath492 are the first hitting time of 1 by a reflected brownian motion , and have expectation 1 . thus @xmath493 and @xmath494 . this means that @xmath495 in proposition [ nala ] . we can deduce an estimation of @xmath250 when the process has infinite variation . however , can not be directly applied ; one has to use the associated continuous path , since times @xmath86 and @xmath85 can be jump times . we now want to integrate some bounded function @xmath496 against @xmath2 . first suppose that @xmath2 is continuous and @xmath61 . let us remember ( proposition [ finitevar ] ) that if @xmath2 has finite variation , then @xmath172 and @xmath175 are finite measures , and are the positive and negative parts of the lebesgue stieltjes measure @xmath30 ; moreover , since the images of the finite length measure @xmath20 by @xmath31 and @xmath32 are respectively @xmath172 and @xmath175 ( proposition [ lebesgue ] ) , we have @xmath497 so @xmath498 if @xmath253 , we extend @xmath2 to @xmath117 $ ] as in , put @xmath499 for @xmath500 $ ] , and we can use the same formula to define the integral ; actually , with the notation , we have @xmath501 } \bigl(\rho ( \tau ^\nearrow ) -\rho(\tau^\nwarrow)\bigr)\lambda(d\tau)\nonumber\\[-8pt]\\[-8pt ] & & { } -\int_{[a , o]}\rho(\tau^\nwarrow)\lambda(d\tau ) + \int_{[o , b]}\rho(\tau^\nearrow)\lambda(d\tau).\nonumber % \end{aligned}\ ] ] in this form , one can notice that the integral on @xmath6 $ ] depends on @xmath9 and @xmath2 on @xmath6 $ ] , and not on the extension of @xmath2 out of @xmath6 $ ] . more generally , even if @xmath2 has infinite variation , we can define the integral by the right - hand side of or , provided @xmath502 notice that the right - hand side of is the limit as @xmath140 of the integral on the trimmed tree @xmath41 which is the tree of @xmath84 defined by , so @xmath503 is the limit of @xmath504 . this means that in this sense our approach is similar to other approaches using a regularization of @xmath2 ; another example for which there has been a lot of work recently is the russo vallois approach @xcite . [ young ] assume that @xmath2 is continuous . one has @xmath505 for some @xmath506 , as soon as @xmath15 . thus is satisfied as soon as @xmath507 , and in this case we can define @xmath35 by the right - hand side of or . it satisfies @xmath508 for @xmath15 . moreover , this integral coincides with the riemann stieltjes integral constructed by young @xcite ( see also @xcite ) ; this means that @xmath509 for @xmath510 , as the mesh of the subdivision @xmath5 of @xmath6 $ ] tends to 0 . the integral @xmath511 can be defined similarly by replacing @xmath9 by @xmath512}$ ] ; it satisfies the chasles relation , and @xmath513 let us first assume @xmath61 . it follows from the disintegration formula @xmath514 of proposition [ lambda12 ] that @xmath515 define @xmath516 by @xmath517 . then @xmath518\\[-8pt ] & \le&\frac1{2^{r / p}a^r}v_q(\rho)^{1/q}v_p(\omega)^{r / p}\nonumber\end{aligned}\ ] ] from hlder s inequality and . consequently , @xmath519 is of order @xmath520 and is integrable with respect to @xmath93 near 0 ; more precisely , with @xmath521 , @xmath522 where we have used @xmath523 in the last line . if @xmath524 , we decompose @xmath19 into @xmath126 $ ] and @xmath131 $ ] ; we can apply the above procedure to the integral on the latter part , and again prove , but without the factor 2 . on the other hand , @xmath126 $ ] has finite length so the integral is finite on it ; more precisely , @xmath525}|\rho(\tau^\nearrow)-\rho(\tau^\nwarrow ) ( d\tau ) & = & \int_{[a , o]}|\rho(\tau^\nwarrow)|\lambda(d\tau ) + \int_{[o , b]}|\rho(\tau^\nearrow)|\lambda(d\tau)\\ & \le&\delta(0,1 ) \sup|\rho|\le2v_p(\omega)^{1/p } \sup|\rho|.\end{aligned}\ ] ] the result follows by adding these two estimates . thus we can define the integral @xmath8 by ; this integral satisfies , and similarly , @xmath526 where the @xmath1-variation of @xmath2 is limited to @xmath58 $ ] . one easily deduces by applying @xmath527 more precisely , by considering the variations of @xmath9 and @xmath2 on @xmath58 $ ] , @xmath528 thus @xmath529 by applying and the similar estimate for @xmath9 , we get @xmath530 which converges to 0 since @xmath2 is continuous . in the proof , we have considered separately the arc @xmath126 $ ] . actually , @xmath531}\bigl(\rho(\tau^\nearrow)-\rho(\tau^\nwarrow ) \bigr)\lambda ( d\tau ) = \int\rho\,d\underline{\omega}\ ] ] with @xmath532}\omega\vee\inf_{[t,1]}\omega.\ ] ] in the framework of theorem [ young ] , the fact that our integral is a riemann stieltjes integral implies that it is linear with respect to @xmath2 ; this property was not evident on our definition , since the tree associated to the sum of two paths is not simply related to the trees of the two paths . actually , we do not know whether the space of @xmath2 satisfying is linear . the tree is associated to a continuous path @xmath533 , as it has been explained in proposition [ graphtree ] , and @xmath116 has the same variations as @xmath22 . one can also consider @xmath534 which has the same variations as @xmath9 . then the left - hand side of is the integral for @xmath535 and @xmath116 , so holds true . for the riemann sums , we modify in the previous proof by introducing @xmath536 such that @xmath537 ; then @xmath538 so that @xmath539 we have to prove that the supremum tends to 0 as the mesh of the subdivision tends to 0 . for any @xmath165 , let us consider @xmath540 then @xmath541 and the number of jumps greater than @xmath418 is finite , so from the continuity of @xmath9 at these points , @xmath542 we deduce the convergence from these two properties . this theory can be applied to paths of fractional brownian motions with hurst parameter @xmath17 , or to lvy processes without brownian part and such that @xmath543 is integrable with respect to the lvy measure for some @xmath16 . a limitation of the young integral concerns its iteration . if @xmath2 and @xmath9 have respectively @xmath1- and @xmath14-finite variation for @xmath15 and @xmath2 is continuous , then we can consider the function @xmath544 and implies that @xmath144 has @xmath1-finite variation ; however , it generally does not have @xmath14-finite variation so , unless @xmath16 , one can not construct @xmath545 . nevertheless , we now check that this is possible with our framework ( for a continuous one - dimensional path @xmath2 ) . the idea is to look for a weaker condition than @xmath38 for . for instance , if @xmath546 , holds for any bounded @xmath547 and any continuous @xmath2 , and @xmath548 for a primitive function @xmath549 of @xmath547 ; this is because the integral on @xmath131 $ ] in is 0 ( @xmath550 ) , and the integral on @xmath126 $ ] is easily computed from . however , in this case , the integral is not always the limit of riemann sums , as it is easily seen for @xmath551 . we want to generalize this example . [ eyoung ] let @xmath2 be continuous . the integrability condition holds as soon as @xmath555 moreover , if @xmath15 and @xmath2 fixed with @xmath556 , the space of bounded functions @xmath9 such that @xmath39 is a banach space @xmath557 for the norm @xmath558 and we have @xmath559 for some @xmath506 . in the estimation , we can use @xmath560 and @xmath561 for @xmath562 . thus is replaced by @xmath563 this proves the first statement . the banach property is easily verified from the lower semicontinuity of @xmath564 with respect to uniform convergence . by applying on @xmath58 $ ] , we estimate @xmath511 , and deduce that @xmath565 and @xmath566 are bounded by the right - hand side of [ for the estimation of the @xmath1-variation , we use ] . the last property which has to be proved in order to conclude is . to this end , we are going to check that @xmath567 as soon as @xmath568 ; then follows by applying on the intervals @xmath569 $ ] in order to estimate @xmath570 . the left - hand side of is written as an integral on the tree ; the integral on @xmath131 $ ] is estimated by the right - hand side of as in ; for the integral on @xmath126 $ ] , it can be written as @xmath571}\bigl(\rho(\tau^\nearrow)-\rho(\tau^\nwarrow ) \bigr)\lambda ( d\tau ) = \int_{\inf\omega}^{\omega(0)}\bigl(\rho(\beta_2(x))-\rho(\beta _ 1(x))\bigr)\,dx\ ] ] with @xmath572 this expression is also easily estimated by the right - hand side of . [ edo ] for @xmath15 and @xmath573 , consider a continuous real - valued map @xmath2 with finite @xmath1-variation , and let @xmath574 be the banach space of functions @xmath9 such that @xmath575 with values in @xmath576 and with finite @xmath14-variation , and let @xmath577 with values in @xmath578 . let @xmath547 be a @xmath579 function with bounded derivatives from @xmath580 into the space of linear maps @xmath581 . consider , for @xmath582 in @xmath580 , the equation @xmath583 where the integral should be understood as the sum of integrals with respect to each component , each one being given by an expression of type or . then this equation has a unique solution in the banach space @xmath584 . in this proof , the constants @xmath46 may depend on @xmath547 and @xmath582 , but not on @xmath585 . it is not difficult to deduce from the lipschitz property of @xmath547 that @xmath586 maps @xmath587 into @xmath588 and has at most linear growth : @xmath589 let us prove that @xmath549 is locally lipschitz . it is easy to verify @xmath590 and let us estimate @xmath591 . let @xmath5 be a subdivision satisfying @xmath553 , and use the notation @xmath592 . it follows from the boundedness of the derivatives of @xmath547 that latexmath:[\ ] ] in order to conclude , we have to estimate this @xmath786 norm . the formula for @xmath547 can be differentiated , so @xmath547 is smooth and @xmath787 for @xmath788 . in particular , @xmath789 for any @xmath619 and some @xmath790 . on the other hand [ recall the definition of @xmath791 in ] , @xmath792 for @xmath793 in particular , @xmath547 satisfies the assumption of lemma [ cameron ] . moreover , @xmath794 and @xmath795 where the last estimate is easily obtained by considering separately the integrals on @xmath796 $ ] and @xmath797 $ ] . thus we have obtained an estimate for @xmath798 , and we deduce that @xmath799 for any @xmath619 and some @xmath800 . we still have to prove that the moments of @xmath801 are bounded ; but , from jensen s inequality , @xmath802\,ds,\ ] ] so , since @xmath803 is gaussian , this expression has bounded expectation provided @xmath804 , and therefore if @xmath722 is small enough . our aim is to describe a part of the rough paths theory through a point of view which is well adapted to our approach ( theorem [ intfbm ] ) . our result ( theorem [ intrough ] below ) is in particular comparable to @xcite , and we include for completeness a short proof which is sufficient for our purpose . let @xmath805 be a path with finite @xmath1-variation , for @xmath806 . in this case , we learn from the theory of rough paths that @xmath585 is not sufficient for the construction of an integral calculus , but we also need its double integrals . more precisely , let @xmath805 and @xmath807 take their values respectively in @xmath808 and @xmath809 . we suppose that latexmath:[\[\label{xixi } paths with finite @xmath1-variation can be reduced to this case by a change of time ) . the path is supposed to be multiplicative in the sense @xmath811 for @xmath369 . if @xmath812 , then @xmath686 is necessarily the young integral @xmath813 but if @xmath814 , the function @xmath686 , when it exists , is not unique ; one can add to it @xmath815 for any @xmath816-hlder continuous @xmath622 . let us now explain how one can define integrals @xmath817 , in a way which coincides with the tree approach of theorem [ intfbm ] . [ intrough ] consider paths @xmath688 satisfying and @xmath818 , @xmath9 with values in @xmath581 ( the space of linear maps ) , and @xmath819 with values in the space @xmath820 . we suppose that @xmath821 and @xmath822 for any @xmath216 and any subdivision @xmath823 of @xmath58 $ ] , put @xmath824 then @xmath825 converges as @xmath826 tends to 0 , and the limit @xmath827 satisfies @xmath828 for some @xmath829 . we use the simple notation @xmath817 though the integral actually depends on @xmath831 and @xmath688 . notice , however , that if @xmath832 for almost any @xmath181 ( and this is the case for an @xmath25-fractional brownian motion and @xmath833 ) , then @xmath819 is uniquely determined by @xmath9 . proof of theorem [ intrough ] in the proof we will use the following result taken from young integration . let @xmath825 be a function defined on finite subdivisions @xmath823 of @xmath58 $ ] and let @xmath834 be the subdivision with @xmath835 removed . we suppose that @xmath836 . then @xmath825 converges as the mesh of @xmath837 tends to 0 , and @xmath838 where the trivial subdivision @xmath839 . let @xmath736 be the functional of . then @xmath840 where we have used the multiplicative property of @xmath686 . the condition is satisfied with @xmath841 and @xmath842 , so the result is proved . in particular , we can compute the integral @xmath843 of a one - form by considering @xmath844 and @xmath845 ; the property implies that the integral is the limit of generalized riemann sums , so it coincides with the standard rough paths approach .

we consider a real - valued path ; it is possible to associate a tree to this path , and we explore the relations between the tree , the properties of of the path , and integration with respect to the path . in particular , the fractal dimension of the tree is estimated from the variations of the path , and young integrals with respect to the path , as well as integrals from the rough paths theory , are written as integrals on the tree . examples include some stochastic paths such as martingales , lvy processes and fractional brownian motions ( for which an estimator of the hurst parameter is given).=-1 . .

the study of finite group actions on topological spaces has a long and distinguished history . a frequent theme is to try and understand the topology of the fixed point set , both in it s intrinsic form , and as a subspace of the original space . the classic work of smith shows that for finite @xmath1-groups acting on spheres , the fixed point set has the @xmath5 cohomology of a sphere . however , there are examples of ` exotic ' actions on spheres , where the fixed point set is not _ homeomorphic _ to a sphere ( indeed , does not even have the @xmath3 cohomology of a sphere ) . in this short paper , we are interested in relating actions on a hyperbolic group with the induced action on its boundary at infinity . we will start by relating the fixed subgroup of an automorphism with the fixed subset of the induced action on the boundary at infinity . in particular , this will allow us to use the classic theorem of smith to prove that if one starts with a @xmath2-hyperbolic poincar duality group over @xmath3 , and the group that is acting is a finite @xmath1-group ( @xmath1 a prime ) , then the fixed subgroup is a poincar duality group over @xmath4 . we will then use the strict hyperbolization technique due to charney and davis [ 6 ] to construct examples of involutions of a poincar duality group over @xmath3 whose fixed subgroup fails to be a poincar duality group over @xmath3 ( and in fact , are nt even duality groups over @xmath3 ) . these examples also provide examples of ` exotic ' involution on a sphere ( the boundary at infinity ) which can be realized geometrically ( i.e. by an isometry of a @xmath6 space ) . they also show that , in general , one could have involutions of @xmath6 spaces having a sphere as the boundary at infinity , where the induced involution on the boundary has a fixed point set which is _ not _ an anr . * remark : * this paper was motivated by the following more specific questions ( each of which is still open ) . let @xmath7 where @xmath8 is a closed negatively curved riemannian manifold , and let @xmath9 be an automorphism with @xmath10 . * question 1 * : is the fixed subgroup @xmath11 a poincar duality group over @xmath3 ? * question 2 * : is @xmath12 induced by an involution of @xmath8 ? that is to say , does there exist a self - homeomorphism @xmath13 with @xmath14 , and @xmath15 ? * question 3 * : let @xmath16 be the induced involution of the sphere at infinity of the universal cover of @xmath8 . is the fixed point set of @xmath17 ( when non - empty ) an anr ? * acknowledgements . * the authors would like to thank the referee for several helpful comments , in particular for pointing out the existence of [ 5 ] and [ 8 ] , and for suggesting the extension of theorem 2.2 that is included at the end of section 2 . we would also like to thank the members of the geometric and function theory seminar at the university of michigan for pointing out a substantial simplification in our original proof of proposition 2.1 . [ fixedsgpsbdry ] let @xmath18 be a @xmath2-hyperbolic group , @xmath19 an automorphism of @xmath18 of finite order @xmath20 , and @xmath21 the induced action of @xmath19 on @xmath22 . then @xmath23 is homeomorphic to @xmath24 . let @xmath25 be a symmetric generating set for @xmath18 , and consider the action of @xmath26 on @xmath18 . observe that if we define a new generating set @xmath27 , then @xmath19 acts by isometries on the cayley graph @xmath28 of @xmath18 with respect to these generators . indeed , we note that given any pair of elements @xmath29 in @xmath18 , we have that : @xmath30 taking a minimal such expression , and applying @xmath19 to it , we see that : @xmath31 but by invariance of @xmath32 under @xmath19 , we immediately get an expression for @xmath33 as a product of @xmath34 elements of @xmath35 . this forces @xmath36 . but now @xmath19 , by hypothesis , has finite order @xmath20 . so by iterating our inequality we get that : @xmath37 which implies that all the inequalities are in fact equalities , and hence that @xmath19 does indeed act by isometries on @xmath28 . from now on , we will omit the subscript @xmath32 from our distance function in order to simplify notation . our next step is to define certain subsets of the cayley graph @xmath28 in terms of their behavior under @xmath19 , and to control the distance between these subsets . we let @xmath38 , and observe that @xmath39 . neumann [ 15 ] has shown that , for each i , there exists a @xmath40 such that @xmath41 . next we observe that , by neumann [ 15 ] , the subgroup @xmath42 is quasi - convex in @xmath18 . in particular , @xmath43 embeds in @xmath22 . now note that , trivially , we have that @xmath44 is in fact a subset of @xmath23 . to prove equality , we need to show the reverse inclusion . so let us take a point @xmath45 , and let @xmath46 be a geodesic ray based at the identity and with @xmath47 . now by our choice of generators , we know that @xmath48 will also be a geodesic ray ( since @xmath19 acts isometrically on the cayley graph ) , and since the point @xmath49 is fixed by @xmath50 , we must have @xmath51 for some constant @xmath52 . our next claim is that , for each @xmath53 , the inequality @xmath54 holds ( and hence , as @xmath55 , forces @xmath56 ) . in order to see this , we consider the following construction : given an integer @xmath53 , we define @xmath57 to be an integer satisfying @xmath58 ( note that both @xmath59 and @xmath60 correspond to elements in @xmath18 ) . we claim that @xmath61 for all @xmath53 . by way of contradiction , assume that @xmath62 . the triangle inequality gives us : @xmath63 a contradiction ( recall that @xmath64 are both the identity element in @xmath18 ) . the case @xmath65 can be dealt with in an analogous manner . we now know that , if @xmath66 is an arbitrary geodesic ray originating at the identity , and having @xmath67 , then @xmath68 . however , we also have that @xmath69 for some constant @xmath70 . in particular , we can find a geodesic ray in @xmath71 which has uniformly bounded distance from @xmath66 , which forces @xmath72 , completing the proof of the proposition . we say that a topological space @xmath73 is an @xmath53-dimensional cech cohomology sphere with @xmath74 coefficients ( where @xmath74 is a pid ) provided that @xmath75 for all @xmath76 , and @xmath77 ( @xmath78 refers to reduced cech cohomology ) . we say that a torsion - free group @xmath0 is a _ duality group of dimension @xmath53 over @xmath74 _ ( where again , @xmath74 is a pid ) , provided that there is a right @xmath79-module @xmath52 such that one has natural isomorphisms @xmath80 for all @xmath81 and all @xmath79-modules @xmath82 ( naturality is taken with respect to @xmath82 , and @xmath0 acts diagonally on the tensor product @xmath83 ) . if in addition we have that @xmath84 , then we say that @xmath0 is a _ poincar duality group of dimension @xmath53 over @xmath74_. finally , if @xmath0 is a poincar duality group of dimension @xmath53 over @xmath74 , and the @xmath0 action on @xmath85 is trivial , we say that @xmath0 is an _ orientable poincar duality group of dimension @xmath53 over @xmath74_. for background material on duality groups and poincar duality groups , we refer to the lecture notes by bieri [ 2 ] . next , we quote the following result from bestvina and mess ( corollary 1.3 in their paper [ 1 ] ) : [ begm ] let @xmath18 be a torsion - free @xmath2-hyperbolic group . then @xmath18 is a poincar duality group of dimension @xmath53 over @xmath86 if and only if @xmath87 is an @xmath88-dimensional cech cohomology sphere with @xmath86 coefficients . using their result , we obtain an immediate corollary to our previous proposition : [ pdgrpsbdry ] let @xmath18 be a torsion - free @xmath2-hyperbolic poincar duality group of dimension @xmath53 over @xmath5 . let @xmath0 be a finite @xmath1-group ( @xmath1 prime ) acting by automorphisms on @xmath18 . then there is a @xmath89 such that the subgroup @xmath90 is a poincar duality group of dimension @xmath91 over @xmath5 . let us first consider the case where @xmath0 is @xmath4 . then consider the induced action of @xmath0 on the boundary at infinity @xmath22 . notice that , by bestvina and mess result , @xmath92 is a compact @xmath88-dimensional cech cohomology sphere with @xmath5 coefficients . so we can use a version of smith theory ( see theorem iii.7.11 in bredon [ 4 ] ) , to get that the fixed point set of the action on the boundary at infinity must be a @xmath93-dimensional cech cohomology sphere with @xmath94 coefficients ( for some @xmath95 ) . now our proposition [ fixedsgpsbdry ] along with bestvina and mess result immediately implies that the group @xmath90 is a poincar duality group of dimension @xmath91 over @xmath5 ( where @xmath96 ) . for the more general case , we note that , since every @xmath1-group is solvable , one can find a normal subgroup @xmath97 . finally one uses induction , since we have that @xmath98 . this gives the general case . as was pointed out to the authors by the referee , corollary 2.1 also follows from the result announced by chang and skjelbred in [ 5 ] , where they explain why the fixed set of a finite @xmath1-group action on a poincar duality space over @xmath4 is still a poincar duality space over @xmath4 . we conclude this section by mentioning that a poincar duality group over @xmath3 is automatically a poincar duality group over @xmath4 , but that the converse does not necessarily hold . in theorem 2.2 , the group @xmath99 will be an example of this with @xmath100 . one can now ask the question of whether the previous result can be strengthened to obtain that the fixed subgroup is a poincar duality group over @xmath101 . this turns out to be false , and in this section , we will construct counterexamples . as was pointed out by the referee , similar examples were constructed by davis & leary [ 8 ] . their construction used the reflection trick method ( as opposed to our use of hyperbolization ) and served a somewhat different purpose . we now proceed to state our main theorem . let @xmath102 be a pl involution of a sphere @xmath103 whose fixed point set is a submanifold @xmath104 which is _ not _ a homology sphere ( with @xmath3 coefficients ) , and has dimension @xmath105 . let @xmath73 be the strict hyperbolization of the suspension of @xmath103 , and @xmath106 the induced involution on @xmath73 . let @xmath18 be the fundamental group of @xmath73 , and @xmath19 the induced involution on @xmath18 . then @xmath19 is an involution of a ( @xmath2-hyperbolic ) orientable poincar duality group over @xmath101 whose fixed subgroup @xmath99 is not a duality group over @xmath101 . before starting with the proof , let us note that examples of involutions of spheres whose fixed point sets are not homology spheres do exist . in fact , jones [ 11 ] has proved that every closed pl manifold that has the @xmath107 homology of a sphere can be realized as the fixed point set of a pl involution of some larger dimensional sphere . for a more concrete example , we can consider brieskorn spheres : for @xmath108 , define two complex functions @xmath109 , and @xmath110 . using these two functions , define a pair of manifolds @xmath111 and @xmath112 by considering the intersection of @xmath113 and @xmath114 with a small enough ball centered at the origin in the appropriate complex vector space . it is known that @xmath111 is pl homeomorphic to the sphere @xmath115 , while @xmath112 is a @xmath116-dimensional manifold that does not have the @xmath101-homology of a sphere ( combine lemma 8.1 with the comments on pg . 72 in milnor [ 14 ] ) . furthermore , observe that the involution @xmath117 on @xmath111 has fixed point set @xmath112 , giving us an infinite family of examples . we start by recalling that the strict hyperbolization procedure given by charney and davis ( section 7 in [ 6 ] ) takes a simplicial complex and functorially assigns to it a topological space ( in fact , a union of compact hyperbolic manifolds with corners ) that supports a metric of strict negative curvature . let us apply this procedure to the suspension of the sphere @xmath118 ( respectively @xmath119 ) , and call the resulting space @xmath120 ( respectively @xmath121 ) . we will omit the dimension of the spaces unless we explicitly require them for computations . we now list out some properties of the spaces @xmath73 and @xmath122 . observe that , by a result of illman [ 9 ] , there exists a triangulation of the pair @xmath123 such that the involution @xmath102 is a simplicial map . in particular , the involution on the suspension will still be simplicial , and @xmath119 is a subcomplex of @xmath118 . functoriality of the strict hyperbolization procedure now implies that @xmath122 is a totally geodesic subspace of @xmath73 , invariant under the induced involution @xmath106 on @xmath73 . since hyperbolization preserves the local structure , @xmath73 will be an orientable manifold , while @xmath122 will have a pair of non - manifold points ( corresponding to the two vertices of the suspension ) . now take a basepoint @xmath124 , and let @xmath125 , @xmath126 . the involution @xmath106 will give an order two automorphism @xmath19 of the group @xmath18 . we note that , since @xmath18 is the fundamental group of a closed orientable aspherical manifold , it is automatically an orientable poincar duality group over @xmath101 . now consider the fixed subgroup @xmath99 . in order to get information about this group , we consider a lift of the action to the universal cover @xmath127 of @xmath73 . let @xmath128 be a preimage of the point @xmath129 , and let us lift the involution @xmath106 to the universal cover . note that the fixed point set of the lifted involution is precisely the path connected lift @xmath130 of @xmath122 that contains the point @xmath131 . furthermore , the action @xmath19 on @xmath18 is compatible with the lift @xmath132 of @xmath106 , in the sense that @xmath133 . next we note that @xmath134 . indeed @xmath86 is automatically fixed by @xmath19 , hence we have a containment @xmath135 . on the other hand , for an arbitrary @xmath136 , we have that @xmath137 . in particular , @xmath138 must be fixed under @xmath132 , which implies @xmath139 . since @xmath130 is a path connected , totally geodesic subset , we can connect @xmath131 to @xmath138 by a path which lies entirely within @xmath130 . looking at the projection of this path in @xmath73 , we observe that it is a closed loop based at @xmath129 , representing the element @xmath140 , and lying entirely in @xmath122 . hence @xmath141 , giving us the reverse containment . we conclude that the two groups are equal . [ graph ] so in particular , @xmath122 is a topological space which happens to be a @xmath142 . in particular , the group cohomology of @xmath99 is related to the compactly supported cohomology of @xmath130 . so we have now reduced our claim to analyzing the properties of @xmath143 . in order to do this , we consider the zeeman spectral sequence ; let us first introduce some terminology . we will denote by @xmath144 the @xmath145 local homology sheaf for @xmath122 , and by @xmath146 the corresponding sheaf for @xmath130 . for @xmath147 ( respectively , in @xmath148 ) , we will denote by @xmath149 ( respectively @xmath150 ) the stalk at the point @xmath151 . recall that @xmath130 is the hyperbolization of an @xmath152-dimensional complex @xmath153 ; we will use @xmath154 to denote the subspace of @xmath122 obtained from the hyperbolization of the @xmath34-skeleton of @xmath119 . observe the following facts about the local homology sheaf : * if @xmath155 and @xmath156 , then @xmath157 . * if @xmath158 , then @xmath159 . * there exists a point @xmath160 and an integer @xmath161 such that @xmath162 and @xmath163 all of the previous remarks are clear , with the possible exception of the third : let @xmath1 be one of the two vertices of the suspension . since the original link of @xmath1 was _ not _ a homology sphere , and as hyperbolization does not change the link , there must exist an @xmath164 which yields the desired fact . note that the sheafs we are considering are given by local data , so that we have that @xmath165 , whenever @xmath166 is a lift of the point @xmath151 . hence we have that the three facts mentioned above for the stalks of the local homology sheaf @xmath144 on @xmath122 also hold for the stalks of the local homology sheaf @xmath146 on @xmath130 . now the zeeman spectral sequence ( see section 2 of mccrory [ 13 ] , based on previous work of zeeman [ 17 ] ) states that : @xmath167 with differentials @xmath168 . observe that , by the properties listed above for the @xmath169 local homology sheaf , @xmath170 if @xmath171 and @xmath172 . so in particular , all the terms vanish except those in the @xmath173 row and those in the @xmath174 column ( see figure on previous page ) . we now plan on working with this spectral sequence . observe from the shape of the spectral sequence that one has isomorphisms : @xmath175 @xmath176 and that the differential @xmath177 maps @xmath178 to @xmath179 . however , we know that @xmath180 , so the differential must be an isomorphism . this yields : @xmath181 ( since we are dealing with complexes , cech cohomology coincides with standard cohomology ) . furthermore , @xmath130 is simply - connected and has dimension @xmath182 , hence @xmath183 is the trivial @xmath3 sheaf over @xmath184 . this implies : @xmath185 now focusing on the left hand term , we note that @xmath186 for all @xmath187 , which gives us : @xmath188 but now observe that if @xmath189 is a vertex in @xmath190 which is a lift of @xmath1 ( one of the vertex points of the suspension ) , then @xmath191 for every element @xmath192 . since all the points @xmath193 lie in @xmath190 , and since @xmath99 is an infinite group , this implies that @xmath194 is not finitely generated . so in particular , @xmath195 is not finitely generated . since @xmath122 is a finite complex which happens to be a @xmath196 , we conclude that @xmath197 is not finitely generated . by bieri and eckmann s criterion ( see bieri [ 2 ] , section 9.10 ) , this implies that @xmath198 can not be a poincar duality group over @xmath199 . in order to see that @xmath99 is not even a duality group over @xmath3 , it is sufficient to show that the cohomological dimension of @xmath99 is greater than @xmath200 . we first note that , since @xmath201 , we have that @xmath202 , so it is sufficient to show that @xmath99 has non - trivial cohomology in some dimension that is strictly greater than @xmath20 . observe that , by construction , we have that @xmath99 is the fundamental group of the finite aspherical @xmath152-dimensional space @xmath122 , which implies that the cohomological dimension of @xmath99 is at most @xmath203 . we would be done provided we can show that the cohomological dimension of @xmath99 is exactly @xmath203 . looking back at the construction of @xmath122 , we observe that the submanifold @xmath104 we started with is a @xmath204 homology sphere . suspending the manifold , we obtain an @xmath152-dimensional space which is a @xmath205 homology manifold . now @xmath122 is the hyperbolization of this space , and since the hyperbolization procedure preserves the local structure , @xmath122 is also an @xmath152-dimensional @xmath205 homology manifold . this implies that @xmath206 , which forces the cohomological dimension of @xmath99 to be at least @xmath203 . this completes our proof . as was pointed out to the authors by the referee , the argument in theorem 2.2 can also be used to show that the condition that @xmath0 be a p - group in corollary 2.1 really is necessary . namely , there are examples of a @xmath207 action on an orientable poincar duality @xmath2-hyperbolic group over @xmath3 whose fixed subgroup is not a duality group over _ any _ pid ( in which @xmath208 ) . indeed , note that the unit tangent bundle @xmath209 of an @xmath88-dimensional sphere can be identified with the stiefel manifold @xmath210 of orthonormal @xmath211-frames in @xmath212 . the latter can be embedded in @xmath213 via the map @xmath214 ( where @xmath215 are orthonormal vectors ) . note that since @xmath216 are orthonormal , we have that @xmath217 , and also that : @xmath218 this implies that @xmath209 is diffeomorphic to the brieskorn variety for the polynomial @xmath219 . in particular , we see that @xmath209 is the fixed point set of the @xmath207 action on the brieskorn variety for the polynomial @xmath220 , where the action is given by @xmath221 , where @xmath222 . furthermore , since odd dimensional spheres have a non - zero vector field , we have that @xmath223 . now let @xmath112 be the fixed point set of the above mentioned action of @xmath207 on the brieskorn variety @xmath111 for the polynomial @xmath224 . as we mentioned earlier , the brieskorn variety @xmath111 is pl - homeomorphic to @xmath115 , while by the previous paragraph , @xmath112 is diffeomorphic to @xmath225 . suspending the spaces and hyperbolizing gives us a @xmath207 action on a @xmath6 space @xmath73 , where now the fixed subset @xmath122 is the hyperbolization of the suspension of @xmath225 . the proof that @xmath226 is not poincar duality over any pid @xmath74 is almost a verbatim repetition of that given for theorem 2.2 . in particular , the local homology sheaf for the space @xmath122 will have three distinct indices ( namely @xmath227 ) for which @xmath228 ( where again , @xmath1 is one of the suspension points ) . working through the zeeman spectral sequence , we again find indices ( @xmath229 ) where the cohomology of @xmath226 is not finitely generated . the only substantial change is in the argument showing that the cohomological dimension of @xmath226 over @xmath74 is @xmath230 . to do this , we merely note that the hyperbolization map @xmath231 induces a surjection on integral homology , together with the fact that @xmath232 . we finish our paper with a few remarks . firstly , we note that the results we obtain are , in some sense , dealing with exceptional automorphisms of @xmath2-hyperbolic groups . indeed , levitt & lustig [ 12 ] have shown that , in a suitable sense , ` most ' automorphisms of a @xmath2-hyperbolic group have very simple fixed point sets for their induced actions on the boundary at infinity ( in fact , their fixed point sets consist of a pair of points ) . also , if we start with a torsion free group , then the group of inner automorphisms will also be torsion free , hence any automorphism of finite order in some sense ` lives ' in the outer automorphism group , which tends to be small . secondly , we should point out that , in the counterexamples we constructed , the groups @xmath18 all have boundary at infinity which is in fact _ homeomorphic _ to a sphere . this follows from the fact that the link of every vertex in the space @xmath127 is pl - homeomorphic to the standard sphere , so by a result of davis & januszkiewicz [ 7 ] , the boundary at infinity of @xmath127 is homeomorphic to a sphere . thirdly , we can ask related questions in a somewhat more general setting . more precisely , given an arbitrary topological space @xmath122 , we can consider the question of what type of actions can be realized _ algebraically _ or _ geometrically_. by a geometric action , we mean one that is induced by an isometry of a @xmath2-hyperbolic space @xmath73 whose boundary is homeomorphic to @xmath122 . by an algebraic action , we mean one that is induced by an automorphism of a @xmath2-hyperbolic group @xmath18 whose boundary is homeomorphic to @xmath122 . note that , at the cost of changing the set of generators for the group @xmath18 ( as in the proof of proposition 2.1 ) , we can always view an algebraic action as a geometric one ( given by an isometry of the cayley graph ) . the fact that this question is non - trivial , even in the more general setting , can be seen by considering the situation of a menger manifold . it is well known that there are numerous @xmath2-hyperbolic groups whose boundary at infinity are menger manifolds . now a result of iwamoto [ 10 ] states that every closed subset of a menger manifold can be realized as the fixed point set of an involution . on the other hand , if an involution can be realized algebraically via an involution @xmath106 of a group @xmath18 , then the fixed point set on the boundary at infinity must coincide with the boundary at infinity of the subgroup @xmath233 . however , the latter set can not have any cutpoints ( see bowditch [ 3 ] and swarup [ 1 ] ) . this gives a necessary condition for a closed subset of a menger manifold to be the fixed point set of an algebraically realizable involution . what are the sufficient conditions ? finally , we mention that these examples give involutions of a @xmath2-hyperbolic group @xmath18 where the fixed point set of the induced involution on the boundary at infinity is not an anr , although @xmath234 . one could ask whether the fixed point set could display other complicated behavior . for instance , does there exist an involution of a @xmath2-hyperbolic group @xmath18 , with fixed subgroup @xmath86 , with the property that @xmath234 , @xmath235 , and the embedding @xmath236 is a locally flat , non - trivial knot ? [ 1 ] bestvina , m. & mess , g. _ the boundary of negatively curved groups_. j. amer . math . soc . 4 ( 1991 ) , no . 3 , pp . 469481 .

in this paper , we show that if @xmath0 is a finite @xmath1-group ( @xmath1 prime ) acting by automorphisms on a @xmath2-hyperbolic poincar duality group over @xmath3 , then the fixed subgroup is a poincar duality group over @xmath4 . we also provide a family of examples to show that the fixed subgroup might not be a poincar duality group over @xmath3 . in fact , the fixed subgroups in our examples even fail to be duality groups over @xmath3 .

suppose we are interested in the physics of events localized in a region @xmath1 of the space . the hilbert space of states can be decomposed accordingly as a tensor product @xmath2 of the spaces of the states localized in @xmath1 and in the complementary region @xmath3 . take now the vacuum @xmath4 as a global state of the system , with density matrix @xmath5 . the state @xmath6 relevant to the algebra of operators acting on @xmath7 follows from the partial trace of @xmath8 over the complementary hilbert space @xmath9 . this gives the local reduced density matrix @xmath10 the global state @xmath8 is generally entangled in the bipartite system @xmath11 and in consequence this matrix is mixed . the corresponding entropy @xmath12 is usually called entanglement or geometric entropy . the entanglement entropy is one of the most prominent candidates to explain the intriguing entropy of the black holes @xcite . however , in this proposal the role of quantum gravity is fundamental to produce a finite entropy , and the whole subject is still controversial . on the other hand , this and other measures of entanglement have also been extensively studied in condensed matter and low dimensional systems , partially motivated by advances in quantum information theory and the density matrix renormalization group method . as a result it was uncovered that a variety of phenomena such as quantum phase transitions have an interesting correlate in the entanglement properties of fundamental states @xcite . from the point of view of quantum field theory ( qft ) the function @xmath13 can be considered as a non local variable with interesting non perturbative properties @xcite , which can be defined for any theory disregarding the field content . in this context , we can mention among the applications the description of topological order @xcite , and the renormalization group irreversibility in two dimensions @xcite . in the continuum limit described by a quantum field theory the entanglement entropy is divergent due to the presence of an unbounded number of local degrees of freedom . the divergent terms must be proportional to quantities which are local and extensive on the boundary of @xmath1 . this can be seen as a consequence of the local nature of the ultraviolet divergences , and that the boundary is shared between @xmath1 and @xmath3 which have the same entropy ( for any pure global state ) @xmath14 on a technical level this characteristic of the divergent terms is due the fact that the entanglement entropy is the variation of the euclidean free energy with respect to conical singularities located on the boundary of @xmath1 @xcite . thus , in @xmath15 spatial dimensions , we expect to have an expansion of the form @xmath16 \,\epsilon^{-(d-1 ) } + g_{d-2}[\partial v]\,\epsilon^{-(d-2 ) } + ... + g_0[\partial v]\,\log ( \epsilon \lambda)+ s_0(v)\ , , \label{div}\ ] ] where @xmath17 is a finite part , @xmath18 is a short distance cutoff , and the @xmath19 are local and extensive functions on the boundary @xmath20 , which are homogeneous of degree @xmath21 ( in a different context a similar expansion was used in @xcite ) . the leading divergent term coefficient @xmath22 $ ] is proportional to the @xmath23 power of the size of @xmath1 . this was noted since the earliest papers on the subject @xcite and is usually referred to as the area law for the entanglement entropy . however , strictly speaking , @xmath24 depend on the regularization procedure and it is not proportional to the area if this later is not rotational invariant . for example , if we draw a square of side @xmath25 on a two dimensional lattice , we would have @xmath26 , where @xmath18 is the lattice spacing , but where the dimensionless constant @xmath27 strongly depends on the relative angle between the square sides and the lattice symmetry axes . moreover , all the terms @xmath19 with @xmath28 are not physical within qft since they are not related to continuum quantities . on the contrary , the dimensionless coefficient @xmath29 of the logarithmic term is expected to be universal . logarithmic divergent terms in the entropy have been previously found in four dimensional black hole space - times @xcite . they are present generically in even dimensions for sets with smooth curved boundaries . this follows from the heat kernel expansion for conical manifolds with smooth singularity surface @xcite . in this work we show that there is also a logarithmic term in three dimensions for sets @xmath1 with non - smooth boundary . in particular , we consider the case of spatial polygonal sets ( in a different scenario , a logarithmic contribution to the entanglement entropy was also reported in @xcite ) . since @xmath29 is dimensionless , extensive and local on the boundary we conclude that for @xmath1 a polygon it must be of the form @xmath30 where the sum is over all vertices @xmath31 and @xmath32 is the vertex angle . on general grounds one also expects point - like vertex induced logarithmic terms in any dimensions . all these considerations apply to the alpha - entropies ( or rnyi entropies ) as well @xmath33 these measures of information behave very much like the entanglement entropy , as discussed in several papers @xcite . for integer @xmath34 they are also more suitable for explicit calculation , since the traces @xmath35 involved can be represented by a functional integral on an n - sheeted space with conical singularities located at the boundary of @xmath1 @xcite . the entanglement entropy follows from @xmath36 by analytical continuation . thus , in 2 + 1 dimensional theories the alpha - entropies contain a logarithmic term @xmath37 analogous to the one in ( [ div ] ) for the entropy . we also have @xmath38 . in eq . ( [ cinco ] ) @xmath39 is a parameter with the dimensions of an energy , depending on @xmath1 and on the particular theory . for a massless field it is the inverse of any typical dimension @xmath40 of @xmath1 and when the mass dominates , @xmath41 , it can be taken as @xmath42 . we find the analytic expression of @xmath43 for a free scalar field , and check our results by numerical simulations on a two dimensional lattice . the outline of the paper is the following . in section 2 , we obtain the analytic expression for @xmath35 for the local density matrix associated to a plane angular sector for a massive scalar field in @xmath44 dimensions . the calculation is reduced to the one of the trace of the green function on a two dimensional sphere , where boundary conditions are specified on a segment of a great circle . we use the method introduced in @xcite to obtain this trace exactly . finally , we identify the logarithmic coefficients @xmath43 of ( [ cinco ] ) . we also compare them with the results given by simulations in a square lattice finding a perfect accord . in section 3 , we find the small angle limit of @xmath45 for free fields and relate it to the two dimensional entropic c - functions . finally , in section 4 we present our conclusions . we have also included in the appendix a a detailed derivation of the formulas concerning the green function in the sphere with a cut presented in section 2.1 , and in appendix b , the method we have used to compute numerically the geometric entropy in a two dimensional lattice . to compute the coefficients @xmath43 of the logarithmic term in ( [ cinco ] ) for a scalar field we choose for convenience to study the entropy associated to a set @xmath1 given by a plane angular sector of angle @xmath46 . the traces @xmath47 involved in ( [ cuatro ] ) , with @xmath48 , can be represented by a functional integral on an @xmath49-sheeted three - dimensional euclidean space with conical singularities located at the boundary of the set @xmath1 @xcite . to be explicit , calling @xmath50 and @xmath51 to the upper and lower faces of the plane angular sector , the replicated space is obtained considering @xmath49 copies of the three - space cut along the angular sector , and sewing together the upper side of the cut @xmath52 with the lower one @xmath53 , for the different copies @xmath54 , and where the copy @xmath55 coincides with the first one . the trace of @xmath56 is then given by the functional integral @xmath57 $ ] for the field in this manifold , @xmath58}{z[1]^{n}}\ , . \label{dd}\ ] ] then , following @xcite we consider a free massive complex scalar field on this manifold , and map the problem to an equivalent one in which we deal with @xmath49 decoupled and multivalued free complex scalar fields . first we arrange the values of the field in the different copies in a single vector field living in a three dimensional space , @xmath59 where @xmath60 is the field on the @xmath61 copy . note that in this way the space is simply connected but the singularities at the boundaries of @xmath1 are still there since the vector @xmath62 is not singled valued . in fact , crossing the plane angular sector from above ( side @xmath50 ) or from below ( side @xmath51 ) , the field gets multiplied by a matrix @xmath63 or @xmath64 respectively . here @xmath65 which has eigenvalues are @xmath66 , with @xmath67 . this is because in the fermionic case it is @xmath68 . ] @xmath66 , with @xmath69 . then , changing basis by a unitary transformation in the replica space , we can diagonalize @xmath63 , and the problem is reduced to @xmath49 decoupled fields @xmath70 living on a single three dimensional space . these fields are multivalued and defined on the euclidean three dimensional space with boundary conditions imposed on the two dimensional set @xmath1 given by @xmath71 here @xmath72 and @xmath73 are the limits of the field as the variable approaches @xmath1 from each of its two opposite sides in three dimensions . in this formulation we have @xmath74\,,\label{sumz}\ ] ] where @xmath75 $ ] is the partition function corresponding to a field which acquires a phase @xmath76 when the variable crosses @xmath1 . we note that the coefficient of the logarithmic divergent term in @xmath77 $ ] corresponds to the integrated trace anomaly @xmath78 , or equivalently the @xmath79 coefficient of the heat kernel expansion @xcite , for a massless scalar in three dimensions with the boundary conditions ( [ bc ] ) . our calculation of @xmath75 $ ] is based on the relation between the free energy and the green function for a free massive scalar @xmath80=-\int dr^3 g_a(\vec{r},\vec{r})\ , . \label{green}\ ] ] here @xmath81 is the green function for a complex scalar of mass @xmath82 in three dimensions subject to the boundary conditions ( [ bc ] ) . to be explicit , we have @xmath83 where @xmath84 is orthogonal to the plane of @xmath1 . the laplacian and the boundary conditions allow the separation of angular and radial equations in polar coordinates . using standard methods we arrive at the expression ( see for example the similar calculation in @xcite ) @xmath85 where @xmath86 is the standard bessel function . here the sum is over the normalized eigenvectors @xmath87 of the angular equation @xmath88 where @xmath89 is the laplacian on the sphere with domain given by the functions satisfiyng the boundary conditions inherited from ( [ cator ] ) . specifically we can choose the cut on the equatorial plane @xmath90\,.\label{boun}\ ] ] the functions @xmath91 are the spherical harmonics if @xmath92 , otherwise they are known to satisfy lam differential equations @xcite . also , the values of @xmath93 are not integer if @xmath94 . the precise expressions for @xmath91 and @xmath93 will not be relevant in what follows . taking the trace @xmath95 in eq . ( [ ga ] ) gives @xmath96=-\frac{1}{2m^2}\sum_{\nu}(\nu+1/2)=-\frac{1}{2m^2}\textrm{tr}\sqrt{-\delta_{\omega}+\frac{1}{4}}\ , . \label{trsqrt}\ ] ] though this expression is divergent , the piece we are interested in , which is the one dependent on the angle @xmath46 , is finite . to proceed , we find convenient to express the trace of the square root of the operator in ( [ trsqrt ] ) in terms of the corresponding green function . this is done by using the expression for the powers of an elliptic operator @xmath97 in terms of the resolvent given in @xcite , @xmath98 where @xmath99 is a curve depending on the particular operator @xmath97 . in the present case @xmath99 begins at infinity , pass along the negative real axes on the upper complex plane , encircles the origin and goes back to infinity on the lower half of the complex plane along the negative real axes . this gives for the trace @xmath100 the problem is then reduced to the calculation of the trace of the two dimensional green function on a sphere with a cut of angle @xmath46 , where the boundary conditions ( [ boun ] ) are imposed . in a previous paper @xcite we have solved the analogous problem on the plane with boundary conditions imposed on an interval , using a generalization of a method originally developed in @xcite . it essentially consists in exploiting the symmetries of the helmholtz equation even in the presence of symmetry breaking boundary conditions by analyzing the behavior of the green function at the singular points . following the recipe of @xcite step by step ( with the additional algebraic complications corresponding to the spherical case ) we find the analytic expression for the trace of the green function as a solution of a system of ordinary differential equations . the details of the derivation are given in the appendix a. explicitly we find @xmath101 the function @xmath102 is the solution of the following set of ordinary non linear differential equations ( we omit the subscript @xmath103 and the dependence on @xmath46 of the variables for notational convenience ) @xmath104 where @xmath105 , @xmath106 , @xmath107 , @xmath108 , @xmath109 are functions of @xmath46 given in terms of @xmath110 , @xmath111 , @xmath112 , @xmath113 , @xmath114 , and @xmath50 by the following set of algebraic equations @xmath115 the boundary conditions at @xmath116 are @xmath117 - \frac{\pi}{2 } \sinh \left ( \frac{\pi\mu}{2 } \right ) \right)}{2^{2a}\mu \left ( \cos \left ( 2 a \pi\right)+\cosh ( \pi \mu)\right ) \gamma ( 1+a ) \left| \gamma \left ( \frac{1}{2}-a+\frac{i\mu}{2}\right ) \right|^2 } \,,\label{xx1}\\ x_2(\pi)&=&x_1(\pi)\left|_{\,a\rightarrow ( 1-a ) } \right . \,,\label{xx2 } \\ u(\pi)&=&0\ , , \label{upi}\\ b ( \pi)&=&\frac { 2^{1 - 2a } a ( 1-a ) \left| \gamma\left ( \frac{1}{2 } + a+\frac{i\mu}{2}\right)\right|^2}{m\gamma^2 ( 1+a)}\,,\label{treintaytres}\\ c(\pi)&=&b(\pi)\left|_{\,a\rightarrow ( 1-a)}\right . \,,\label{treintaydos}\end{aligned}\ ] ] where @xmath118 and @xmath119 is the digamma function . the meaning of the extra variables @xmath105 , @xmath106 , @xmath107 , @xmath111 , @xmath112 , @xmath50 , @xmath114 , @xmath113 , @xmath108 and @xmath109 is the same as in @xcite and is given in appendix a. the trace in ( [ quince ] ) is regularized such that it vanishes when @xmath120 , where there is no vertex point and no logarithmic term is present in the entropies . in @xcite the partition function for a dirac fermion on the poincar disk with boundary conditions analogous to ( [ boun ] ) imposed on a geodesic segment has been written in terms of a solution of the painlev vi differential equation . though we were not able to find an explicit relation , it is likely that our results for @xmath121 could have also an expression in terms of solutions of these type of equations . [ tbp ] gathering all the results together , using eqs . ( [ cuatro ] ) , ( [ cinco ] ) with @xmath122 , ( [ sumz ] ) , ( [ trsqrt ] ) , ( [ tr ] ) , and ( [ quince ] ) , we arrive at the following result for a real scalar @xmath123 where we have made explicit the @xmath124 dependence of @xmath110 . this equation , together with ( [ hprima]-[treintaytres ] ) gives our final expression for the coefficient of the logarithmic term in @xmath36 for a real scalar field ( half the complex scalar result ) , which is in position to be evaluated by solving the ordinary differential equations numerically . the functions @xmath43 satisfy @xmath125 , which is a consequence of the symmetry in the entropies @xmath126 due to the purity of the vacuum state . in the figure ( 1 ) we have plotted @xmath127 and @xmath128 for @xmath129 $ ] . the values of @xmath130 , @xmath131 and @xmath132 for @xmath133 , @xmath134 and @xmath135 obtained by lattice simulations are also plotted . they show a perfect accord ( less than one percent error ) with the analytical results . these particular values of the angle are the ones for which the coefficient can be calculated in absolute terms with very small error on a square lattice of limited size ( in the present case it was @xmath136 points ) . the numerical methods consist of evaluating the entropy for a massless real scalar ( see appendix b and @xcite ) for a given shape ( square , triangle , etc . ) and different overall size @xmath137 , and then fitting the result as @xmath138 . it is also possible to evaluate very accurately @xmath139 for specific combinations of angles using rectangular triangles . in this way we have computed in the lattice @xmath140 with @xmath141 , where @xmath142 and @xmath143 are small integers . we also obtain in this case a perfect accord with the analytical results . we have also checked that @xmath139 does not depend on the orientation of the polygon with respect to the lattice simmetry axes . from ( [ final ] ) and the differential equations for @xmath110 , it is possible to derive expansions for @xmath43 and @xmath45 for small @xmath46 or @xmath144 . in particular , in the small @xmath46 limit the green function on the cut sphere is related to the corresponding one for the flat space problem with boundary conditions imposed on an interval . taking into account the results of @xcite , which deal with this later problem , we can show directly from the differential equations that for small @xmath46 @xmath145 where @xmath146 is the one dimensional entropic c - function for a free scalar field @xcite . the c - function for a theory in @xmath147 dimensions is defined as @xmath148 where @xmath149 is the entanglement entropy corresponding to an interval of length @xmath150 in @xmath147 dimensions . for free fields @xmath151 where @xmath152 , and @xmath124 is the field mass . we can obtain the formula ( [ 35 ] ) ( and the analog one which is valid for free fermions ) with a less technical derivation . this also sheds light on the origin and the necessity of the logarithmic term . in a previous paper we have shown that the entropy corresponding to a spatial rectangle in @xmath44 dimensions with a short side @xmath25 and long side @xmath40 , @xmath153 , contains the universal term ( included in the finite term @xmath154 in eq.([div ] ) ) @xcite @xmath155 where @xmath156 is a dimensionless function of @xmath25 and the renormalized parameters of the theory . for a free massless theory it is @xmath157 where @xmath158 and @xmath159 are the one dimensional entropic c - functions for a real scalar and a majorana fermion , and @xmath160 and @xmath161 are the multiplicity of the boson and fermion degree of freedom . this gives @xmath162 for a real scalar and @xmath163 for a @xmath44 dimensional dirac fermion . now , we may approximately decompose a plane angular sector with small angle @xmath46 as a union of thin and long rectangles with bigger size as we move further from the angle vertex . thus , there is a term in the entropy for the angular sector which is due to the sum of the terms ( [ tete ] ) for the rectangles ( the term ( [ tete ] ) is extensive in the direction in which lie the different rectangles ) . in the limit of an infinite partition this gives the desired result @xmath164 [ tb ] we have shown that there is a logarithmic divergent term with universal coefficient for the entanglement entropy corresponding to spatial polygonal sets in 2 + 1 dimensions . we have found the analytic expression for the coefficient in the alpha entropies for integer @xmath34 and a free scalar field , and the small angle limit for the corresponding coefficient on the geometric entropy for general free fields . interestingly , since this paper first appeared as a preprint , a vertex induced logarithmically divergent term in three dimensions has also been found in the conjectured geometrical form of the entanglement entropy for some conformal field theories arising in the ads / cft duality context @xcite . the overall form of the logarithmic coefficient as a function of the angle is similar to our exact results , and the authors have also been able to show the relation between @xmath156 and @xmath165 ( eqs . ( [ tete ] ) and ( [ 38 ] ) ) for their entropy functions , which presumably correspond to some interacting theory . technically , the calculation in this paper amounts to the one of the conformal anomaly for a free scalar on a three - dimensional manifold with a conical singularity located on the boundary of a plane angular sector . this is a non - trivial result , since there is no known general method applicable to this case involving conical singularity surfaces which are themselves non - smooth . we have shown that this particular problem can be mapped to the one of the calculation of the trace of the green function for a massive scalar field on a two dimensional sphere where boundary conditions are imposed on a segment of a great circle . we have found this trace analytically by a method introduced in @xcite which allows one to exploit the rotational symmetries of the sphere even in the presence of the symmetry breaking boundary conditions . our results for the green function on the cut sphere may find different uses beyond the present one . one possible application is the problem of scattering of waves by a plane angular sector in three dimensions @xcite . remarkably , it also gives the exact entropy functions for a spatial segment in @xmath147 dimensional de sitter space . this space is equivalent to the surface @xmath166 in @xmath44 minkowski space . the euclidean de sitter space then corresponds to a sphere with radius @xmath40 . the formulas corresponding to ( [ sumz ] ) and ( [ green ] ) in this case take us again to the green function on the cut sphere . the result can then be readily written off , @xmath167 where @xmath150 is the physical size of the geodesic interval , and @xmath82 is the field mass . the additive constant is logarithmically divergent with the cutoff , in order to keep @xmath36 positive for small values of @xmath150 . the formulas ( [ 35 ] ) , ( [ tete ] ) and ( [ 38 ] ) show a remarkable relation between universal terms in the entropies for different dimensions . the function @xmath168 , with @xmath149 the one dimensional entropy corresponding to an interval of length @xmath150 , plays the role of the zamolodchikov s c - function in the entanglement entropy c - theorem @xcite . this theorem states that there is a universal dimensionless quantity in two dimensions which is decreasing under scaling and has a well defined value at the fixed points ( proportional to the virasoro central charge)@xcite . the c - function introduced by zamolodchikov is constructed from a correlator of stress tensor traces , and we have shown that it can be taken to be @xmath169 as well . there has been a substantial effort to extend the c - theorem to higher dimensions , but a definitive result in this direction is still missing @xcite . at fixed points the function @xmath169 is a constant given by the coefficient of the term proportional to @xmath170 in the two dimensional entropy . then , one would be tempted to speculate that a running function which takes the value @xmath45 at fixed points could be a good candidate for a c - function in three dimensions . on the other hand , the logarithmic term in the entropy induced by the vertices is , remarkably , the only obstacle in the following simple argument attempting to prove the c - theorem in any dimensions . consider the mutual information @xmath171 between two non intersecting sets @xmath172 and @xmath173 . this is dimensionless and universal ( all boundary terms get subtracted ) , and it is also positive and increasing with the size of each set separately @xcite . thus , if we take @xmath172 and @xmath173 as anti - starshaped ( see figure ( 2 ) ) @xmath174 is decreasing under dilatations . however , it fails to have a well - defined value at the ultraviolet fixed point , since in that limit @xmath172 and @xmath173 go to angular sectors with a common vertex , and @xmath174 has a logarithmic divergence due to the mismatch of the vertex induced terms in ( [ mar ] ) . we warmly thank c.d.fosco for very useful discussions and important input in the early phase of this project , and r.trinchero for pointing us to ref . @xcite . in this appendix we derive the set of non linear differential equations ( [ hprima]-[treintaytres ] ) which give the trace of the green function on a sphere with a cut of angle @xmath46 ( see figure 3 ) , where the boundary conditions ( [ boun ] ) are imposed . we will follow here the same steps as in @xcite where we have solved the analogous problem on the plane with boundary conditions imposed on a finite interval . we use a generalization of a method originally developed in @xcite . the green function @xmath175 , where @xmath176 and @xmath177 are points on the sphere , is uniquely defined by the following three requirements : a.- it satisfies the helmholtz equation on the sphere @xmath178 where , in polar coordinates , @xmath179 b.- the boundary condition is ( it also holds for the green function derivatives ) @xmath180\,.\ ] ] here the points with polar coordinates @xmath181 , @xmath182 $ ] , form the cut location . we need to specify the cut through its two endpoints with angle @xmath183 and @xmath184 . at the end of the calculation we can set as in section 2 @xmath185 . from now on we also choose @xmath186 $ ] . c.- @xmath187 is bounded everywhere ( including the cut ) except at @xmath188 . several properties of the green function can be derived from its definition . first it is hermitian @xmath189 and the two reflection symmetries of the problem give @xmath190 where @xmath40 and @xmath63 are the reflexion operations given by @xmath191 in what follows we study the structure of singularities of some functions related to @xmath192 and use repeatedly the fact that a nonsingular solution of the homogeneous helmholtz equation must be zero due to the uniqueness theorem . according to the boundary conditions , near the end points of the cut the green function must have branch cut singularities . the requirement that the function must remain bounded on the cut and the equation ( [ equ1 ] ) imply that the leading terms of @xmath187 for @xmath193 near @xmath194 ( and fixed @xmath177 ) have to be of the form @xmath195^{a } s_{1}(z^{\prime})+[(\varphi-\varphi_1)+i(\theta-\pi/2)]^{1-a } s_{2}(z^{\prime } ) \ , , \label{equ2}\ ] ] for some functions @xmath196 and @xmath197 . we have written explicitly only the terms with powers of @xmath198 $ ] with exponent smaller than one . these are the ones of interest in the sequel . note that the contributions at this order must be analytic or anti - analytic in @xmath176 in order to cancel the laplacian term in ( [ equ1 ] ) . the most singular contributions to @xmath199 for @xmath200 follow from the derivative of ( [ equ2 ] ) @xmath201^{a -1}s_{1}(z^{\prime})+(a -1)[(\varphi-\varphi_1)-i(\theta-\pi/2)]^{-a } s_{2}(z^{\prime } ) \,.\label{deri}\ ] ] the function @xmath199 satisfies the homogeneous helmholtz equation and the boundary conditions , and it is not singular at @xmath202 . it has only one singularity located at @xmath183 whose expression is given by ( [ deri ] ) . thus , an adequate linear combination of this function for different values of @xmath203 must be a nonsingular solution of the helmholtz equation , and therefore identically zero . following the same argument as in the flat case @xcite , this leads to the fundamental relation @xmath204 where we are using vectorial notation @xmath205 and @xmath172 is an hermitian matrix . the function @xmath206 satisfies the homogeneous helmholtz equation with the same boundary conditions as @xmath207 with @xmath177 fixed , but it is unbounded around @xmath208 . in fact , it follows from ( [ deri ] ) and ( [ c ] ) that it has singular terms proportional to @xmath198^{-a } $ ] and @xmath209^{a -1}$ ] . however , these disappear if we derive with respect to @xmath210 . on the other hand @xmath206 behaves around @xmath211 as an ordinary wave , that is , it vanishes proportionally to @xmath212^{a } $ ] and @xmath213^{1-a } $ ] . then @xmath214 has singular terms around @xmath211 which are proportional to @xmath212^{a -1}$ ] and @xmath215^{-a } $ ] . this structure of singularities and the relation ( [ sime1 ] ) gives place to @xmath216 where the matrix @xmath217 is a function of @xmath218 . [ tb ] using ( [ c ] ) , ( [ a ] ) and ( [ b ] ) to calculate @xmath219 we get @xmath220 and that the matrix @xmath172 must be a constant , @xmath221 . thus , it can be evaluated by the knowledge of the solutions for @xmath120 or using the flat space limit @xmath222 @xcite . it is given by @xmath223 where @xmath224 is the pauli matrix . the equation ( [ deri ] ) leads to the behavior @xmath225^{-a } \\ \frac{1}{(1-a ) } [ ( \varphi-\varphi_1)+i(\theta-\pi/2)]^{a -1 } \end{array } \right ) \label{wr}\ ] ] for @xmath176 in the vicinity of @xmath226 . using ( [ b ] ) we get a similar expression for s(rz ) @xmath227^{1-a } \\ \left[(\varphi -\varphi_1)+i(\theta-\pi/2)\right]^{a } \end{array } \right ) \ , . \label{wrr}\ ] ] in order to use eq . ( [ c ] ) to compute the trace of the green function we need more information on @xmath206 . with this aim we exploit the symmetries of the problem as follows . we consider two rotations around the axes lying on the equator which cut the sphere at the points @xmath228 and @xmath229 . the associated differential operators are given respectively by @xmath230 we use the notation @xmath231 . from ( [ defl1 ] ) and ( [ defl2 ] ) we check that they satisfy the algebra of angular momentum operators @xmath232&=&\partial_{\varphi}\,,\\ \left[l_1,\partial_{\varphi}\right]&=&-l_2\,,\\ \left[l_2,\partial_{\varphi}\right]&=&l_1\,,\\ l_1 ^ 2+l_2 ^ 2+\partial^2_{\varphi}&=&\delta_{\omega}\,,\end{aligned}\ ] ] and the following relations @xmath233 they commute with the laplacian operator and thus the functions @xmath234 with @xmath235 are non singular at @xmath188 and satisfy the homogeneous helmholtz equation and the boundary conditions . however , due to the @xmath236 terms , they are singular at @xmath226 and @xmath211 as a function of @xmath176 . near @xmath226 we have from ( [ equ2 ] ) , ( [ wr ] ) , ( [ defl1 ] ) and ( [ defl2 ] ) @xmath237^{a -1}s_{1}(z^{\prime})-i(1-a ) [ ( \varphi-\varphi_1)+i(\theta-\pi/2)]^{-a } s_{2}(z^{\prime})\right ] \nonumber \\ & \sim & -i\cos(\varphi^{- } ) s(z^{\prime})a \sigma_3 s(tz)\,,\\ l_2 g(z , z^{\prime})&\sim & i \sin(\varphi^{-})s(z^{\prime})a \sigma_3 s(tz)\,.\end{aligned}\ ] ] this gives from the uniqueness of the solution of the homogeneous helmholtz equation for non - singular functions @xmath238\,,\label{dl1}\\ l_2 g(z , z^{\prime})+l_2^{\prime } g(z , z^{\prime})&= & -i\sin(\varphi^{- } ) \left[s^{\dagger}(z ) \sigma_3a s(z^{\prime})- s^{\dagger}(rz ) \sigma_3a s(rz^{\prime})\right]\,.\label{dl2}\end{aligned}\ ] ] from ( [ wr ] ) we see that the most divergent terms of @xmath239 exactly cancel . but this combination should also have a contribution to order @xmath212^{a -1}$ ] and @xmath215^{-a } $ ] for @xmath176 in the vicinity of @xmath226 . thus we have @xmath240 where @xmath241 is a matrix which depends on @xmath242 . a similar argument hold for @xmath243 @xmath244 giving place to @xmath245 with @xmath246 depending on @xmath242 . taking the derivative with respect to @xmath183 of ( [ dl1 ] ) and ( [ dl2 ] ) , and extracting the coefficients of the divergent terms , we get @xmath247\,,\\ \label{tata } l_2 s(z)&=&-\xi_1s(z)-i\cos(\varphi^{-})\sigma_3 s(z)+i\sin(\varphi^{-})\left[\left[\sigma_3,\gamma\right]s(rz)+\sigma_3\partial_{\varphi}s(z)\right]\,.\end{aligned}\ ] ] we can also write the reflected equation for ( [ bata ] ) and ( [ tata ] ) @xmath248 \,,\label{bata1}\\ l_2 s(rz)=\xi_1s(rz)+i\cos(\varphi^{-})\sigma_3 s(rz)-i\sin(\varphi^{-})\left[\left[\sigma_3,\gamma\right]s(rz)-\sigma_3\partial_{\varphi}s(z)\right]\,.\label{tata1}\end{aligned}\ ] ] subtracting the hermitian conjugate of the equation ( [ bata ] ) and ( [ bata1 ] ) and using @xmath249 , @xmath250 , we obtain @xmath251 a similar relation for @xmath246 follows by subtracting the hermitian conjugate of ( [ tata ] ) and ( [ tata1 ] ) , @xmath252 the expansion of @xmath206 can be extended to the following order around the singular point @xmath253 introducing a real matrix @xmath254 @xmath225^{-a } \\ \frac{1}{(1-a ) } [ ( \varphi-\varphi_1)+i(\theta-\pi/2)]^{a -1 } \end{array } \right ) + n \left ( \begin{array}{l } \left[(\varphi-\varphi_1)-i(\theta-\pi/2)\right]^{1-a } \\ \left[(\varphi-\varphi_1)+i(\theta-\pi/2)\right]^{a } \end{array } \right)\,.\label{2orden}\ ] ] this general expansion for @xmath255 inserted in ( [ chi ] ) and ( [ chi1 ] ) allows us to solve for some components of @xmath254 and obtain a relation between @xmath241 and @xmath246 @xmath256 with @xmath257 it also gives additional relations which , together with the algebraic equations ( [ alge1 ] ) , ( [ alge2 ] ) , give the general form of @xmath258 and @xmath241 in terms of the following parametrization @xmath259 where @xmath50 , @xmath114 , @xmath113 , @xmath260 , and @xmath261 are real functions of @xmath124 and @xmath46 . taking derivatives respect to @xmath262 and @xmath263 of ( [ bata ] ) and ( [ tata ] ) and combining them to reconstruct the helmholtz equation for @xmath206 , and using the expansions of @xmath206 and @xmath264 , we get from the series around the singular points the differential equations ( [ cprima ] ) , ( [ bprima ] ) , ( [ uprima ] ) for @xmath114 , @xmath113 and @xmath50 , and the algebraic equations ( [ 4a ] ) and ( [ minuscula ] ) which determine @xmath108 and @xmath109 . the equation ( [ c ] ) gives for the trace of the green function @xmath265 where @xmath266 and @xmath267 . to find this quantity , we use the information obtained in 5.1 and 5.2 . we first define the following auxiliary integrals @xmath268 these are convergent . they are also real since the relation ( [ sime1 ] ) implies that @xmath269 . the equations for these quantities are obtained basically combining conveniently the components of ( [ bata ] ) and ( [ tata ] ) multiplied by the components of @xmath206 and @xmath264 , and integrating on the sphere . in this way the combination @xmath270 integrated on the sphere gives place to the equation ( [ 29 ] ) . similarly , from the combination @xmath271 we get equation ( [ 30 ] ) and from @xmath272 we get equation ( [ 31 ] ) . differential equations for the integrated variables follow by taking the derivative of ( [ hh ] ) , ( [ x1 ] ) and ( [ x2 ] ) with respect to @xmath210 . these correspond to ( [ hprima ] ) , ( [ x1prima ] ) and ( [ x2prima ] ) of section 2.1 . in the flat space analog of the present calculation it is possible to give a closed algebraic expression of @xmath110 in terms of @xmath50 @xcite . this simplification seems to be lost in the sphere . the homogeneous helmholtz equation on the cut sphere for a function @xmath273 can be solved exactly by separation of variables when @xmath120 . the solution has the general form @xmath274 where we have put now the end points of the cut at the poles @xmath275 and @xmath276 of the polar coordinates . here @xmath277 is given by @xmath278 with @xmath279 and @xmath280 the standard legendre functions , @xmath281 and @xmath282 . evaluating the limits @xmath283 and @xmath284 and comparing with the expansion ( [ wr ] ) for @xmath206 we get for @xmath120 @xmath285e^{i\varphi a } \ , , \\ s_2&=&\frac{2^{(a-1)/2}}{4\pi ( 1-a)}\frac{1}{2^{\frac{-(a+1)}{2}}\cos(a\pi+i\frac{\pi}{2}\mu ) \csc ( a\pi)\gamma(1-a)}\left[\frac{\pi}{2}\cot((1-a)\pi)p_{\frac{1}{2}(-1+i\mu)}^{a-1}(w)\right.\nonumber \\ & -&\left.q_{\frac{1}{2}(-1+i\mu)}^{a-1}(w)\right]e^{i\varphi ( a-1 ) } \ , . \label{s2pi}\end{aligned}\ ] ] using these expressions , the eqs . ( [ a ] ) and ( [ b ] ) , and the expansion of @xmath264 we find @xmath286 , @xmath287 and @xmath288 given in ( [ upi ] ) , ( [ treintaytres ] ) and ( [ treintaydos ] ) . the results ( [ hache ] ) , ( [ xx1 ] ) and ( [ xx2 ] ) for @xmath289 , @xmath290 and @xmath291 follow directly from ( [ hh ] ) , ( [ x1 ] ) and ( [ x2 ] ) , and the explicit form of @xmath292 and @xmath293 , where we have used some formulae for the integrals of a product of two legendre functions given in @xcite . we use the method presented in @xcite to give an expression for @xmath294 in terms of correlators for free bosonic discrete systems . take a free hamiltonian for bosonic degrees of freedom with the form @xmath295 where @xmath296 and @xmath297 obey the canonical commutation relations @xmath298=i\delta _ { \vec{r}\vec{r}^\prime}$ ] , @xmath82 is a hermitian positive definite matrix , and the sums are over the lattice sites @xmath299 . the vacuum ( ground state ) correlators are given by @xmath300 let @xmath301 and @xmath302 be the correlator matrices restricted to the region @xmath1 , that is @xmath303 . the entropies can be evaluated as @xmath304 where @xmath305 are the eigenvalues of @xmath306 . thus , to compute the entropy numerically we need to diagonalize the matrix @xmath307 . we took the lattice hamiltonian for a real massless scalar in three dimensions as @xmath308 we have set the lattice spacing to one . the correlators ( [ x ] ) and ( [ p ] ) are @xmath309 it is relevant to the accuracy of the entropy calculation to evaluate the correlators with enough precision . this can be very time - consuming . we have found it is much faster to evaluate one of the two integrals in the correlators analytically ( in terms of polynomials times elliptic functions ) using a program for analytic mathematical manipulations , and then doing the last integral numerically . t. j. osborne , m. a. nielsen , phys . a * 66 * , 032110 ( 2002 ) [ arxiv : quant - ph/0202162 ] ; a. osterloh , l. amico , g. falci , r. fazio , nature * 416 * , 608 ( 2002 ) [ arxiv : quant - ph/0202029 ] ; g. vidal , j. i. latorre , e. rico and a. kitaev , phys . lett . * 90 * , 227902 ( 2003 ) [ arxiv : quant - ph/0211074 ] . a. kitaev and j. preskill , phys . lett . * 96 * , 110404 ( 2006 ) [ arxiv : hep - th/0510092 ] ; m. levin and xiao - gang wen , phys . lett . * 96 * , 110405 ( 2006 ) [ arxiv : cond - mat/0510613 ] ; p. fendley , m. p. a. fisher and c. nayak , [ arxiv : cond - mat/0609072 ] . h. casini and m. huerta , phys . b * 600 * , 142 ( 2004 ) [ arxiv : hep - th/0405111 ] . see also j. gaite , phys . lett . * 81 * , 3587 ( 1998 ) [ arxiv : hep - th/9710241 ] ; j. i. latorre , c. a. lutken , e. rico and g. vidal , phys . rev . a * 71 * , 034301 ( 2005 ) [ arxiv : quant - ph/0404120 ] ; r. orus , phys . rev . a * 71 * , 052327 ( 2005 ) [ arxiv : quant - ph/0501110 ] ; huan - qiang zhou , t. barthel , j.o . fjaerestad , u. schollwoeck , phys . a * 74 * , 050305(r ) ( 2006 ) [ arxiv : cond - mat/0511732 ] . s. n. solodukhin , phys . rev . d * 51 * , 618 ( 1995 ) [ arxiv : hep - th/9408068 ] ; s. n. solodukhin , phys . d * 51 * , 609 ( 1995 ) [ arxiv : hep - th/9407001 ] ; r. b. mann and s. n. solodukhin , nucl . b * 523 * , 293 ( 1998 ) [ arxiv : hep - th/9709064 ] . d. v. fursaev , phys . b * 334 * , 53 ( 1994 ) [ arxiv : hep - th/9405143 ] ; j. s. dowker , class . * 11 * , l137 ( 1994 ) [ arxiv : hep - th/9406002 ] . a. t. abawi , r. f. dashen , and h. levine , j. math . phys . * 38 * , 1623 ( 1997 ) ; a. t. abawi , r. f. dashen , phys . e * 56 * , 2172 ( 1997 ) ; l. kraus and l. m. levine , comm . * 14 * , 49 ( 1961 ) . j. palmer , m. beatty and c. a. tracy , commun . phys . * 165 * , 97 ( 1994 ) [ arxiv : hep - th/9309017 ] ; b. doyon , nucl . b * 675 * , 607 ( 2003 ) [ arxiv : hep - th/0304190 ] ; b. doyon and p. fonseca , j. stat . * 0407 * , p002 ( 2004 ) [ arxiv : hep - th/0404136 ] . j. l. cardy , phys . b * 215 * , 749 ( 1988 ) ; d. anselmi , annals phys . * 276 * , 361 ( 1999 ) [ arxiv : hep - th/9903059 ] ; a. cappelli and g. dappollonio , phys . b * 487 * , 87 ( 2000 ) [ arxiv : hep - th/0005115 ] ; s. forte and j. i. latorre , nucl . b * 535 * , 709 ( 1998 ) [ arxiv : hep - th/9805015 ] ; a. cappelli , r. guida and n. magnoli , nucl . b * 618 * , 371 ( 2001 ) [ arxiv : hep - th/0103237 ] ; a. h. castro neto and e. h. fradkin , nucl . b * 400 * , 525 ( 1993 ) [ arxiv : cond - mat/9301009 ] . h. bateman ( manuscript project ) , _ higher trascendental functions _ , vol . 1 , mcgraw - hill , new york ( 1953 ) .

we show that the entanglement entropy and alpha entropies corresponding to spatial polygonal sets in @xmath0 dimensions contain a term which scales logarithmically with the cutoff . its coefficient is a universal quantity consisting in a sum of contributions from the individual vertices . for a free scalar field this contribution is given by the trace anomaly in a three dimensional space with conical singularities located on the boundary of a plane angular sector . we find its analytic expression as a function of the angle . this is given in terms of the solution of a set of non linear ordinary differential equations . for general free fields , we also find the small - angle limit of the logarithmic coefficient , which is related to the two dimensional entropic c - functions . the calculation involves a reduction to a two dimensional problem , and as a byproduct , we obtain the trace of the green function for a massive scalar field in a sphere where boundary conditions are specified on a segment of a great circle . this also gives the exact expression for the entropies for a scalar field in a two dimensional de sitter space . keywords : entanglement entropy , conformal anomaly , three dimensional field theory . pacs : 03.70.+k , 03.65.ud , 04.62.+v , 05.50.+q

wolf - rayet ( wr ) stars without hydrogen in their spectra are helium stars ( * ? ? ? * for a review see ) . wr / x - ray binaries are composed of a helium star and a compact star , which can be a neutron star or black hole . at present seven wr / x - ray binaries are known or suspected , all except one in external galaxies ; in six cases the orbital periods range from 0.2 days to about 1.5 day ; one system has a period of about 8 days @xcite.these systems are expected to be the outcome of the later evolution of high - mass x - ray binaries ( hmxbs ) when the evolved massive donor stars in these systems have started to overflow their roche lobes . if the systems do not merge , the outcome of this later evolution is expected to be a very close binary consisting of a helium star - the helium core of the original donor- and the compact star ( see , who suggested that the peculiar 4.8-hour period x - ray binary cyg x-3 is such a system , ( which is indeed the case , as its companion is a wr star of type wn5 , as shown by @xcite ) . when this prediction of spiral - in was made ( we use here the expression `` spiral - in '' for a drastic decrease of the orbital period , due to any kind of mechanism ) , it was thought that helium stars with masses @xmath2@xmath3would be observable as wr stars , but measurements of masses of wr stars in binaries have since shown that , in order to show the typical wr spectral characteristics ( strong emission lines of ionized he , c and o , produced by a dense high - velocity stellar wind outflow ) , the helium stars must have a mass @xmath4@xmath3 . this implies that the hydrogen - rich progenitor of the wr star must have had a mass @xmath5 . there are very few hmxbs known with such a large donor star mass : in our galaxy only 4u1223 -62 ( gx301 - 4 ) and 4u1700 - 37 . unfortunately , the compact stars in these systems have rather low masses , of order 1.5 and @xmath6@xmath3 , respectively , which will make their survival of spiral - in impossible , as will be shown later in this paper . we will show that , on the other hand , if the compact star is a black hole , with a mass @xmath7 , the systems may survive spiral - in , and become close binaries consisting of a wr star and a black hole . for this reason , one expects in general that in wr / x - ray binaries the compact star is a black hole . we show here that there are two different mechanisms by which the final episode of mass transfer in the black - hole hmxb can take place : * in systems in which the donor star has a radiative envelope when it begins to overflow its roche lobe , no common envelope will form , provided the mass ratio of donor star and accretor is not larger than about 3 to 4 @xcite . ( we will assume in this paper that this limiting mass ratio is 3.5 ) . the donor then keeps slightly overfilling its roche lobe and transfers matter on a thermal timescale at a high rate . this results in the formation of a thick accretion disk around the compact star , from where the mass is lost from the system at a high rate , as is shown by the system of ss433 @xcite . in most cases with a black hole compact star this type of evolution leads to spiral in , but the orbital shrinking is moderate and depends on the initial mass ratio of the system . * in systems with a donor with a radiative envelope , but mass ratio larger than 3.5 , or in which the donor has a convective envelope at the time it begins to overflow its roche lobe , a common envelope will form in which the compact star and the dense evolved core of the donor spiral down towards each other due to the large frictional energy dissipation by their motions in the envelope . we show here that this first mode can be important for the formation of short - period wr / x - ray binaries and , subsequently , close double black holes , and may occur over quite a wide range of initial orbital periods of binaries , from one week to well over one year . we show that also the second mode , for donors with radiative envelopes , can lead to double black holes , in this case with very short orbital periods of less than one day . section 2 considers the first above - mentioned mode of spiral - in in more detail , and section 3 considers common envelope evolution , and examines in which regime of binary periods each of the modes is expected to be dominant . in this section some examples are given of how a number of well - known observed wr+o binaries with relatively short orbital periods are expected to evolve in the future , and are expected to produce close double black holes . the ss433 system consists of a roche - lobe filling a4 - 7i supergiant donor star with an estimated mass of @xmath8 and a luminosity of about 3800@xmath9 , plus a compact star with a mass of @xmath10 , in a 13.1-day period binary @xcite . the compact star is surrounded by an extended and luminous accretion disk , about an order of magnitude brighter than its a - supergiant companion . this disk ejects the famous precessing relativistic jets with a velocity of 0.265c , in which neutral hydrogen is ejected at a rate of some @xmath11@xmath12 , while in a strong disk wind with a velocity of about 1500@xmath13of order some @xmath14@xmath12is ejected , as is seen in the form of the ` stationary ' h@xmath15 line and broad absorption lines @xcite . the total mass loss from the disk is basically all the matter that the a - supergiant donor is transferring to the compact object by roche - lobe overflow on its thermal timescale of @xmath16 years ( see also * ? ? ? * ) . the observed radiative accretion luminosity of the compact star with its disk does not exceed the eddington luminosity @xmath17@xmath18of the compact star ( which corresponds to a real accretion rate onto the compact star of order only a few times @xmath19@xmath12 ) , although when seen along the jets the uv luminosity might be as large as perhaps @xmath20@xmath18@xcite , which would correspond to an accretion rate of order @xmath21@xmath12 . this mass loss has been going on for thousands of years , as can be seen from the large radio nebula w50 that surrounds the system and has been produced by the precessing jets and the strong disk wind . even though this mass transfer has been going on for thousands of years , the system has not entered in a common - envelope ( ce ) state . the reason why ss433 has not gone into a ce phase is , as explained by @xcite and @xcite , the fact that the a - supergiant star has a radiative envelope . if one takes away mass from a star with radiative envelope , this envelope responds by shrinking on a dynamical timescale , followed by a re - expansion on the thermal timescale of the envelope . as a result , this star can keep its radius close to that of its roche lobe and will transfer matter to its companion on the thermal timescale of its envelope , without going into a ce phase . there is , however , the caveat that this thermal timescale mass - transfer from a radiative donor envelope may itself become unstable , if the mass ratio of donor and companion star is larger than about 3 to 4 ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? the fact that ss433 did avoid going into a ce therefore implies that its mass ratio must be below this limit , which we will here assume to be 3.5 . with the above given ranges for the estimated masses of donor and accretor of s433 , a combination of masses that is expected to fulfill this condition is : a 10@xmath3 donor and a 3.6@xmath3compact star . in this paper we will , for the sake of argument , assume that these are the masses of the components of this system . on the other hand , for systems with a donor with a radiative envelope and mass ratio larger than 3.5 or if the donor star has a convective envelope , the formation of a common envelope is unavoidable . the reaction of a convective envelope to mass loss is expansion on a dynamical ( pulsational ) timescale , and thus the envelope becomes violently unstable , which leads to runaway mass transfer and the formation of a common envelope . as shown by @xcite , in the case of roche - lobe overflow from donor stars with a radiative envelope , a further condition for avoiding the formation of a common envelope is that the spherization radius @xmath22 of the accreting compact object remains smaller than its roche lobe , where @xmath22 is given by : @xmath23 where @xmath24 is the schwarzschild radius of the compact object . in the case of ss433 , @xmath25 is of order @xmath26 and @xmath27@xmath28 , so @xmath29@xmath28@xmath30@xmath31 , which is deep inside the roche lobe of the compact star , and a ce will be avoided . in all hmxbs with orbital periods upward on one day the same will hold . so in all cases of hmxb systems with a donor with a radiative envelope and mass ratio less than about 3.5 , one expects the system to go into normal roche - lobe overflow evolution similar to that of ss433 . the `` ss433-mode '' of mass transfer is what we have in the past called `` isotropic re - emission '' . with the ss433-mode of mass transfer , followed by mass loss from the disk , which has the specific orbital angular momentum of the compact object , it is simple to calculate how the orbit of the system will change . in case that a fraction of the transferred matter is ejected from the compact star and its disk with the specific orbital angular momentum of this star , and a fraction @xmath32 is accreted by this star , the orbital angular momentum loss leads to a change of the orbital radius a given by : @xmath33^{-3 - 2/(1-\beta)},\ ] ] where q is the mass ratio of donor and compact star , and subscript zero indicates the initial situation at the onset of roche - lobe overflow . for the case in which in which @xmath34 , as is in fact the case in ss433 , as the accreted amount is @xmath14 to @xmath35 times the transferred amount , this equation in the limit of @xmath36 approaching unity simplifies to : @xmath37 using kepler s third law the corresponding equation for the change of the orbital period is : @xmath38 in the case of ss433 , assuming the initial mass of the a - supergiant donor to have been @xmath39 to @xmath40 , the mass of its helium core is about 2.7@xmath3 . this means that at the end of the roche - lobe overflow phase @xmath41 , while at present @xmath42 . inserting these values into equation ( 4 ) , one finds that at the end of the roche - lobe overflow the orbital period of the system will be @xmath43 days . so , ss433 will with these assumed component masses finish as a detached binary consisting of a 2.7@xmath3 helium star and a 3.6@xmath3compact star . figure [ fig:1 ] depicts how the orbital period of ss433 is expected to change as a function of its decreasing donor mass . the entire process will take place on the thermal timescale of the envelope of the 10@xmath3a - supergiant which is between @xmath44 and @xmath16 years . donor star ( with a 2.7@xmath3helium core ) , and a 3.6@xmath3 compact star , consistent with the values measured by hillwig et al . ( 2004 ) , the mass ratio chosen to assure stable roche - lobe overflow . for cyg x-1 we adopted a donor mass of 19@xmath3(with a 10@xmath3helium core , and a 14.8@xmath3compact star ) in a 5.6day orbit . [ fig:1 ] , scaledwidth=45.0% ] the helium star in the resulting system will during helium shell burning go through a second mass transfer phase and finally explode as a supernova , likely leaving a neutron star . if the system remains bound in response to the natal kick of the neutron star , a close eccentric binary will result , consisting of the present @xmath45@xmath3 compact star plus a neutron star . one may wonder what is so special about ss433 and why we do not see more ss433-like systems . we propose that the answer is : the very unusual combination for a hmxb of a rather low donor mass ( presently @xmath46@xmath3and initially @xmath47@xmath3 ) plus a quite massive compact star ( @xmath45@xmath3 ) . an 11 to 12@xmath3initial donor mass and an orbital period of @xmath48days are typical for a be / x - ray binary , the most common type of hmxb , containing a b - emission ( be ) line star . there are known some 200 be / x - ray binaries in our galaxy and the magellanic clouds . list 160 in our galaxy and the two magellanic clouds and lists 141 in our galaxy plus the smc alone . since these papers appeared , swift , integral and other satellites have discovered several tens more , bringing the total presently known number to about 200 . in all but one of the known be / x - ray binaries , the compact stars are neutron stars which have a typical mass of about 1.4@xmath3 . only one b - e / x - ray binary is known to harbor probably a black hole companion , with a mass of 3.6 to 6.9@xmath3 @xcite . if the companion of a 12@xmath3be star is a 1.4@xmath3neutron star , the value of @xmath49 in equations ( 3 ) and ( 4 ) is 8.57 , and the formation of a common envelope is unavoidable , and the two stars will merge , as will be shown below . only if the compact star has a mass @xmath50@xmath3or larger , the system will spiral in slowly and survive the ss433-like mass - transfer process . therefore , out of the @xmath51 be / x - ray binaries , only the one system with an ( alleged ) black hole companion will in the future evolve like ss433 and survive , all the others will have merged after transferring only a small amount of mass . so , the birthrate of ss433-like systems is at most about 0.5 per cent of the birthrate of be / x - ray binaries . ( for a model for the formation and future evolution of the possible be / black - hole binary , see * ? ? ? the simple reason why ss433 can stably survive this type of spiral - in process for @xmath52 to perhaps @xmath53 years is because of its unique combination for a be / x - ray binary progenitor of a quite massive compact star and a relatively moderate - mass donor star . the @xmath45@xmath3compact object in ss433 must be a low - mass black hole , because causality allows neutron stars to have masses not larger than about 3@xmath3 @xcite . in order to determine the limiting orbital period for having a radiative envelope , we notice that for an effective temperature @xmath54k ( spectral type earlier than @xmath55a7 ) , stars have a radiative envelope ( e.g. see * ? ? ? if the luminosity of a donor star of a given mass m is known , its radius @xmath56 can be found from the relation @xmath57 , where @xmath58 is the stefan - boltzmann constant . if the mass @xmath59 of the compact companion of the star is known , and we set the stellar radius @xmath56 equal to the radius @xmath60 of the roche lobe of the donor , then the equation for the roche lobe given by @xcite : @xmath61 allows one to calculate the orbital radius @xmath62 of the binary in which the donor fills its roche lobe , as the mass ratio @xmath63 is known . using kepler s third law , one also calculate the corresponding orbital period @xmath64 . to calculate the maximum orbital periods up to which donor stars still have a radiative envelope , we used the luminosities of post - main - sequence evolutionary tracks , for solar metallicity , of rotating stars with masses up to 50@xmath3given by , and the limiting effective temperature of @xmath65k . post - main - sequence stars originating from stars more massive than about 50@xmath3become luminous blue variables , stars that experience strong eruptive mass loss episodes . because of their strong mass loss they lose most of their hydrogen - rich envelope and always stay at effective temperatures above 8100k . therefore these post - main sequence stars are expected to have radiative envelopes , so for them we used as maximum radius just their maximum post - main - sequence stellar radius . we made these calculations for compact companions with masses of 1.5@xmath3 , 5@xmath3 , 10@xmath3and 15@xmath3 . figures [ fig:2 ] and fig:3give these upper limiting orbital periods as a function of donor mass for initial donors with masses between 9@xmath3and 85@xmath3 . ( in the mass range between 40@xmath3and 60@xmath3there are no tracks by available . it is known from evolutionary calculations with similar assumed wind mass loss rates that for masses above 50@xmath3the stars at the end of hydrogen burning have lost most of their h - rich envelopes and their radii drop rapidly as a function of mass . we have , for the sake of argument , assumed that in the mass range between 40 and 50@xmath3the orbital periods at @xmath66k are constant and after that they linearly decrease towards the orbital period of the 60@xmath3star ) . the figures show that for donor masses up to about 50@xmath3these upper limiting orbital periods range from about 50 to 400days , and beyond 50@xmath3they go down rapidly . below these limiting curves , the donor stars in hmxbs will transfer mass to their compact companions according to the ss433-type of mass transfer , and the systems will not go into common envelope ( ce ) evolution , provided the mass ratio of donor and compact star is less than about 3.5 . the regions where this ss433-like evolution will occur are indicated in the figure 3 by the blue - colored parts of the diagrams.(notice that for calculating the curves of the limiting orbital periods , as well as the limiting donor masses for mass ratio 3.5 , we used the real post - main - sequence masses of the stars , which are considerably reduced with respect to the intitial masses , due to stallar wind mass loss on the main sequence ) . to the right of these blue regions and below the radiative boundary periods , systems will go into common envelope evolution with a donor with radiative envelope . above the radiative boundary periods they will go into common envelope evolution with a convective envelope ( or if the period is too large they will not experience mass transfer at all ) . , @xmath67 and @xmath68 , respectively . ( compare with figure [ fig:2 ] , where we showed the case of a @xmath69@xmath3neutron star companion ) . the region is bound by the condition that the mass ratio between the donor and the compact object is not too extreme , i.e. smaller than @xmath70 ( vertical dashed gray line ) . systems in the part to the right of the blue region will go into ce evolution . the upper limiting period for mass transfer from a donor with a radiative envelope ( blue line with circles , see section [ uplim_rad ] ) and the the lower - limiting orbital - period curves for survival of ce evolution as wr / x - ray binaries are shown for two values of the ce parameters ( light and dark red lines , see section [ uplim_ce ] ) . [ fig:3],scaledwidth=50.0% ] the next question is : which systems with donor stars in this radiative - envelope regime , will survive as binaries after the onset of mass transfer by roche - lobe overflow ? it turns out that none of the systems with a 1.5@xmath3compact star ( neutron star ) will , as was already mentioned above . because of their mass ratios of far above the mass ratio upper limit of 3.5 , they go into ce evolution and for initial donor masses below 40 @xmath3 they all merge . this can be seen from the lower - limit curve for survival of ce evolution ( the red curve ) in figure [ fig:2 ] . ( the way this curve was calculated is explained in the next section ) . although for donor masses larger than 40@xmath3according to the figure systems might survive ce - evolution ( grey hashed region ) , such massive donors with ns companions may not be very likely , for stellar evolution reasons . in figure [ fig:2 ] we have indicated also the ranges of orbital periods and masses of the bulk of the known supergiant hmxbs with neutron star companions , and of the b - emission x - ray binaries with neutron star companions . one observes that the vast majority of the known supergiant hmxbs with neutron stars does not survive spiral - in . it can be seen from this figure that among the known types of hmxbs only the b - emission x - ray binaries with a neutron star companion and orbital periods ranging from larger than 220d with a 9@xmath3donor to larger than 370d with a 20@xmath3donor ( the red region of the diagram ) , will survive spiral in and can later form double neutron stars , after explosion of the helium star . this is a well - known result . while none of the systems in the radiative - donor regime with a 1.5@xmath3neutron star survives the mass transfer , the systems with a 5@xmath3 , 10@xmath3and 15@xmath3compact companion star ( black hole ) , in the blue - colored parts of figure[fig:3 ] , do survive ss433-like mass transfer and can produce helium - star binaries with relatively short orbital periods : wr / x - ray binaries . it should be kept in mind that for the case with a 5@xmath3 , 10@xmath3and 15@xmath3compact star , the donors which we consider , overflow their roche lobes after leaving the main sequence , which means that they evolve according to case b of close binary evolution @xcite . the lower limit for the orbital period for this case is about one week . an example of a system that will survive ss433-like evolution is cygnus x-1 ( which has 5.6 day orbital period , a @xmath71 black hole and a @xmath72 donor , which according to its luminosity must have started out as a 30@xmath3star @xcite . after ss433-like mass transfer the donor in cygnus x-1 will leave a helium star of about 10@xmath3 , which might either leave a neutron star or a low - mass black hole , with an orbital period of about 9 days , depending on the direction and magnitude of the birth kick of the compact object . so cyg x-1 will not terminate as a close system . in figure 1 the expected future evolution of its orbit is depicted . systems with orbital periods above the upper - limit line for radiative envelope in figures [ fig:2 ] and [ fig:3 ] will have convective envelopes and will , when their donor stars overflow their roche lobes , go into ce evolution and spiral in , as described e.g. by @xcite , , @xcite and later papers and recently by @xcite . the same holds for systems with donors in the radiative region and mass ratios larger than about 3.5 . we use here the formalism for the orbital change in the case of ce evolution as given by @xcite and @xcite , which yields a ratio of the final and initial orbital radii @xmath73 and @xmath74 , respectively , given by : @xmath75 where @xmath76 is the mass of the helium core of the donor , @xmath77 is the mass of the hydrogen - rich envelope of the donor at the moment when roche - lobe overflow begins , @xmath78 is the so - called `` efficiency factor '' of ce evolution which indicates the efficiency with which the release of orbital gravitational binding energy that occurs during spiral in of the compact star towards the core of the donor is converted into kinetic energy required to eject the common envelope , @xmath79 is a parameter that depends on the stellar mass density distribution and @xmath80 is the ratio of the roche - lobe radius @xmath81 and the orbital radius @xmath82 at the onset of ce evolution . the value of @xmath80 is typically of order 0.5 . there are many factors that influence the precise value of the product @xmath83 , such as energy sources like recombination energy and accretion energy release ( e.g. see the discussion in @xcite , and in ) . we assume here two values of the common envelope efficiency : @xmath84 and @xmath85 ( e.g. @xcite , @xmath86 , and - while we know this an oversimplification - for our calculations we assume @xmath87 . to calculate the outcome of ce evolution for stars in the mass range 10 to 85@xmath3we used the rotating evolutionary models for solar metallicity of . these models were calculated with stellar wind mass loss , such that at the end of core - hydrogen burning , when the stars leave the main sequence , their masses are lower than their initial values , and the mass of the helium core is known . using for @xmath88 then these reduced post - main - sequence masses , and the corresponding @xmath76 values , one can calculate , for given values of @xmath59 and of the combination @xmath89 , what the values of the ratio @xmath90 will be . given the mass @xmath76 of the helium core one knows the radius of this helium star ( for these we used the interpolation formula from onno pols given in @xcite ) . together with the mass @xmath59 of the compact star this radius gives the minimum final orbital radius @xmath73 for systems that survive ce - evolution , as the helium star is not allowed to be larger than its roche - lobe radius in the final system . if @xmath73 would be smaller than this minimum value , the system does not survive and merges . using the above - given values of @xmath83 and @xmath80 , and the values of @xmath90 calculated according to the above given recipe for each initial donor mass and @xmath59 , one can then calculate the lower - limits to the initial orbital radii @xmath74 required for the systems to survive ce evolution . these @xmath74 values give one also the minimum initial orbital periods for surviving ce evolution , as a function of initial donor mass and @xmath59 . in figure [ fig:2 ] the lower limiting period curve for survival of ce evolution for @xmath91 is indicated for systems with a neutron star companion with @xmath92@xmath3 . for the case of @xmath93@xmath3 , 10@xmath3and 15@xmath3 , the calculated lower limiting period curves for ce evolution with the above - given values of @xmath94 and 0.45 are given . ( the latter value was derived from orbits of post - ce binaries by ) . systems above these curves in the parts of the panels in figure [ fig:3 ] , to the right of the blue - colored regions are expected to survive ce - evolution as wr / x - ray binaries with short orbital periods . table [ table1 ] shows as an example how we expect six well - known observed massive wr+o spectroscopic binaries with well - determined masses and orbital periods to evolve in the future . the masses and orbital periods of these systems were taken from the catalogue of @xcite . we have calculated the future evolution of these systems semi - empirically according to a recipe given in the appendix . columns 2 , 3 and 4 list the observed parameters of the systems . column 5 lists the masses of the components after the wr star has terminated its evolution and has become a black hole , and subsequently the companion has evolved for 3 million years to leave the main sequence and start roche - lobe overflow . we calculated the masses in column 5 by taking into account the wind mass - loss of both stars , and by assuming that at the end of the life of the wolf - rayet star as a wc - type star , 90% of the mass of the wc star disappeared into the black hole ( cf . * ) , meaning that just only the gravitational binding energy is lost . as there is no mass lost , we haave assumed that the black holes did not receive anatal kick . we calculated how the orbits changed due to the wind mass loss and the core - collapse gravitational - mass loss , and assumed that after this the orbits tidally synchronized again . we then assumed the o - type companions of the bhs still to live another 3 million years as core - hydrogen burning stars , losing mass by stellar wind in this period . we subsequently calculated the spiral in of the resulting hmxb consisting of the o - star and the bh , using the ss433-type of spiral in . this resulted into the wr+bh systems with masses and orbital periods listed in columns ( 6 ) and ( 7 ) , respectively . one notices that several of these wr x - ray binaries have orbital periods of order 1 to 2 days , and that their compact stars are black holes . after taking account of the wind mass loss of the wr stars in these systems during their further evolution and , again assuming that 90 per cent of the mass of the final wc star disappears into the black hole , one obtains masses and orbital periods of the double black hole binaries resulting from these systems listed in columns 8 and 9 , respectively . one observes from this table that some well - known wr+o spectroscopic binaries with orbital periods of order one week can produce close wr+bh x - ray binaries , and subsequently produce close double black hole binaries . since the first four systems in table [ table1 ] have a mass ratio of donor and black hole before spiral - in mass transfer very close to the 3.5 limit , it is not completely sure whether the mass transfer from the radiative envelope will not become unstable . to mark this uncertainty , we have a put a colon after the post - spiral - in orbital periods in table [ table1 ] . for the last two systems in table [ table1 ] the mass ratio before spiral - in is below 3.5 and the transfer is expected to certainly be stable . these systems , however , do not leave close wr / x - ray binaries , and also not close double black holes . they leave wr / x - ray binaries resembling the one observed system with an orbital period of about 8 days , mentioned in the introduction . in the case of the first four systems in table [ table1 ] : if their mass transfer becomes unstable , a common envelope will form and the outcome of the evolution must be calculated using equation [ eq : af_ai ] . it turns out that with the above assumed value of @xmath95 , all these four systems do not survive ce - evolution and will merge , as the roche - lobe radius of the post - ce helium star is smaller than the radius of this star . it is , however , very well possible that the value of @xmath96 is larger than 0.25 . using the value @xmath97 derived by , three of the top - four systems in table [ table1 ] will still merge , but the system of wr97 will survive and produce a wr / x - ray binary with an orbital period of @xmath98 hours , which will produce a double bh with orbital period @xmath99 hours . [ cols="<,^,^,^,^,^,^,^,^,^ " , ] one sees that the orbital periods of wr / xrbs produced by stable roche lobe overflow are systematically considerably larger than those produced by ce evolution . in order to estimate the galactic formation rate of double black holes from the channel discussed here , we just give an order of magnitude estimate . in a later paper we will make more refined estimates by means of binary population synthesis . we consider here the number of stars in the galaxy with masses larger than 25@xmath3 , as these are the stars that are expected to produce wr stars ( see section 1 ) . @xcite estimated the number of o - b2 stars in the galaxy as 60 000 . as the mass of a b2 main - sequence star is 10@xmath3 , one finds that with a salpeter imf , combined with a massive star lifetime proportional to @xmath100 , that there are about 3000 stars more massive than 25@xmath3 in the galaxy . one obtains about the same number if one assumes the number of 6000 o - stars in the galaxy estimated by @xcite , taking into account that main - sequence o - stars have masses upwards from 20@xmath3 . however , this number of 3000 stars can not be correct , because the number of wr stars in the galaxy is already at least 2000 . @xcite estimates this number to be even as large as 6500 . this is probably an overestimate , but the new 2016 wr catalogue of paul crowther , which is an update of @xcite , lists 634 wr stars of which of order 500 are in our galaxy . assuming the incompleteness factor of this catalogue for our galaxy to be at least a factor of 4 ( as interstellar extinction obscures a considerable number of the stars at large distances , despite the large intrinsic brightness of wr stars ) , the total number of wr stars in our galaxy is at least 2000 . since wr stars live about 10 per cent of the 4 million years lifetime of their o - star progenitors , one then expects , in a steady state of star formation , the number of o - stars more massive than 25@xmath3 in the galaxy to be at least 20 000 : some 7 times larger than the above - cited earlier estimates . we will assume here that the number of stars more massive than 25 @xmath3 in the galaxy is 20 000 . @xcite find that the percentage of interacting binaries among massive stars is close to 100 per cent . from the orbital period distribution of 50 unevolved o - type spectroscopic binaries that these authors determined , one sees that some 9 have orbital periods between 7 and 20days . these are expected to evolve following case b of close binary evolution . about 50 per cent of the binaries have mass ratios @xmath101 , such that they evolve with close to conservative mass transfer , producing wr+o binaries with orbital periods in the range roughly between 7 and 40days , which produce in the end two black holes that do not differ much in mass . so , in total 9 per cent ( 50 per cent of 18 per cent ) of the massive stars produce wr+o binaries with relatively short orbital periods , between 7 and 40 days . this 9 per cent means : 1800 o - type binaries , which produce - in a steady state- at any time 180 wr+o binaries ( as the lifetime of wr stars is 10 per cent of the lifetime of o - stars ) , which then produce a steady population 1350 bh+o binaries ( assuming the post - mass - transfer o - star companions to live 3 million years on the main sequence ) . the x - ray lifetime of the latter systems ( when the o - stars have left the main sequence and almost fill their roche lobe and produce a strong stellar wind ) is of order 50 000 to 100 000 years , such that in a steady state there will be between 22 and 45 bh - hmxbs in our galaxy . ( notice that this number predicts - with an even spread of these hmxbs throughout the galactic disk with a radius of 14kpc- that there should be about 0.5 to 1 bh - hmxbs within 2kpc from the sun . we do , indeed , observe one such system within 2kpc from the sun : cygnus x-1 ) . the bh+o binaries in which the mass ratio of o - star and bh is @xmath102 , are expected to evolve later in life according to the ss433-mode of spiral after their x - ray binary phase . figure 4 suggests , with a 10@xmath3bh companion , that perhaps about 10 per cent of the bh+o systems are located in the blue region of the diagram and make wr / x - ray binaries byisotropic re - emission " mass transfer . of the remaining systems , with mass ratios @xmath103 , which go into ce evolution , some 10 per cent may have orbital periods long enough to survive ce ( as illustrated by tables 1 and 2 and figure 4 ) with very short orbital periods , of less than a day . assuming thus that some 10 per cent of the bh+o binaries survive as wr / x - ray binaries - half of them with very short orbital periods- this amounts to 135 of the bh+o systems present in a steady state . as the wr / x - ray binaries live 7.5 times shorter than the 3 million year lifetime of their progenitors , one expects in a steady state at any time 18 wr / x - ray binaries to be present in our galaxy , half of them with periods less than a day . these systems will produce double black holes with typical masses around 10@xmath3 , half of them with orbital periods less than a day , which will merge within a hubble time . with 9 such wr - systems with a typical lifetime of 400 000 years one expects this merger rate in the galaxy to be of order once per 50 000 years . if we compare the predicted number of wr / x - ray binaries in our galaxy with the observations , we notice that we know only one such system in our galaxy : cygnus x-3 , which has an orbital period of 0.2 days . this system is an extremely strong x - ray source , with an x - ray luminosity @xmath104@xmath18 . it is close to the galactic plane , at a distance of about 10kpc , behind 3 spiral arms , as determined from 3 different hi - absorption lines during its giant radio outbursts . the x - ray luminosity of a wind - accreting system like cyg x-3 is proportional to @xmath105 , where @xmath106 is the stellar wind velocity near the compact star and @xmath62 is the orbital radius . the reason why cyg x-3 is such a strong x - ray source is two - fold : ( i ) due to the small distance of the compact star from the surface of the wr star , causing it to be in the low - velocity part of the stellar wind , and ( ii ) its small orbital radius . a wr / x - ray binary with the same masses of the components , but an orbital period of one day will have a 3 times wider orbit , such that already alone because of this its x - ray luminosity will be an order of magnitude smaller . since its accretor will also be in a higher - velocity part of the wind , its x - ray luminosity will be still smaller than that , probably not more than @xmath107@xmath18 . therefore , other wr / x - ray binaries in our galaxy , if they exist , will in general be much less prominent as x - ray sources than cyg x-3 , and due to the large extinction near the galactic plane , hard to distinguish from other types of x - ray sources . it thus seems that our predicted number of such system of about 18 is not ruled out by the observations . the estimates of the relative importance of this formation channel of double black holes made in the last section are very rough and need , of course , refinement through full binary population synthesis evolution simulations . this is a next step to be carried out . also , we only considered here evolution with solar metallicity , as the galactic wr binaries are young and therefore have high metallicity . it is obvious that if evolution with lower metallicity would be considered , the outcome could have been quite different , as more massive black holes will be produced due to the weaker stellar wind mass loss during the evolution of massive stars . this is also a refinement that will have to be included in the binary population synthesis models in the future . further , as mentioned above , it is crucial to make detailed binary evolution calculations for black - hole hmxbs with different mass ratios of donor and accretor , in order to estimate , as a function of donor mass , the precise upper limit of the mass ratio up to which the ss433-like mass transfer from a companion with a radiative envelope can proceed stably . this limit is of crucial importance for estimating the precise birth rate of wr x - ray binaries and double black holes resulting for this type of evolution . nevertheless , as we have shown , massive binaries of all kinds of orbital periods may contribute to the formation of close double black hole binaries , and not only the very wide systems that go into common - envelope evolution ( e.g. see * ? 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( eds . ) 2003 , , vol . 212 of _ iau symposium _ , k. a. 2001 , , 45 , 135 , m. h. , charles , p. a. , geballe , t. r. , king , d. l. , miley , g. k. , molnar , l. a. , van den heuvel , e. p. j. , van der klis , m. , van paradijs , j. 1992 , , 355 , 703 , r. f. 1984 , , 277 , 355 , j.2012 , , 054 to calculate the evolution of the observed wr+o binaries in table 1 , we made the following assumptions : * on the basis of evolution calculations , we assumed the wr stars to spend 70% of their helium - burning lifetime as a wn star and 30% as a wc star . * we assumed an observed wn star to be half - way its wn lifetime , that is : at 35% of its helium star lifetime . similarly , we assumed observed wc stars to be at 85% of their helium star lifetime , so still to have 15% of this lifetime to go . * we assumed their wind mass loss rates over their entire lifetime to correspond to the ones stars of their presently observed wr type . for these rates we used wind mass loss rates 1.4 times lower than assumed by , for the following reasons : schaller et al . found for their adopted wind mass loss rates for solar metallicity that even stars up to 120@xmath3 finished with a mass of only 8@xmath3 . it is evident that this can not be correct , since the population i x - ray binary cygnus x-1 is a 14.8@xmath3black hole(@xcite,@xcite ) . assuming some 90 per cent of the final wc star ( basically a co core ) to become the black hole ( heger , 2012 ) , the final mass of the wr progenitor of cyg x-1 must have been 16.4@xmath3 . the formation of a black hole with the mass of cyg x-1 is possible only if the real wind mass loss rates at solar metallicity are about 1.4 times lower than the ones used by schaller et al . in that case the progenitor of cyg x-1 was a star of about 80@xmath3with solar metallicity . * also for the o - stars we used wind loss rates of 1.4 times lower than the ones used by . * as total lifetime of the wr stars ( massive helium stars ) we used 400 000 yrs .

it is shown that black - hole high - mass x - ray binaries ( hmxbs ) with relatively short orbital periods , of order one week to a month may survive spiral in as wolf - rayet ( wr ) x - ray binaries with an orbital period of order a day to a few days , and can later evolve into close double black holes . the reason why black - hole hmxbs with these orbital periods may survive spiral in is : the combination of a radiative envelope of the donor star , and a high mass of the compact star . when the donor star has a radiative envelope when it begins to overflow its roche lobe , and the mass ratio of donor and accretor is not larger than about 3.5 , the formation of a common envelope ( ce ) is avoided , and the systems spiral in slowly with rapid stable roche - lobe overflow , where the transferred mass is ejected from the vicinity of the compact star ( so - called `` isotropic re - emission '' mass loss mode , or `` ss433-like mass loss '' ) . if the mass ratio of donor and black hole is @xmath0 , these systems will go into ce evolution and , if they survive , produce wr / x - ray binaries with orbital periods of only a few hours . if the compact star is a neutron star , the mass ratio of donor and accretor is very large and hmxbs with the same orbital periods do never survive the mass transfer and merge . we therefore find that wr / x - ray binaries are expected to only harbor black holes . several of the well - known wr+o binaries in our galaxy and the magellanic clouds , with orbital periods in the range between a week and several months will later evolve into wr+bh binaries and will produce close double black holes . the galactic formation rate of close double black holes resulting from these systems is estimated to be of order @xmath1 per year . the reason why ss433-like systems are so rarely observed is explained . stars : wolf - rayet stars , x - ray binaries , black holes , black - hole binaries

studies of stars in the central galactic region are essential for an understanding of the milky way . for example , the questions of the structure and extent of the galactic bar , the star formation history , and the distance to the center of our galaxy can all be answered by analyzing various stellar tracers . knowledge of the galactic extinction is mandatory to begin investigations of these stars in a quantitative fashion . a good understanding of extinction is particularly important for the direction toward the galactic bulge and within the galactic plane , where extinction is very significant . extinction @xmath3 changes as a function of wavelength @xmath4 . the shape of the extinction curve is characterized by the quantity @xmath5 , the ratio of total to selective extinction ; @xcite have shown that the family of curves characterized by this single parameter provides a good fitting approximation to most extinction curves . thus , if the value of @xmath5 can be determined ( e.g. , from optical and ir photometry ) , then one can approximately find the properties of the entire uv - ir extinction curve . there are a few suggestions that the coefficient of selective extinction in the direction of the galactic bulge may be non - standard ( e.g. * ? ? ? anomalous extinction , i.e. , extinction described by smaller @xmath6 or @xmath7 ratio than the standard value of @xmath6 = 3.1 , can significantly affect analysis of the stellar populations in the galactic bulge . here , @xmath6 is the coefficient of selective extinction @xmath8 , with reference to the @xmath9 and @xmath0 filters , and @xmath10 , alluding to the @xmath0 and @xmath11 filters . especially with the onset of a new generation of large telescopes , this problem becomes increasingly pressing and important , and one that can be addressed with this study . here , optical @xmath0 and @xmath11-band data for 3256 rr0 lyrae stars from the macho microlensing experiment are used to find both the coefficient of selective extinction and to compute the absolute reddening near the galactic bulge / bar . the value of @xmath6 is known to vary in certain directions , and reddening laws are normally derived from differential measurements of reddened and unreddened luminous early - type stars . however , as such stars are rare in the galactic bulge , the normal derivation of @xmath6 can not be carried out . the @xmath6 values determined from the rr lyrae stars are compared with other reddening relations . the data used in this analysis are based on the macho photometry of 3674 rr0 lyrae stars toward the galactic bulge . cook , kunder , & popowski ( 2007 , in preparation ) . cook et al . ( 2007 , in preparation ) divide the macho bulge rr0 lyrae variables into those associated with the galactic bulge ( 3256 stars ) , and the ones associated with the neighboring sagittarius dwarf galaxy ( 409 stars ) . their classification is assumed in this work . the structure of this paper is the following . the method and resultant coefficient of selective extinction toward the bulge is discussed and compared to other estimates in 2 . reddenings of the bulge rr0 lyrae stars are determined in 3 using their minimum v - band light colors these reddenings are then compared to @xcite reddening estimates and the outliers are discussed . as the bulge rr lyrae stars span a much larger range in color than sgr ones , a coefficient of selective extinction is determined using only this subsample . assuming that the extinction affects the @xmath0 magnitudes much more significantly than the distance spread , the intrinsic scatter in absolute magnitudes , and the scatter due to various observational errors , the slope of the stellar locus on a @xmath12 ( mean v magnitudes of the rr lyrae stars ) versus @xmath13 ( mean color of the rr lyrae stars ) color - magnitude diagram ( cmd ) yields @xmath14 @xcite . in other words , if in the distance modulus , @xmath15 we assume @xmath16 and @xmath17 is approximately constant , then the apparent magnitude , @xmath12 , is proportional to @xmath18 : @xmath19 taking the @xmath20 of each rr0 lyrae star to be approximately constant , @xmath21 which is in the form of a general linear equation with @xmath12 as the independent variable , @xmath13 as the dependent variable , and @xmath7 as the slope . it is shown in the following section that the above conditions are approximately met . figure [ plotone ] is a plot of mean @xmath12 , versus mean @xmath13 . artificial star tests from @xcite indicate that the limiting magnitude of the photometry in the @xmath0 and @xmath11 band is about 21.5 , which is 0.5 magnitudes fainter than the faintest rr lyrae star in this macho bulge sample . the monte carlo analysis discussed later in this section indicates the limiting magnitude has little effect on this paper s analysis . the sources contributing to the intrinsic dispersions of the average @xmath0 and @xmath11 apparent magnitudes are now discussed and quantified following the approach from @xcite . photometric errors . since many macho fields partly overlap on the sky , some stars may be counted in the database twice . using 184 double - represented stars , the internal photometric uncertainty in @xmath22 , @xmath0 , and @xmath11 was found by comparing colors and magnitudes evaluated in different fields . for the macho data toward the galactic bulge , the individual error in @xmath12 is @xmath23 0.15 mag and the individual error in @xmath13 is 0.04 ( cook et al . 2008 , in preparation ) . as the error in @xmath12 and @xmath13 are correlated , upon multiplying the error in @xmath13 , ( 0.04 mags ) , by the slope of the fit , ( 4.28 ) , this value is added to the error in @xmath12 . the photometric error in the points in figure [ plotone ] is thus 0.32 mags . absolute magnitude of rr lyrae stars . it is generally assumed that the absolute magnitude of an rr lyrae star has a linear dependence on @xmath24 $ ] , and that @xmath25 / @xmath26 = $ ] 0.2 mag @xmath27 . the dispersion around the mean value of @xmath28 $ ] = -1.00 dex found in galactic bulge rr lyrae stars is @xmath29 mag . ( kunder et al . 2007 , in preparation ) . this corresponds to an absolute magnitude dispersion of about 0.06 mag . the level of evolution off the zero - age horizontal branch ( zahb ) can also affect at rr lyrae star s absolute magnitude . from the vertical height of the hb of a number of globular clusters of different metallicities , the dispersion around the average rr lyrae luminosity due to the evolution off the zahb of each individual rr lyrae is estimated as @xmath30 0.08 mag @xcite . the scatter in rr lyrae absolute magnitude due to evolutionary phase , helium content , and alpha - element abundance [ @xmath31/fe ] , are less significant and are not accounted for . \3 . bulge distance spread . using the standard hernquist model density profile with a scale length , characteristic density , and characteristic velocity of the bulge from @xcite table 2 , the size of the bulge is @xmath30 1 kpc . as the galactic latitude and longitude of the macho bulge fields ranges from -1.5@xmath32 to - 10@xmath32 and 0@xmath32 to 10@xmath32 , respectively , a correction between the line of sight of the rr lyrae stars in the bulge fields and the plane of the galaxy is negligible . a distance spread of 1 kpc corresponds to a @xmath33 of @xmath30 0.29 mags . adding in quadrature all dispersion contributions , the scatter in @xmath0 for a fixed @xmath22 is 0.44 mags , with the major source of this scatter resulting from the extent of the bulge along the line of sight . fitting a straight line to the data must be done with caution , since errors are present in both independent and dependent variables and are furthermore correlated with each other . ordinary least squares ( ols ) , which assumes no error in the independent variable , can therefore not be used here . instead a straight line fit to the data is obtained using an ols bisector fit to the individual data points . the ols - bisector method has been used when both variables are subject to measurement error and it is not clear which variable should be treated as the independent and which as the dependent . it has been shown to outperform other approaches in such cases @xcite . using the slopes program @xcite , the data are fit using the ols - bisector method . the slope found is @xmath34 and is shown as a solid line in figure [ plotone ] . upon restricting the fit to those stars with @xmath35 , a slope of @xmath36 is obtained . a further cut to remove the very red stars with @xmath37 , yields a slope of @xmath38 . these fit values are given in table [ tab : rvtab ] ; the @xmath7 chosen as the optimal regression is @xmath39 4.28 , as it accounts for all data values . table [ tab : rvtab ] shows that the slope is quite sensitive to just a few points , although these formal errors for the ols - bisector method are quite small . the error in @xmath7 of @xmath40 is chosen to encompass the range of slopes . the ols - bisector fit to our data is further investigated with a simple monte carlo analysis . to duplicate the observed rr0 lyrae data in this paper , 3500 @xmath41 values between 0 and 1.5 are randomly selected . @xmath41 is converted to av using a selective extinction coefficient , @xmath7 , which we vary . assuming an rr lyrae absolute magnitude of @xmath16 = 0.6 and a distance to the bulge of 8 kpc , an observed @xmath0 magnitude is found . as the observed @xmath0 magnitudes in the macho data have an error of @xmath30 0.5 mags , gaussian random numbers are generated with a dispersion of 0.5 mags , and added to the simulated @xmath0 magnitudes . using @xmath42 , and assuming the uncertainty in the measured @xmath22 colors is 0.05 magnitudes , the simulated v magnitudes can be plotted as a function of simulated ( v - r ) magnitudes . we make sure this simulated plot looks similar to figure [ plotone ] . next a cut is made to remove all stars which have v and r magnitudes less than 21.5 . the ols - bisector method is then used to derive the slope of the fit before and after the magnitude cut . we find there is no significant change in the value of the slope after the cut . however , the slope found using the ols - bisector method is not always the value of the slope that was input . this has implications on the derived @xmath7 value . apparently the errors in the photometry in this paper are sufficient to throw off the ols - bisector fit for a given value . figure [ plottwo ] shows the difference in the true @xmath7 value and the ols - bisector slope , @xmath43-ols slope , as a function of the true @xmath7 value . the scatter in this plot is mainly due to the errors in the @xmath0-magnitudes and @xmath22 colors . a slope with a value of @xmath30 4.6 exhibits the least amount of deviation between the true and calculated @xmath7 value . the @xmath7 value found in this paper is close to 4.6 , so the slope found by the ols - bisector method should not be significantly biased . even values well above and below 4.6 do not differ from the true @xmath7 value by more than 0.2 mags . other simulations with somewhat different input parameters ( e.g. , uncertainty in photometry ) are performed and results similar , but not identical to figure [ plottwo ] , are found . thus , we can not simply correct the slope that is obtained from the real data using these simulations . thus , an uncertainty of 0.2 in the @xmath7 is adopted . it is an interesting question how this @xmath7 value from the macho rr lyrae stars compares with other reddening relations . reddening relations determined from broad band photometric data depend on the intrinsic spectral energy distributions of the objects used to determine them as well as the particular filters . the macho @xmath44 and @xmath45 filters are considerably different from johnson @xmath0 and kron - cousins @xmath11 . to compare our results to the standard extinction curve , the @xcite extinction curve with a constant total - to - selective extinction parameter , @xmath46 , is used with the phoenix model synthetic spectra @xcite . the johnson bvr transmission curves from @xcite yield @xmath47 for @xmath48 k , @xmath49 for @xmath50 k and @xmath51 for @xmath52 k. an effective temperature of 6000 k - 7000 k is typical of rr lyrae stars , and an effective temperature of 30000 k is typical of early type stars . the synthetic spectra adopted is one with @xmath28 = -1.0 $ ] and @xmath53 , but varying @xmath28 $ ] and surface gravity does not significantly affect the @xmath7 values . the macho @xmath44 and @xmath45 filters are non - standard , but were transformed into the johnson @xmath0 and kron - cousins @xmath11 as described in @xcite . we simulated this procedure by using the macho @xmath44 and @xmath45 transmission curves and our synthetic spectra to obtain macho @xmath44 and @xmath45 magnitudes . these were then converted using the @xcite transformation . it is found that @xmath54 for @xmath48 k , and @xmath55 for @xmath50 k. both these values are in agreement with the value obtained in this analysis . in comparison , a @xmath6 = 2.5 would yield @xmath56 . reddening laws are normally derived from differential measurement of reddened and unreddened luminous early - type stars . but such stars are absent in the galactic bulge . this analysis uses population ii type stars and indicates that the reddening law in the bulge is on average similar to the standard solar neighborhood value of @xmath46 . changing @xmath6 to e.g. a non - standard value , does significantly affect the measured @xmath7 . quantitatively , it is found that the change in @xmath7 is 68% of the change in @xmath6 . using galactic bulge red clump giants from the ogle dataset , @xcite present substantial evidence that the ratio of total to selective absorption , @xmath57 , is much smaller toward the galactic bulge than the value corresponding to the standard extinction curve and that @xmath6 varies considerably along different lines of sight . their value of @xmath58 , as opposed to the standard value of about 2.5 , corresponds to an @xmath6 of about 2.6 . @xcite constructed extinction and reddening maps for baade s window from color - magnitude diagrams obtained by ogle which showed that extinction in baade s window is quite irregular , varying between 1.3 and 2.8 mag in @xmath59 , with an estimated error of @xmath30 0.1 mag . @xcite use 70 galactic planetary nebulae observed using narrow - band filters to find observed @xmath60 = 2.0 toward the bulge . the @xmath7 value determined from the rr lyrae stars above , indicates that the standard galactic reddening law can in general be adopted in studies of objects toward the bulge . we caution that the value of @xmath6 can vary in certain directions but that on average , the standard reddening law can be used . @xcite and @xcite argue that apparent @xmath61 colors of rr lyrae stars at phases close to the minimum light can be utilized to measure the amount of interstellar reddening along the line of sight to the star since the intrinsic @xmath62 colors seem constant in the phase interval 0.5 0.8 . @xcite suggest that @xmath63 colors might also be used , perhaps with a smaller or negligible metallicity correction . we investigate this further for @xmath22 colors below . the colors at minimum light , @xmath20 , can be computed via two different methods . in the first method , the fourier fits to both @xmath0 and @xmath11-band data are used to average the colors , in magnitude units , for phases between 0.5 and 0.8 . however , we find that when @xmath20 is determined using this method , it correlated with amplitude . in the second method , the @xmath22 color is found at minimum @xmath0-band light . @xcite used this technique and found that intrinsic color at minimum @xmath0-band light is nearly independent of period and @xmath0-band variation amplitude . since this method yields @xmath20 values that are very weakly correlated with amplitude , this is the method employed here . table [ tab : chabtab ] presents our determination of unreddened @xmath20 colors for eleven well observed field rr0 variables ( the data in table [ tab : chabtab ] come from liu & janes 1989 ; cacciari et al . 1987 ; and layden 2005 , private communication ) . the color excess , @xmath41 , is computed by multiplying the published @xmath64 values by the ratio of @xmath65 , where the values @xmath66 and @xmath67 are taken from @xcite . value . ] a plot of @xmath41 vs @xmath22 has a linear character . this is quite remarkable , considering some of these stars exhibit the blazhko effect ( e.g. , aw and ) , some are affected by both the blazhko effect and a `` phase - lag '' problem ( e.g. , ss for ) , and some have evidence of shock waves in their atmospheres ( e.g. , rv phe , v440 sgr ) as discussed by @xcite , @xcite . it has been shown that in many rr lyrae stars the blazhko effect does not affect colors at minimum light , making this method robust . we conclude that the mean @xmath20 color of rr0 lyrae stars at minimum @xmath0-band light is @xmath68 . table [ tab : chabtab ] suggests that the intrinsic color at minimum light is independent or at least very insensitive to metallicity . the _ observed _ and _ intrinsic _ colors at minimum @xmath0-band are put forth to determine the color excess of the bulge rr0 lyrae stars : @xmath69 the fourier fits performed to find @xmath22 color at minimum @xmath0-band light are sensitive to the quality of a lightcurve . to insure high @xmath41 accuracy without sacrificing the sample size , the fourier fits in both passbands for all stars are individually examined . only those with fits that approximated the rr lyrae light curve nicely are taken ( i.e. , if the fourier method mimicked features in the rr lyrae lightcurve well , such as the dip and sharp rise of the curve at minimum light ) . this criterion results in 3525 rr0 lyrae stars with well determined reddening values . example entries of this reddening catalog are presented in table 3 ; the complete listing is available in the electronic version . multiplying the 3525 @xmath70 values by the selective extinction coefficient , @xmath7 , allows the determination of the visual extinction . a map of the visual extinction , @xmath18 , is shown in figure [ plotthree ] . as @xcite also remark , it is immediately obvious that on large scales , extinction is regularly stratified parallel to the galactic plane . to determine the distance modulus to the galactic bulge from the rr0 lyrae stars , an rr lyrae absolute magnitude of @xmath71 at @xmath28=-1.5 $ ] as compiled by @xcite is adopted . if a metallicity dependence of @xmath72 + constant$ ] is adopted ( consistent with the range of slopes found in the literature ) , then using the average metallicity of @xmath28=-1 $ ] from above , an @xmath73 for rr lyrae stars in the bulge is obtained . this predicts @xmath74 for bulge rr lyrae . adopting the rr lyrae statistical parallax calibration of @xmath75 at @xmath28=-1.6 $ ] from @xcite , then using the same metallicity dependence , an @xmath76 for rr lyrae stars in the galactic bulge is predicted . this is identical to the galactocentric distance modulus measured by @xcite from the orbit of a star around the central black hole . in this subsection the reddenings obtained here are tested against @xcite values , which are based on photometry from the macho bulge fields . the macho data of the galactic bulge are based on observations of 94 bulge fields , each field covering an area of 43 by 43. @xcite use 4 x 4 sky regions ( tiles ) as resolution elements . they show that mean colors for these tiles can be converted to extinction , and can thus be used to derive a visual extinction map . each rr0 lyrae star is matched with the corresponding tile , and the color excess , @xmath77 , of the line of sight to the rr0 lyrae star is compared to the @xcite , @xmath78 , as calibrated to @xcite extinction , as calibrated to @xcite extinction is about @xmath79 redder than @xmath78 calibrated to @xcite . ] . figure [ plotfour ] shows this linear relationship , with the line of slope unity over - plotted . the @xmath77 values become larger than @xmath78 when reddening increases , which could mean the @xcite extinction map underestimates the extinction in areas with high reddening . performing the @xcite calibration to the macho rr lyrae lightcurve data does not alter this result . an increase in scatter is expected in stars with high reddenings , since generally a redder rr0 lyrae star is fainter and is more likely to have noisy photometry and lightcurves . especially near the galactic center , differential reddening due to nonuniform distribution of the intervening dust on small spatial scales makes the study of the stellar population difficult even in the infrared ( see , e.g. , * ? ? ? * ; * ? ? ? * ) . @xcite show in their figure 5 , that the amount of differential reddening in a field is directly proportional to the average reddening : the more reddening present in a field , the more patchy that reddening is . as the @xcite extinction map is based on the average color of 4 x 4 sky regions , their average colors and hence reddenings would be susceptible to differential reddening , explaining the slight discrepancy between their reddenings and ours at higher reddenings . what could cause the systematic bias in the @xmath77 values to become larger than the @xmath78 as the reddening increases ? @xcite perform a calibration to @xcite extinction and this calibration assumes a uniform @xmath6 value . this assumption does not hold , however , since the extinction toward the bulge has nonstandard properties ( popowski 2000 ) . @xcite find that the difference between the measured line of sight @xmath57 , and the standard value of @xmath57 depends on color . furthermore , they find that this difference is larger for redder @xmath80 colors and practically negligible in the range where the ogle data were calibrated by standards @xmath81 . similarly , the difference in the coefficient of selective extinction @xmath7 and the standard @xmath7 would also increase with increasing @xmath82 and hence with increasing @xmath83 ( since @xcite uses the average color to determine @xmath83 ) . as the @xcite @xmath83 calibration neglects a change in @xmath7 with increasing color , at higher @xmath83 the values will be underestimated . the most obvious outlier in figure [ plotfour ] is the rr0 lyrae star that is designated in the macho database as 175.30921.95 . it has a negative @xmath70 , which is unphysical . it has a period , amplitude and light curve shape that confirm it is a fundamental mode pulsator . however , as is clear from its position on a color - magnitude diagram , with @xmath84 it is one of the bluest stars in the dataset ( see figure [ plotone ] ) . @xcite suggests that as many as 3% of the rr0 lyrae stars may have an unusual blueness associated with them . the only stars with a color bluer than this star and with an r - band lightcurve suitable for reddening determination , are stars 125.23192.175 and 117.26079.4609 , with @xmath85 and @xmath86 , respectively . the bluer of these two stars @xmath70 deviates by 0.3 mag from @xmath83 , while the other one by only 0.06 mag . other very blue stars include 129.26623.1139 with @xmath87 and 146.28412.88 with @xmath88 , but their @xmath70 is in agreement with @xmath83 . one more characteristic that distinguishes the blue outlying stars 175.30921.95 and 125.23192.175 from their blue non - outlying counterparts , is the fact that these stars have periods that lie in the range 0.469 - 0.472 days , a range in which unstable light curves are common @xcite . spectra of unusually blue stars in the period interval of 0.469 - 0.472 days , ar her and bb vir , were taken by @xcite and @xcite . it was found that these stars have systematically stronger hydrogen lines at minimum light than all the other stars , indicating a higher temperature . the cause of the higher temperature , however , remains unknown . @xcite suggested that bb vir could be a rr0 lyrae star that for some reason has evolved a long way beyond the fundamental blue edge without yet changing mode . the `` hysteresis '' effect is well - known in globular clusters and it is possible that stars 175.30921.95 and 125.23192.175 are also extreme examples of it . stars at the blue edge of the instability strip should have a larger amplitude @xcite . stars 175.30921.95 and 125.23192.175 , the blue stars with deviating @xmath41 values , have about twice as large @xmath0 amplitudes as the other two blue stars , 129.26623.1139 and 146.28412.88 . it is also possible that the macho photometry of the star 175.30921.95 and 125.23192.175 is incorrect . follow - up observations would help clarify this point . in conclusion , although rare , reddening values for rr0 lyrae stars determined by finding the @xmath22 color at minimum @xmath0-band light can be in error if the star is unusually blue . this can be corrected if the temperature of the rr0 lyrae star is known . a quantitative analysis of reddening toward the galactic bulge is performed . assuming that the extinction affects the @xmath0 magnitudes more significantly than the distance spread and the intrinsic scatter in absolute magnitudes , we interpret the slope of the stellar locus on the @xmath0 versus @xmath22 diagram as the selective extinction coefficient , @xmath89 . from the macho rr0 lyrae sample toward the galactic bulge , we derive @xmath90 . this corresponds to the average value observed in the solar neighborhood @xmath2 , and indicates that the optical reddening law toward the bulge is consistent with the standard extinction law . we show that the mean minimum - light @xmath20 color of an rr0 lyrae star is nearly constant at minimum @xmath0-band light and equals @xmath68 , regardless of metallicity . using this property , reddening values to 3525 rr0 lyrae stars are found . parameters derived for individual stars are presented in the electronic version . we would like to thank the referee , derck massa , whose thorough report has led to substantial improvements to this paper . andrea kunder thanks the max - planck - institut fr astrophysik and lawrence livermore national laboratory , where part of this work was completed , for their wonderful people and hospitality , especially martin jubelgas , david syphers , and andreas hamm . khc s and part of amk s work was performed under the auspices of the u.s . department of energy by lawrence livermore national laboratory in part under contract w-7405-eng-48 and in part under contract de - ac52 - 07na27344 . alcock , c. , et al . 2000 , , 542 , 257 alcock , c. , et al . 1999 , , 111 , 1539 babu , j. & feigelson , e. d. 1992 , commun . simulation 21 , 533 bessell , m.s . & germany , l.m . 1999 , pasp , 111 , 1421 bessell , m.s . 1990 , pasp , 91 , 589 blanco , v. 1992 , , 104 , 734 cacciari , c. , clementini , g. , prevot , l. , lindgren , h. , lolli , m. , & oculi , l. 1987 , , 69 , 135 cacciari , c. , clementini , g. , prevot , l. , & buser , r. 1989a a&a 209 , 141 cacciari , c. , clementini , g. , & buser , r. 1989b a&a 209 , 154 cacciari , c. & clementini , g. , 2003 , a&a , 635 , 105 cardelli , j.a . , clayton , g.c . & mathis , j.s . 1989 , , 345 , 245 clementini , g. , raffaele , g. , bragaglia , a. , carretta , e. , di fabrizio , l , & maio , m. ( 2003 ) aj , 125 , 1309 davidge , t. j. 1998 , , 115 , 2374 dutra , c. m. , santiago , b. x. , bica , e. l. d. , & barbuy , b. 2003 , mnras , 338 , 253 ( dsbb ) eisenhauer , f. , schdel , r. , genzel , r. , ott , t. , tecza , m. , abuter , r. , eckart , a. & alexander , t. 2003 , , 597 , l121 feigelson , e. d. & babu j. 1992 , apj , 397 , 55 ferley , j. & barnes , t.g . 1996 a&a 312 , 957 fitzpatrick , e.l . 1999 , pasp , 111 , 63 . fogel , j.a . , tiede , g.p . & kuchinski , l.e . 1999 117 , 2296 gould , a. & popowski , p. , 1998 , , 508 , 844 hauschildt , p.h . , allard , f. , baron , e. , 1999 , 512 , 377 isobe , t. , feigelson , e. d. , akritas , m. g. & babu , g. j. 1990 364 , 104 kanbur , s. & fernando , i. 2004 , preprint ( astro - ph/0407485 ) kinman , t.d . & carretta , e. 1992 pasp 104 , 111 kunder , a. m. , popowski , p. , cook , k. h. , nikolaev , s. & chaboyer , b. 2006 , aspc 349 , 395 liu , t. & janes , k.a . 1989 apjs , 69 , 593 mateo , m. , udalski , a. szymanski , m. , kauny , j. , kubiak , m. & krzemiski , w. 1995 , , 109 , 588 narayanan , v. k. , gould , a. , & depoy , d. l. 1996 , , 472 , 183 popowski , p. 2000 , , 528 , l9 popowski , p. , cook , k.h . , & becker , a. 2003 , , 126 , 2910 ruffle , p. m. e. , zijlstra a. a. , walsh j. r. , gray m. d. , gesicki k. , minniti d. , & comeron f. 2004 , mnras , 353 , 796 sandage , a. 1981 , , 244 , l23 sandage , a. 1990 , , 350 , 603 schlegel , d.j . , finkbeiner , d.p . , & davis , m. 1998 , , 500 , 525 stanek , k.z . 1996 , , 460 , l37 sturch , c. 1966 , , 143 , 774 udalski , a. 2003 , , 590 , 284 widrow l.m . & dubinski , j. 2005 , 631 , 855 derived using @xmath22 at minimum light , @xmath77 , for 2605 rr lyrae stars with color - magnitude based @xmath78 from @xcite . the relation is linear , with @xmath77 values slightly larger than @xmath78 at high extinction . [ plotfour],width=604 ]

we present mean reddenings toward 3525 rr0 lyrae stars from the galactic bulge fields of the macho survey . these reddenings are determined using the color at minimum @xmath0-band light of the rr0 lyrae stars themselves and are found to be in general agreement with extinction estimates at the same location obtained from other methods . using 3256 stars located in the galactic bulge , we derive the selective extinction coefficient @xmath1 . this value is what is expected for a standard extinction law with @xmath2 .

the interests of vendors and customers seem antagonistic _ a priori _ , the former aiming at decreasing quality and increasing price , whereas the latter wishing exactly the opposite . the situation is fortunately more complex , the interests of both sides being sometimes compatible . intuitively , a vendor may sell more items by increasing their perceivable quality , making everybody happier . but the situation is more subtle because of asymmetric information : the vendor knows much better than his prospective customers the real quality of his products . in akerlof s famous lemon problem , the customers have no means to assertain the quality of products , which leads to a no - trade paradox @xcite . when the customers are better equipped , optimal quality emerges @xcite . one of the main issues is to understand under which conditions a manufacturer should diversify his production . economics literature has approached this problem mainly with the help of utility functions . several aspects have been studied , among them optimal quality - based product differentiation @xcite , firm competition by quality @xcite and by price @xcite , the relation between product quality and market size @xcite , etc . ( see @xcite for a review ) . we assume that customers decisions , while influenced by perceived properties of the products , are probabilistic in nature . using a probabilistic consumer choice framework makes it possible to avoid utility functions and hence our model can be understood as an alternative to the usual utility - function approach . for other alternatives , known as models of discrete or probabilistic choice , which still use utility theory and yet they are probabilistic see @xcite . in our work we take the point of view of a monopolistic vendor faced to consumers deciding to buy one of his products according to their perception of its quality . the resulting complex complex system with one vendor , several product variants , and many heterogeneous buyers , is investigated by numerical techniques . the paper is organized as follows . in section [ sec : one_var ] we introduce our framework and determine the optimal quality of a single product proposed to homogeneous or heterogeneous customers . in section [ sec : multiple ] we examine the conditions under which a vendor should segment the market by manufacturing several products of different quality . in section [ sec : price ] we allow the vendor to optimize the price as well . we leave to the appendices a deeper discussion of our assumptions and more technical results on the economics of spamming . we assume that the only difference between products lies in their quality @xmath4 which is therefore the main quantity of interest here . with a suitable choice of units , one can write the profit of a vendor per item sold as @xmath5 where @xmath6 is an increasing function and @xmath7 . for the sake of simplicity , we take @xmath8 ; @xmath9 is also found in the literature but does not alter qualitatively our results . while @xmath10 could in principle be greater than one , a vendor would never choose it , hence our analysis is restricted to @xmath11 $ ] . we assume that a customer buys a product of quality @xmath10 with acceptance probability @xmath12 . while there are many possible choices , e.g. those of refs . @xcite or piece - wise linear functions as in ref . @xcite , we shall mainly use @xmath13 where @xmath14 is the acceptance parameter : for small @xmath15 , @xmath16 is mostly flat , resulting in a lack of quality discrimination ; as @xmath15 grows , the core of @xmath16 shifts towards higher quality , which reflects enhanced discrimination abilities ( see fig . [ fig : p_a ] ) . we will use the shorthands `` ignorant '' for buyers with a small @xmath15 and `` informed '' for those with a large @xmath15 ; an ignorant buyer is quite likely to reject even a perfect product . since for @xmath14 and @xmath17 $ ] is @xmath18 , by eq . ( [ p_a ] ) we implicitly assume that for the considered product there are substitutes which can satisfy needs of consumers . for various values of the acceptance parameter @xmath15 . ] if there are @xmath19 buyers with acceptance parameters @xmath20 ( @xmath21 ) , faced with a single product of quality @xmath10 , the vendor s expected profit @xmath22 is @xmath23 where @xmath24 represents the fixed part of production costs due , for instance , to the initial investment needed to setup the manufacturing plant . assuming that @xmath19 is large , the fluctuations of @xmath22 can be neglected . the structure of this expression is similar to the profit function introduced in @xcite . since @xmath25 , @xmath26 , and @xmath27 is a continuous function , there is at least one @xmath10 maximizing @xmath22 in @xmath28 . in the following we take the point of view of the vendor and hence optimize his expected profit @xmath22 . when there is only one type of buyers , the expected profit simplifies to @xmath29 which reaches its maximum at @xmath30 expectedly , @xmath31 increases when the buyers have a sharper eye . the total optimal profit reads @xmath32 in fig . [ fig : xopt_k ] we report the expected optimal profit per customer @xmath33 as a function of @xmath15 for @xmath34 . when @xmath35 , a vendor only makes a profit when the quality is not too high or too low . accordingly @xmath36 has a maximum at @xmath37 . therefore , if the vendor can not easily change @xmath10 , he should target a population with @xmath38 , or strive to modify the abilities of his prospective customers to detect quality , thereby increasing his profit . when @xmath39 , both the consumers and the vendor benefit from an increase in @xmath10 ; we shall call it the cooperative region . reversely , when @xmath40 the vendor suffers from excessive quality detection abilities of his customers ; he could try a confusing marketing campain or rebranding so as to lower their abilities this is the defensive region . a similar behaviour has been observed in @xcite . in our case , the fact that the cooperative region is much smaller than the defensive region is a consequence of the shape of @xmath16 . for instance , when the prefactor in @xmath12 changes from @xmath41 to @xmath42 , the size of the cooperative region increases significantly . as a function of @xmath15 for @xmath43 . ] heterogeneity brings in more surprises . let us split the population into two groups , group @xmath44 consisting of @xmath45 buyers with acceptance parameter @xmath20 ; the proportion of group @xmath46 is denoted by @xmath47 . the vendor s expected profit reads @xmath48-z.\ ] ] it is not possible to maximize @xmath22 analytically . the result of numerical investigations is shown in fig . [ fig : qopt-1 - 2 ] as a function of @xmath49 for @xmath50 ( ignorant buyers ) and various choices of @xmath51 . as expected , as the proportion of informed buyers increases , @xmath38 grows . but a surprising behaviour is found for instance when @xmath52 : at @xmath53 the optimal quality changes discontinuously . this is because @xmath27 has two local maxima . while for small @xmath49 the small-@xmath10 peak yields the largest profit , its relative height decreases as @xmath49 increases ; accordingly the discontinuous transition occurs when the heights of the two maxima are equal . in fig . [ fig : qopt-1 - 2 ] we also show the dependence of the optimal profit per buyer @xmath54 on @xmath49 . when group 2 has @xmath55 ( e.g. , @xmath56 ) , adding people with more demands regarding quality is beneficial to the vendor ( eq . ( [ xmax-1 - 1 ] ) ) and @xmath22 is an increasing function of @xmath49 . by contrast , when @xmath57 the optimal profit first decreases as almost nobody of group 2 will buy anything and does so as long as group 2 has less influence on @xmath38 than group 1 . then group 2 supercedes group 1 and imposes its quality demands ; the discussion generalises to an arbitrary number of groups . in other words , when society is too heterogeneous , it is impossible to satisfy all buyer groups with one product . ( left ) and vendor s optimal profit @xmath54 ( right ) versus proportion @xmath49 for various values of @xmath51 ; @xmath50 , @xmath58.,title="fig : " ] ( left ) and vendor s optimal profit @xmath54 ( right ) versus proportion @xmath49 for various values of @xmath51 ; @xmath50 , @xmath58.,title="fig : " ] as a function of quality @xmath10 : one product , two groups of buyers ( @xmath50 , @xmath59 ) . as @xmath49 increases , at @xmath53 the heights of the maxima are equal and a discontinuous change of the optimal quality occurs . ] now we assume that the vendor displays @xmath60 product variants of different quality , at equal prices for the sake of simplicity , and that each buyer buys at most one item . a purchase is a two - step process , as a shopper has also to decide on a variant . the choice is also assumed to be probabilistic : variant @xmath61 is chosen according to @xmath62 here @xmath63 quantifies the selection ability of a given buyer . when @xmath64 is large , the buyer almost surely selects the best variant ; on the contrary when @xmath65 , @xmath66 for all @xmath67 , i.e. , the buyer has no discerning power . since @xmath68 is normalized , each buyer purchases at most one item . similar expressions appear in works on the influence of advertisement @xcite and non - price competition @xcite , but other choices of functions would also be reasonable , such as exponentials as in the logit model @xcite . all @xmath69 have equal weight in eq . ( [ p_s ] ) ; section [ sec : proportions ] generalizes this expression in order to take into account the proportions of displayed items . finally , a more complete discussion on the plausibility of @xmath68 is given in appendix [ app : discussion ] . for various values of the selection parameter @xmath64 . in total three variants are displayed , the qualities @xmath70 and @xmath71 are fixed . ] to summarize , the variant @xmath67 with the quality @xmath72 is bought by buyer @xmath46 with probability @xmath73 . as a consequence , if the vendor displays @xmath60 variants to @xmath19 buyers , his expected profit is @xmath74 this equation can be easily extended to account for special circumstances . for example , when @xmath24 is large , it may be profitable to produce one variant and achieve quality differentiation by artificially damaging a fraction of the production , e.g. by disabling some features @xcite . in this case two variants with qualities @xmath75 are displayed but the profit per item sold is only @xmath76 for both of them and the initial cost is reduced from @xmath77 to @xmath24 . for the sake of simplicity , we focus on two groups of customers consisting of @xmath45 members with acceptance parameter @xmath20 and selection power @xmath78 ( @xmath44 ) . the question is whether the vendor should display one or two products . in our framework , the answer is entirely determined by the respective optimal profit of each possibility , denoted by @xmath79 and @xmath80 . since manufacturing two products requires twice as much initial investment ( by hypothesis ) , the region in which @xmath81 shrinks when @xmath82 increases . this appears clearly in fig . [ fig : differentiation ] where we plot the optimal profits versus @xmath83 for two values of @xmath82 . in addition , when @xmath81 , the two optimal qualities @xmath84 and @xmath85 differ significantly . quite clearly , the lower quality targets the group of ignorant buyers while the higher quality is for informed buyers . remarkably , when @xmath86 , the lower optimal quality is even smaller than the optimal quality @xmath87 corresponding to a homogeneous population of ignorant customers . this downward distortion in a situation of a monopolistic vendor is also reported in @xcite ; it is optimal as it reduces the substitution possibilities of higher - value ( or numerous enough ) customers . the benefits of low quality variants in market competition are discussed in detail in @xcite . ; two - variant profits @xmath88 ( broken and dashdot lines ) are shown only when quality differentiation occurs . b ) optimal quality as a function of proportion @xmath49 , curves for the differentiated qualities @xmath84 and @xmath85 are only shown when @xmath89 . values of parameters are @xmath90 , @xmath59 , @xmath91 , and @xmath92.,title="fig : " ] ; two - variant profits @xmath88 ( broken and dashdot lines ) are shown only when quality differentiation occurs . b ) optimal quality as a function of proportion @xmath49 , curves for the differentiated qualities @xmath84 and @xmath85 are only shown when @xmath89 . values of parameters are @xmath90 , @xmath59 , @xmath91 , and @xmath92.,title="fig : " ] the phase space @xmath93 of optimal production when at most two variants are allowed is reported in fig . [ fig : z_trans]a . at intermediate values of @xmath49 , product differentiation exists if @xmath82 is small enough ( see also @xcite ) . can a further decrease of @xmath82 differentiate further the production ? of optimal production : only @xmath94 variants are allowed ( a ) and without constraints on the maximal number of variants ( b ) . same parameters as in fig . [ fig : differentiation].,title="fig : " ] of optimal production : only @xmath94 variants are allowed ( a ) and without constraints on the maximal number of variants ( b ) . same parameters as in fig . [ fig : differentiation].,title="fig : " ] figure [ fig : z_trans]b reveals the russian - dolls structure of product differentiation . let us consider the case @xmath95 : when @xmath96 , in fact only two products are really different , i.e. @xmath97 . in other words , it pays to duplicate the low quality variant . this is because it decreases the likelihood that an ignorant buyer selects the high quality variant , while informed buyers , thanks to their high selection parameter , are still able to pick the premium variant . this mechanism is at work for a generic @xmath60 : when @xmath82 is small , for the vendor it may be optimal to display @xmath98 low - quality variants with identical qualities and one premium variant . finally we consider the vendor s expected profit for various numbers of displayed variants . as can be seen in fig . [ fig : fine]a , when @xmath99 , the additional gain decreases very fast when @xmath60 increases and vanishes when @xmath100 . by contrast , when @xmath101 , @xmath102 saturates at the much higher @xmath103 and then decreases . displayed variants when @xmath99 ( a ) and when @xmath101 ( b ) . same parameters as in fig . [ fig : differentiation].,title="fig : " ] displayed variants when @xmath99 ( a ) and when @xmath101 ( b ) . same parameters as in fig . [ fig : differentiation].,title="fig : " ] in eq . ( [ p_s ] ) we implicitly assume equal standing of the available variants , which is often not the case in practice . this suggests to introduce variant weights @xmath104 ( @xmath105 ) in the selection probability @xmath68 , taking into account for instance the effective visibility of each product due to advertisement or display position in shops . ( [ p_s ] ) generalizes to @xmath106 an example of the interplay between @xmath104 and @xmath64 is shown in fig . [ fig : illustrations ] : better equipped customers are able to pick the better product even when its effective proportion is small . , @xmath107 ) for various @xmath64 . b ) when qualities are differentiated , informed buyers are able to select the premium variant ( dark symbols ) even when its proportion is small ; ignorant buyers select mostly the low quality variant ( white symbols ) . c ) when only one quality is displayed , the vendor has to compromise between the two groups and a mediocre variant is optimal ( grey symbols).,title="fig : " ] , @xmath107 ) for various @xmath64 . b ) when qualities are differentiated , informed buyers are able to select the premium variant ( dark symbols ) even when its proportion is small ; ignorant buyers select mostly the low quality variant ( white symbols ) . c ) when only one quality is displayed , the vendor has to compromise between the two groups and a mediocre variant is optimal ( grey symbols).,title="fig : " ] , @xmath107 ) for various @xmath64 . b ) when qualities are differentiated , informed buyers are able to select the premium variant ( dark symbols ) even when its proportion is small ; ignorant buyers select mostly the low quality variant ( white symbols ) . c ) when only one quality is displayed , the vendor has to compromise between the two groups and a mediocre variant is optimal ( grey symbols).,title="fig : " ] to study the effects of the proposed generalization we use once again two groups of customers and choose the parameters so as to set the system in the quality differentiation region . results of numerical optimization of the optimal profit are reported in fig . [ fig : proportions ] , @xmath108 denotes the proportion of the premium variant . differentiation occurs in a limited range of @xmath108 : when @xmath108 is either too small or too large , buyers effectively notice only one variant and it is preferable for the vendor to produce only that one . in addition , @xmath54 has a maximum at @xmath109 , which comes from hiding the high quality variant to ignorant buyers while keeping it accessible to informed buyers . ( @xmath90 , @xmath52 , @xmath110 , @xmath111 , @xmath58 , @xmath112 , @xmath113).,title="fig : " ] ( @xmath90 , @xmath52 , @xmath110 , @xmath111 , @xmath58 , @xmath112 , @xmath113).,title="fig : " ] let us now consider the price as a free parameter and investigate how the vendor should fix it optimally . denoting the price by @xmath114 , the profit per item is @xmath115 which means that the maximum quality is @xmath116 . in particular , if the vendor wishes to produce a better product that @xmath117 , the price needs to be increased . the acceptance probability generalizes to @xmath118)$ ] @xmath119 it satisfies two constraints : first , the higher the price , the smaller the acceptance probability . second , because of the @xmath120 term , the sensitivity towards prices decreases as sensitivity to quality increases ; similarly , quality must be judged with respect to price , hence the @xmath121 term . the discussion of the previous sections corresponds to @xmath122 . we restrict our analysis to the simplest case of @xmath19 identical buyers and one product . the expected profit reads @xmath123 with @xmath124 and @xmath125 $ ] , it is maximized by @xmath126 expectedly , the more informed the buyers , the better the products should be , but the vendor can charge a higher price . because @xmath127 is a constant , there is no incentive in this model for exceptionally high prices for high quality variants . in fig . [ fig : prices ] , the resulting optimal profit per buyer @xmath54 is shown together with the optimal profit when the vendor has fixed the price at @xmath128 . the liberty to set the price can increase the profit of the vendor quite considerably . the difference of profit for @xmath129 ( informed buyers ) is due to the fact that the vendor is allowed to charge a higher price for the high quality demanded by the buyers . by contrast , for @xmath130 the main improvement comes from the fact that @xmath131 does not vanish when @xmath132 and @xmath133 . vs acceptance parameter @xmath15 : fixed price ( solid line , the same curve as in fig . [ fig : xopt_k ] ) and variable price ( dashed line ) , @xmath101 . ] due to the complexity of markets and human behaviour , attempts to propose a theory of the whole are illusory . however , simple models can bring insight to elementary mechanisms at work in the real economy . assuming probabilistic buyer behaviour , we formalized buyers abilities , spanning from the zero information to the perfect information limits . adopting the vendor s point of view , we examined the compromise between low quality which minimizes production costs and high quality which maximizes sales . in particular , the fact that customers are heterogeneous forces vendors to diversify their production . in other words , the large variety of products in free - market economies reflects in part the information gathering and processing abilities of customers . in this work we focused on the basic market phenomena but the proposed model is versatile enough to represent more complicated cases . three important extensions seem particularly worth further investigation : including explicitely the price in heterogeneous populations , generalizing the present results to an arbitrary number of consumer groups , and adding more vendors and letting them to compete for customers ( then it can be nash stable that variants with different qualities are provided by different vendors ) . this work was supported by the swiss national science foundation ( project 205120 - 113842 ) and in part by the international scientific cooperation and communication project of sichuan province in china ( grant no . 2008hh0014 ) , the stay of linyuan l in fribourg was supported by sbf ( switzerland ) through project c05.0148 ( physics of risk ) . we acknowledge the early contribution of paolo laureti and the work of our reviewers . in order to better understand the need for both selection and acceptance procedures , it is worthwhile to consider some alternatives . one possibility to simplify our assumptions is to keep only the acceptance process with each displayed variant accepted or not according to the acceptance probability @xmath16 . when @xmath60 variants with the qualities @xmath134 are displayed , the probability @xmath135 that a given customer accepts at least one of them is @xmath136.\ ] ] as @xmath60 increases , @xmath135 converges to one . this means that the vendor can attract the buyers by displaying a large number of very bad products which is generally not the case . however , flooding of customers by low quality occurs under some special circumstances . this _ economics of spamming _ is briefly discussed in the next appendix . another approach is to reduce the model to the best product selection governed by the selection probability @xmath68 . since this probability is normalized to one , when it is applied alone , each buyer surely buys one of the displayed variants and consequently the vendor s profit maximization yields zero quality . obviously , such an optimal solution is pathological . one could eventually consider replacing the unity in the equation @xmath137 by an increasing function of the displayed qualities but this is effectively equivalent to our two step decision process . another possibility is to introduce an artificial non - purchase alternative to eq . ( [ p_s ] ) with an imposed utility as in @xcite which focuses on price differentiation . we see that nor the selection from the available variants is sufficient to model the purchase process . finally , the generalization to diverse proportions of displayed variants , introduced in section [ sec : proportions ] , gives an additional argument . we see that while in the selection step both quality and proportion play their roles , in the final acceptance step it is only quality of the selected variant what matters . thus these two steps are intrinsically different and attempts to merge them are artificial . by the economics of spamming we understand the situation when a low initial cost @xmath24 allows the vendor to produce an abundance of low quality variants . we simplify our considerations to @xmath60 variants with identical qualities @xmath10 and identical buyers with acceptance parameter @xmath15 . assuming that the variants are displayed consecutively , the selection probability plays no role . this situation resembles spam messages arriving into our mailboxes which we reject one by one . according to eq . ( [ paccept ] ) , on average @xmath138^m)$ ] buyers accept one of the displayed variants and the expected vendor profit per buyer is therefore @xmath139^m\big)-mz.\ ] ] since we focus on low quality variants , @xmath140 is small and the approximation @xmath141 $ ] can be used . it follows that for a given quality @xmath10 , the optimal number of displayed variants is @xmath142 however , when buyers perception is limited to a certain number of variants @xmath143 , this value applies instead of @xmath144 . assuming small initial cost @xmath82 , the leading contribution to the optimal quality can be shown to verify @xmath145=q^{\alpha+1}$ ] , where @xmath146 ) . when the resulting @xmath147 , it follows that eq . ( [ mopt ] ) take the simple form @xmath148 ; the optimal quality @xmath38 has to be found numerically . the results are shown in fig . [ fig : spam ] : as @xmath82 gets lower , the optimal quality decreases and the optimal profit increases ; in addition , when acceptance parameters are small ( @xmath149 ) the optimal profit per buyer is almost one which means that nearly all buyers react to the spamming . 99 g. akerlof , _ quarterly journal of economics _ * 84 * , 1970 ( 488500 ) h. s. houthakker , _ the review of economic studies _ * 19 * , 1952 ( 155164 ) c. shapiro , _ the bell journal of economics _ * 13 * , 1982 ( 2035 ) y .- c . zhang , _ physica a _ * 299 * , 2001 ( 104120 ) m. motta , _ the journal of industrial economics _ * 41 * , 1993 ( 113131 ) j. p. johnson , d. p. myatt , _ the american economic review _ * 93 * , 2003 ( 748774 ) p. champsaur , j .- ch . rochet , _ econometrica _ * 57 * , 1989 ( 533557 ) s. berry , j. waldfogel , _ nber working paper no . 9675 _ , 2003 j. sutton , _ sunk costs and market structure _ , mit press , 1991 d. mcfadden , _ journal of business _ * 53 * , 1980 ( 1329 ) i. s. currim , _ journal of marketing research _ * 19 * , 1982 ( 208222 ) s. p. anderson , a. de palma , _ oxford economic papers _ * 44 * , 1992 ( 5167 ) m. medo and y .- c . zhang , _ physica a _ * 387 * , 2008 ( 28892908 ) r. schmalensee , in _ new developments in the analysis of market structure _ ( eds . j. e. stiglitz and g. f. mathewson ) , mit press , 1986 y .- c . zhang , _ physica a _ * 350 * , 2005 ( 500532 ) d. mcfadden , _ frontiers in econometrics _ * 8 * , 1974 ( 105142 ) r. p. mcafee , _ economics : the open - access , open - assessment e - journal _ * 1 * , 2007 - 1 m. mussa , s. rosen , _ journal of economic theory _ * 18 * , 1978 ( 301317 )

we introduce a fully probabilistic framework of consumer product choice based on quality assessment . it allows us to capture many aspects of marketing such as partial information asymmetry , quality differentiation , and product placement in a supermarket . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ @xmath0 department of systems science , school of management , beijing normal university , 100875 beijing , china + @xmath1 lab of information economy and internet research , university of electronic science and technology of china , 610054 chengdu , china + @xmath2 physics department , university of fribourg , prolles , 1700 fribourg , switzerland + @xmath3 institute for scientific interchange , via s. severo 65 , 10113 turin , italy _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

this work aims at providing a framework for reasoning about specifications of deductive systems using _ higher - order abstract syntax _ @xcite . higher - order abstract syntax is a declarative approach to encoding syntax with bindings using church s simply typed @xmath7-calculus . the main idea is to support the notions of @xmath8-equivalence and substitutions in the object syntax by operations in @xmath7-calculus , in particular @xmath8-conversion and @xmath9-reduction . there are at least two approaches to higher - order abstract syntax . the _ functional programming _ approach encodes the object syntax as a data type , where the binding constructs in the object language are mapped to functions in the functional language . in this approach , terms in the object language become values of their corresponding types in the functional language . the _ proof search _ approach encodes object syntax as expressions in a logic whose terms are simply typed , and functions that act on the object terms are defined via relations , i.e. , logic programs . there is a subtle difference between this approach and the former ; in the proof search approach , the simple types are inhabited by well - formed expressions , instead of values as in the functional approach ( i.e. , the abstraction type is inhabited by functions ) . the proof search approach is often referred to as _ @xmath7-tree syntax _ @xcite , to distinguish it from the functional approach . this paper concerns the @xmath7-tree syntax approach . specifications which use @xmath7-tree syntax are often formalized using hypothetical and generic judgments in intuitionistic logic . it is enough to restrict to the fragment of first - order intuitionistic logic whose only formulas are those of hereditary harrop formulas , which we will refer to as the @xmath10 logic . consider for instance the problem of defining the data type for untyped @xmath7-terms . one first introduces the following constants : @xmath11 where the type @xmath12 denotes the syntactic category of @xmath7-terms and @xmath13 and @xmath14 encode application and abstraction , respectively . the property of being a @xmath7-term is then defined via the following theory : @xmath15 @xmath16 where @xmath17 is the universal quantifier and @xmath18 is implication . reasoning about object systems encoded in @xmath10 is reduced to reasoning about the structure of proofs in @xmath10 . mcdowell and miller formalize this kind of reasoning in the logic @xmath19 @xcite , which is an extension of first - order intuitionistic logic with fixed points and natural numbers induction . this is done by encoding the sequent calculus of @xmath10 inside @xmath19 and prove properties about it . we refer to @xmath10 as object logic and @xmath19 as meta logic . mcdowell and miller considered different styles of encodings and concluded that explicit representations of hypotheses and , more importantly , eigenvariables of the object logic are required in order to capture some statements about object logic provability in the meta logic @xcite . one typical example involves the use of hypothetical and generic reasoning as follows : suppose that the following formula is provable in @xmath10 . @xmath20 by inspection on the inference rules of @xmath10 , one observes that this is only possible if @xmath21 and @xmath22 are syntactically equal . this observation comes from the fact that the right introduction rule for universal quantifier , reading the rule bottom - up , introduces new constants , or eigenvariables . the quantified variables @xmath4 and @xmath23 will be replaced by distinct eigenvariables and hence the only matching hypothesis for @xmath24 would be @xmath25 , and therefore @xmath21 and @xmath22 has to be equal . let @xmath26 denote the provability of the formula @xmath27 in @xmath10 . then in the meta logic , we would want to be able to prove the statement : @xmath28 the question is then how we would intrepret the object logic eigenvariables in the meta logic . it is demonstrated in @xcite that the existing quantifiers in @xmath19 can not be used to capture the behaviours of object logic eigenvariables directly . mcdowell and miller then resort to a non - logical encoding technique ( in the sense that no logical connectives are used ) which has some similar flavor to the use of debruijn indices . the use of this encoding technique , however , has a consequence that substitutions in the object logic has to be formalized explicitly . motivated by the above mentioned limitation of @xmath19 , miller and tiu later introduce a new quantifier @xmath2 to @xmath19 which allows one to move the binders from the object logic to the meta logic . a generic judgment in the object logic , for instance @xmath29 is reflected in the meta logic as @xmath30 this meta logic , called @xmath1 @xcite , allows one to perform case analyses on the provability of the object logic . tiu later extended @xmath1 with induction and co - induction rules , resulting in the logic linc @xcite . however , some inductive properties about the object logic are not provable in linc . for example , the fact that @xmath29 implies @xmath31 ( that is , the extensional property of universal quantification ) is not provable in linc . as it is shown in @xcite , this is partly caused by the fact that @xmath32 , where @xmath4 is not free in @xmath5 , is not provable in linc or @xmath1 . in this paper we present the logic @xmath33 , which is an extension of @xmath1 with natural number induction and with the axiom schemes : @xmath34 where @xmath4 is not free in @xmath5 in the second scheme . we show that inductive properties of @xmath7-tree syntax specifications can be stated directly and in a purely logical fashion , and proved in @xmath35 [ [ relation - to - nominal - logic ] ] relation to nominal logic + + + + + + + + + + + + + + + + + + + + + + + + + in formulating the proof system for @xmath33 , it turns out that we can simplify the presentation a lot if we adopt the idea of _ equivariant predicates _ from nominal logic @xcite . that is , provability of a predicate is invariant under permutations of _ names_. this is technically done by introducing a countably infinite set of name constants into the logic , and change the identity rule of the logic to allow equivalence under permutations of name constants : @xmath36 { { \gamma , b\seqsym b ' } } { \pi.b = \pi'.b'}\ ] ] where @xmath37 and @xmath38 are permutations on names . @xmath33 is in fact very close to nominal logic , when we consider only the behaviours of logical connectives . in particular , the quantifier @xmath2 in @xmath33 shares the same properties , in relation to other connectives of the logic , with the @xmath39 quantifier in nominal logic . however , there are two important differences in our approach . first , we do not attempt to redefine @xmath8-conversion and substitutions in @xmath33 in terms of permutations ( or _ swapping _ ) and the notion of _ freshness _ as in nominal logic . name swapping and freshness constraints are not part of the syntax of @xmath35 these notions are present only in the meta theory of the logic . in @xmath33 , for example , variables are always considered to have empty support , that is , @xmath40 for every permutation @xmath37 . this is because we restrict substitutions to the `` closed '' ones , in the sense that no name constants can appear in the substitutions . a restricted form of open substitutions can be recovered indirectly at the meta theory of @xmath33 . the fact that variables have empty support allows one to work with permutation free formulas and terms . so in @xmath33 , we can prove that @xmath41 , where @xmath42 and @xmath43 are names , without using explicit axioms of permutations and freshness . in nominal logic , one would prove this by using the swapping axiom @xmath44 , where @xmath45 denotes a swapping of @xmath42 and @xmath43 , and then show that @xmath46 . the latter might not be valid if @xmath4 is substituted by @xmath42 , for example . the validity of this formula in nominal logic would therefore depend on the assumption on the support of @xmath4 . the second difference between @xmath33 and nominal logic is that @xmath33 allows closed terms ( again , in the sense that no name constants appear in them ) of type name , while in nominal logic , allowing such terms would lead to an inconsistent theory in nominal logic @xcite . as an example , the type @xmath12 in the encoding of @xmath7-terms mentioned previously can be treated as a nominal type in @xmath33 . this has an important consequence that we do not need to redefine the notion of substitutions for the encoded @xmath7-terms . for example , we can define the ( lazy ) evaluation relation on untyped @xmath7-terms as the theory : @xmath47 without having to explicitly define substitutions on terms of type @xmath12 inside @xmath35 substitutions in the object language in this case is modelled by @xmath9-reduction in the meta - language of @xmath35 [ [ outline - of - the - paper ] ] outline of the paper + + + + + + + + + + + + + + + + + + + + section [ sec : lgn ] introduces the logic @xmath48 , which is an extension of first order intuitionistic logic with a notion of name permutation and the @xmath2-quantifier . @xmath48 serves as the core logic for a more expressive logic , @xmath33 , which is obtained by adding rules for fixed points , equality and induction to @xmath49 section [ sec : drv ] examines several properties of derivations , in particular , those that concern preservation of provability under several operations on sequents , e.g. , substitutions . section [ sec : reduct ] defines the cut reduction , used in the cut - elimination proof . the cut elimination proof itself is an adaptation of the cut - elimination proof of @xmath19 by mcdowell and miller @xcite , which makes use of the reducibility technique . section [ sec : norm ] defines the normalizability and the reducibility relations which are crucial to the cut elimination proof in section [ sec : cut - elim ] . finally , in section [ sec : corr ] , we show that the proof system @xmath48 is actually equivalent to @xmath1 ( without fixed points and equality ) with non - logical rules corresponding to the axioms given in ( [ eq1 ] ) above . this paper contains the technical proofs for the results stated in @xcite ; readers are encouraged to consult @xcite for motivations and examples for @xmath6 we first define the core fragment of the logic @xmath33 which does not have fixed point rules or induction . the starting point is the logic @xmath50 introduced in @xcite . @xmath50 is an extension of a subset of church s simple theory of types in which formulas are given the type @xmath51 . the core fragment of @xmath33 , which we refer to as @xmath48 , shares the same set of connectives as @xmath50 , namely , @xmath52 , @xmath53 , @xmath54 , @xmath55 , @xmath56 , @xmath57 , @xmath58 and @xmath59 the type @xmath60 in the quantifiers is restricted to that which does not contain the type @xmath61 hence the logic is essentially first - order . we abbreviate @xmath62 as @xmath63 the sequents of @xmath1 are expressions of the form @xmath64 where @xmath65 is a signature , i.e. , a set of eigenvariables scoped over the sequent and @xmath66 is a local signature , i.e. , list of variables locally scoped over @xmath67 . the introduction rules for @xmath2 , reading the rules bottom - up , introduce new local variables to the local signatures , just as the right introduction rule of @xmath68 introduces new eigenvariables to the signature . the expression @xmath69 is called a local judgment , and is identified up to renaming of variables in @xmath66 . this enforces a limited notion of equivariance : for example @xmath70 is provable , since both local judgments are equivalent up to renaming of local signatures . however , the judgments @xmath71 and @xmath72 are considered distinct judgments , and so are @xmath73 and @xmath74 . these restrictions are relaxed in @xmath49 the sequent presentation of @xmath48 can be simplified , that is , without using the local signatures , if we employ the equivariance principle . for this purpose , we introduce a distinguished set of base types , called _ nominal types _ , which is denoted with @xmath75 . nominal types are ranged over by @xmath76 . we restrict the @xmath2 quantifier to nominal types . for each nominal type @xmath77 , we assume an infinite number of constants of that type . these constants are called _ nominal constants_. we denote the family of nominal constants by @xmath78 the role of the nominal constants is to enforce the notion of equivariance : provability of formulas is invariant under permutations of nominal constans . depending on the application , we might also assume a set of non - nominal constants , which is denoted by @xmath79 we assume the usual notion of capture - avoiding substitutions . substitutions are ranged over by @xmath80 and @xmath81 . application of substitutions is written in a postfix notation , e.g. , @xmath82 is an application of @xmath80 to the term @xmath22 . given two substitutions @xmath80 and @xmath83 , we denote their composition by @xmath84 which is defined as @xmath85 a _ signature _ is a set of variables . a substitution @xmath80 respects a given signature @xmath65 if there exists a set of typed variables @xmath86 such that for every @xmath87 in the domain of @xmath80 , it holds that @xmath88 we denote by @xmath89 the minimal set of variables satisfying the above condition . we assume that variables , free or bound , are of a different syntactic category from constants . a permutation on @xmath90 is a bijection from @xmath90 to @xmath90 . the permutations on @xmath90 are ranged over by @xmath37 . application of a permutation @xmath37 to a nominal constant @xmath42 is denoted with @xmath91 . we shall be concerned only with permutations which respect types , i.e. , for every @xmath92 , @xmath93 further , we shall also restrict to permutations which are finite , that is , the set @xmath94 is finite . application of a permutation to an arbitrary term ( or formula ) , written @xmath95 , is defined as follows : @xmath96 a permutation involving only two nominal constants is called _ swapping_. we use @xmath45 , where @xmath42 and @xmath43 are constants of the same type , to denote the swapping @xmath97 the _ support _ of a term ( or formula ) @xmath22 , written @xmath98 , is the set of nominal constants appearing in it . it is clear from the above definition that if @xmath98 is empty , then @xmath99 for all @xmath37 . the definition of @xmath65-substitution implies that for every @xmath80 and for every @xmath100 , @xmath101 has empty support . therefore @xmath65-substitutions and permutations commute , that is , @xmath102 a sequent in @xmath33 is an expression of the form @xmath103 where @xmath65 is a signature . the free variables of @xmath104 and @xmath105 are among the variables in @xmath65 . the inference rules for the core fragment of @xmath33 , i.e. , the logic @xmath48 , is given in figure [ fig : lg ] . in the rules , the typing judgment @xmath106 denotes the typability of @xmath107 , given the typing context @xmath108 in church s simple type system . in the @xmath109 and @xmath110 rules , @xmath42 denotes a nominal constant . in the @xmath111 and @xmath112 rules , we use _ raising _ @xcite to encode the dependency of the quantified variable on the support of @xmath5 , since we do not allow @xmath65-substitutions to mention any nominal constants . in the rules , the variable @xmath113 has its type raised in the following way : suppose @xmath114 is the list @xmath115 and the quantified variable @xmath4 is of type @xmath60 . then the variable @xmath113 is of type : @xmath116 this raising technique is similar to that of @xmath117 and is used to encode explicitly the minimal support of the quantified variable . its use prevents one from mixing the scopes of @xmath68 ( dually , @xmath118 ) and @xmath2 . that is , it prevents the formula @xmath119 , and its dual , to be proved . looking at the introduction rules for @xmath68 and @xmath118 , one might notice the asymmetry between the left and the right introduction rules . the left rule for @xmath68 allows instantiations with terms containing any nominal constants while the raised variable in the right introduction rule of @xmath68 takes into account only those which are in the support of the quantified formula . however , we will see that we can extend the dependency of the raised variable to an arbitrary number of fresh nominal constants not in the support without affecting the provability of the sequent ( see lemma [ lm : supp1 ] and lemma [ lm : supp2 ] ) . @xmath120 { { \sigma ; \gamma , b \seqsym b ' } } { \pi.b = \pi'.b ' } \qquad \infer[{mc } ] { { \sigma ; \delta_1,\ldots , \delta_n,\gamma \seqsym c } } { { \sigma ; \delta_1 \seqsym b_1 } & \cdots & { \sigma ; \delta_n \seqsym b_n } & { \sigma ; b_1,\dots , b_n,\gamma \seqsym c } } \qquad \infer[{\hbox{\sl c}{\cal l } } ] { { \sigma ; \gamma , b \seqsym c } } { { \sigma ; \gamma , b , b \seqsym c}}\ ] ] @xmath121 { { \sigma ; \gamma , \bot \seqsym c } } { } \qquad \infer[{\top{\cal r } } ] { { \sigma ; \gamma \seqsym \top } } { } \ ] ] @xmath122 { { \sigma ; \gamma , b_1 \land b_2 \seqsym c } } { { \sigma ; \gamma , b_i \seqsym c } } \qquad \infer[{\land{\cal r } } ] { { \sigma ; \gamma \seqsym b \land c } } { { \sigma ; \gamma \seqsym b } & { \sigma ; \gamma \seqsym c}}\ ] ] @xmath123 { { \sigma ; \gamma , b \lor d \seqsym c } } { { \sigma ; \gamma , b \seqsym c } & { \sigma ; \gamma , d \seqsym c } } \qquad \infer[{\lor{\cal r } } , i \in \{1,2\ } ] { { \sigma ; \gamma \seqsym b_1 \lor b_2 } } { { \sigma ; \gamma \seqsym b_i}}\ ] ] @xmath124 { { \sigma ; \gamma , b { \supset}d \seqsym c } } { { \sigma ; \gamma \seqsym b } & { \sigma ; \gamma , d \seqsym c } } \qquad \infer[{{\supset}{\cal r } } ] { { \sigma ; \gamma \seqsym b { \supset}c } } { { \sigma ; \gamma , b \seqsym c}}\ ] ] @xmath125 { { \sigma ; \gamma , \forall_\tau x. b \seqsym c } } { { \sigma , { \cal k } , { \cal c}_{\cal n } \vdash t : \tau } & { \sigma ; \gamma , b[t / x ] \seqsym c } } \qquad \infer[{\forall{\cal r } } , h \not \in \sigma , supp(b ) = \{\vec c\ } ] { { \sigma ; \gamma \seqsym \forall x.b } } { { \sigma , h ; \gamma \seqsym b[h\ , \vec c / x ] } } \ ] ] @xmath126 { { \sigma ; \gamma , \nabla x.b \seqsym c } } { { \sigma ; \gamma , b[a / x ] \seqsym c } } \qquad \infer[{\nabla{\cal r } } , a \not \in supp(b ) ] { { \sigma ; \gamma \seqsym \nabla x.b } } { { \sigma ; \gamma \seqsym b[a / x]}}\ ] ] @xmath127 { { \sigma ; \gamma , \exists x.b \seqsym c } } { { \sigma , h ; \gamma , b[h\,\vec c / x ] \seqsym c } } \qquad \infer[{\exists{\cal r } } ] { { \sigma ; \gamma \seqsym \exists_\tau x.b } } { { \sigma,{\cal k},{\cal c}_{\cal n } \vdash t : \tau } & { \sigma ; \gamma \seqsym b[t / x]}}\ ] ] we now extend the logic @xmath48 with a proof theoretic notion of equality and fixed points , following on works by hallnas and schroeder - heister @xcite , girard @xcite and mcdowell and miller @xcite . the equality rules are as follows : @xmath128 { { \sigma ; \gamma , s = t \seqsym c } } { \{{\sigma\theta ; \gamma\theta \seqsym c\theta } ~ \mid ~ ( \lambda \vec c.t)\theta = _ { \beta\eta } ( \lambda \vec c.s)\theta \ } } \qquad \infer[{{\rm eq}{\cal r } } ] { { \sigma ; \gamma \seqsym t = t } } { } \ ] ] where @xmath129 in the @xmath130 rule . in the @xmath130 rule , the substitution @xmath80 is a _ unifier _ of @xmath131 and @xmath132 . we specify the premise of the rule as a set to mean that every element of the set is a premise . since the terms @xmath21 and @xmath22 can be arbitrary higher - order terms , in general the set of their unifiers can be infinite . however , in some restricted cases , e.g. , when @xmath131 and @xmath132 are _ higher - order pattern _ terms @xcite , if both terms are unifiable , then there exists a most general unifier . the applications we are considering are those which satisfy the higher - order pattern restrictions . to each atomic formula , we associate a fixed point equation , or a _ definition clause _ , following the terminology of @xmath1 . a definition clause is written @xmath133 where the free variables of @xmath5 are among @xmath134 the predicate @xmath135 is called the _ head _ of the definition clause , and @xmath5 is called the _ body_. a _ definition _ is a set of definition clauses . we often omit the outer quantifiers when referring to a definition clause . the introduction rules for defined atoms are as follows : @xmath136 { { \sigma ; \gamma , p\ , \vec t \seqsym c } } { { \sigma ; \gamma , b[\vec t/\vec x ] \seqsym c } } \qquad \infer[{\hbox{\sl def\/}{\cal r } } , p\,\vec x { \stackrel{{\scriptstyle\triangle}}{=}}b ] { { \sigma ; \gamma \seqsym p\ , \vec t } } { { \sigma ; \gamma \seqsym b[\vec t/\vec x]}}\ ] ] in order to prove the cut - elimination theorem and the consistency of @xmath33 , we allow only definition clauses which satisfy an _ equivariance preserving _ condition and a certain positivity condition , so as to guarantee the existence of fixed points . [ def : level ] we associate with each predicate symbol @xmath137 a natural number , the _ level _ of @xmath137 . given a formula @xmath5 , its _ level _ @xmath138 is defined as follows : 1 . @xmath139 2 . @xmath140 3 . @xmath141 4 . @xmath142 5 . @xmath143 . a definition clause @xmath144 is stratified if @xmath145 and @xmath5 has no free occurrences of nominal constants . we consider only definition clauses which are stratified . an example that violates the first restriction in definition [ def : level ] is the definition @xmath146 in @xcite , schroeder - heister shows that admitting this definition in a logic with contraction leads to inconsistency . to see why we need the second restriction on name constants , consider the definition @xmath147 where @xmath42 is a nominal constant . let @xmath43 be a nominal constant different from @xmath42 . using this definition , we would be able to derive @xmath52 : @xmath148 { { \seqsym \bot } } { \infer[{\hbox{\sl def\/}{\cal r } } ] { { \seqsym q\,a } } { \infer[{{\rm eq}{\cal r } } ] { { \seqsym a = a } } { } } & \infer[cut ] { { q\,a\seqsym \bot } } { \infer[id_\pi ] { { q\,a\seqsym q\,b } } { } & \infer[{\hbox{\sl def}{\cal l } } ] { { q\,b\seqsym \bot } } { \infer[{{\rm eq}{\cal l } } ] { { b = a\seqsym \bot } } { } } } } \ ] ] in examples and applications , we often express definition clauses with patterns in the heads . let us consider , for example , a definition clause for lists . we first introduce a type @xmath149 to denote lists of elements of type @xmath8 , and the constants @xmath150 which denote the empty list and a constructor to build a list from an element of type @xmath8 and another list . the latter will be written in the infix notation . the definition clause for _ lists _ is as follows . @xmath151 using patterns , the above definition of lists can be rewritten as @xmath152 we shall often work directly with this patterned notation for definition clauses . for this purpose , we introduce the notion of _ patterned definitions_. a _ patterned definition clause _ is written @xmath153 where the free variables of @xmath154 and @xmath5 are among @xmath134 the stratification of definitions in definition [ def : level ] applies to patterned definitions as well . since the patterned definition clauses are not allowed to have free occurrences of nominal constants , in matching the heads of the clauses with an atomic formula in a sequent , we need to raise the variables of the clauses to account for nominal constants that are in the support of the introduced formula . given a patterned definition clause @xmath155 its raised clause with respect to the list of constants @xmath156 is @xmath157 { \stackrel{{\scriptstyle\triangle}}{=}}b[h_1~\vec c / x_1 , \ldots , h_n~\vec c / x_n].\ ] ] the introduction rules for patterned definitions are @xmath158 { { \sigma ; a , \gamma \seqsym c } } { \{{\sigma\theta ; b\theta , \gamma\theta \seqsym c\theta}\}_\theta } \qquad \infer[{\hbox{\sl def\/}{\cal r } } ] { { \sigma ; \gamma \seqsym a } } { { \sigma ; \gamma \seqsym b\theta}}\ ] ] in the @xmath159 rule , @xmath5 is the body of the raised patterned clause @xmath160 and @xmath161 where @xmath162 is the support of @xmath163 in the @xmath164 rule , we match @xmath165 with the head of the clause , i.e. , @xmath166 these patterned rules can be derived using the non - patterned definition rules and the equality rules , as shown in @xcite , [ [ natural - number - induction . ] ] natural number induction . + + + + + + + + + + + + + + + + + + + + + + + + + we introduce a type @xmath167 to denote natural numbers , with the usual constants @xmath168 ( zero ) and @xmath169 ( the successor function ) , and a special predicate @xmath170 the rules for natural number induction are the same as those in @xmath19 @xcite , which are the introduction rules for the predicate @xmath171 . @xmath172 { { \sigma ; \gamma , nat\,i \seqsym c } } { { \seqsym d\,z } & { j ; d\,j \seqsym d\,(s\,j ) } & { \sigma ; \gamma , d\,i \seqsym c}}\ ] ] @xmath173 { { \sigma ; \gamma \seqsym nat~z } } { } \qquad \infer[nat{\cal r } ] { { \sigma ; \gamma \seqsym nat\,(s\,i ) } } { { \sigma ; \gamma \seqsym nat\,i}}\ ] ] the logic @xmath48 extended with the equality , definitions and induction rules is referred to as @xmath35 in this section we examine several properties of the @xmath2-quantifier and derivations in @xmath33 that are useful in the cut elimination proof . these properties concern the transformation of derivations , in particular , they state that provability is preserved under @xmath65-substitutions , permutations and a restricted form of name substitutions . we first look at the properties of the @xmath2 quantifier in relation to other connectives . the proof of the following proposition is straightforward by inspection on the rules of @xmath49 the following formulas are provable in @xmath48 : 1 . @xmath174 2 . @xmath175 3 . @xmath176 4 . @xmath177 , provided that @xmath4 is not free in @xmath5 . @xmath178 6 . @xmath179 7 . @xmath180 the formulas ( 1 ) ( 3 ) are provable in @xmath50 . the proposition is true also in nominal logic with @xmath2 replaced by @xmath181 [ def : ht ] given a derivation @xmath182 with premise derivations @xmath183 where @xmath184 is some index set , the measure @xmath185 , the _ height _ of @xmath182 , is defined as the least upper bound of @xmath186 we now define some transformations of derivations : weakening of hypotheses , substitutions on derivations , permutations and restricted name substitutions . in the following definitions we omit the signatures in the sequents if it is clear from context which signatures we refer to . we denote with @xmath187 the identity function on @xmath90 . [ def : weak ] _ weakening of hypotheses . _ let @xmath182 be a derivation of @xmath188 let @xmath189 be a multiset of formulas whose free variables are among @xmath65 . we define the derivation @xmath190 of @xmath191 as follows : 1 . if @xmath182 ends with @xmath130 @xmath128 { { \sigma ; s = t , \gamma ' \seqsym c } } { \left\ { \raisebox{-1.5ex}{\deduce{{\sigma\theta ; \gamma'\theta \seqsym c\theta}}{\pi_\theta } } \right\}_\theta } \ ] ] then @xmath192 is @xmath128 { { \sigma ; s = t , \gamma ' , \delta \seqsym c } } { \left\ { \raisebox{-1.5ex}{\deduce{{\sigma\theta ; \gamma'\theta , \delta\theta \seqsym c\theta } } { w(\delta\theta,\pi_\theta ) } } \right\}_\theta } \ ] ] 2 . if @xmath182 ends with @xmath193 @xmath194 { { nat~i , \gamma'\seqsym c } } { \deduce{{\seqsym d~z } } { \pi_1 } & \deduce{{d~i\seqsym d~(s~i)}}{\pi_2 } & \deduce{{d~i , \gamma'\seqsym c}}{\pi_3}}\ ] ] then @xmath192 is @xmath194 { { nat~i , \gamma ' , \delta\seqsym c } } { \deduce{{\seqsym d~z } } { \pi_1 } & \deduce{{d~i\seqsym d~(s~i)}}{\pi_2 } & \deduce{{d~i , \gamma ' , \delta\seqsym c}}{w(\delta,\pi_3)}}\ ] ] 3 . if @xmath182 ends with the @xmath195 rule @xmath196 { { \delta_1,\ldots,\delta_n , \gamma'\seqsym c } } { \deduce{{\delta_1\seqsym b_1}}{\pi_1 } & \ldots & \deduce{{\delta_n\seqsym b_n}}{\pi_n } & \deduce{{b_1,\dots , b_n,\gamma'\seqsym c}}{\pi ' } } \ ] ] then @xmath190 is @xmath196 { { \delta_1,\ldots,\delta_n , \gamma ' , \delta\seqsym c } } { \deduce{{\delta_1\seqsym b_1}}{\pi_1 } & \ldots & \deduce{{\delta_n\seqsym b_n}}{\pi_n } & \deduce{{b_1,\dots , b_n,\gamma ' , \delta\seqsym c}}{w(\delta,\pi ' ) } } \ ] ] 4 . if @xmath182 ends with any other rule and has premise derivations @xmath197 then @xmath190 ends with the same rule with premise derivations @xmath198 @xmath199 @xmath200 [ def : subst ] _ substitutions on derivations . _ if @xmath182 is a derivation of @xmath201 and @xmath80 is a @xmath65-substitution , then we define the derivation @xmath202 of @xmath203 as follows : 1 . suppose @xmath182 ends with @xmath130 : @xmath128 { { \sigma ; s = t , \gamma ' \seqsym c } } { \left\ { \raisebox{-1.5ex}{\deduce{{\sigma\rho ; \gamma'\rho \seqsym c\rho}}{\pi_\rho } } \right\}_\rho } \ ] ] where each @xmath81 is a unifier of @xmath131 and @xmath132 . observe that if @xmath204 is a unifier of @xmath205 and @xmath206 , then @xmath207 is a unifier of @xmath131 and @xmath132 . thus @xmath202 is the derivation : @xmath128 { { \sigma ; s\theta = t\theta , \delta\theta \seqsym c\theta } } { \left\ { \raisebox{-1.5ex}{\deduce{{\sigma\theta\rho ' ; \delta\theta\rho \seqsym c\theta\rho } } { \pi_{\theta\circ \rho ' } } } \right\}_{\rho ' } } \ ] ] 2 . suppose @xmath182 ends with @xmath112 : @xmath208 { { \sigma ; \gamma \seqsym \forall x.b } } { \deduce{{\sigma ; \gamma \seqsym b[h\,\vec c / x]}}{\pi_1 } } \enspace , \ ] ] where @xmath209 let @xmath210 be the support of @xmath211 , which might be smaller than @xmath212 let @xmath81 be the substitution @xmath213 $ ] where @xmath214 is a new variable not already in @xmath65 and not among the free variables in @xmath215 we can assume without loss of generality that @xmath4 is not free in @xmath80 , hence @xmath216)\rho ) \theta = ( b[h'\,\vec d / x])\theta = ( b\theta)[h'\,\vec d / x].$ ] then @xmath202 is @xmath208 { { \sigma\theta ; \gamma\theta \seqsym ( \forall x.b)\theta } } { \deduce{{\sigma\theta , h ' ; \gamma\theta \seqsym ( b\theta)[h'\,\vec d / x]}}{\pi_1(\rho \circ \theta)}}\ ] ] 3 . suppose @xmath182 ends with @xmath111 : this case is dual to the previous one . 4 . if @xmath182 ends with any other rule and has premise derivations @xmath197 , then @xmath202 ends with the same rule and has premise derivations @xmath217,@xmath199 @xmath218 [ def : perm ] let @xmath182 be a proof of @xmath219 and let @xmath220 be a list of permutations . we define a derivation @xmath221 of @xmath222 as follows : 1 . suppose that @xmath182 ends with @xmath223 @xmath120 { { \sigma ; b_1 , \ldots , b_n \seqsym b_0 } } { \pi.b_j = \pi'.b_0 } \enspace .\ ] ] obverse that @xmath224 hence @xmath221 ends with the same rule . suppose @xmath182 ends with @xmath195 : @xmath196 { { b_1,\ldots , b_n\seqsym b_0 } } { \deduce{{\delta_1\seqsym d_1}}{\pi_1 } & \ldots & \deduce{{\delta_m\seqsym d_m}}{\pi_m } & \deduce{{d_1,\ldots , d_m,\delta_{m+1}\seqsym b_0}}{\pi ' } } \ ] ] where @xmath225 are partitions of @xmath226 suppose that for each @xmath227 , @xmath228 for some index @xmath229 let @xmath230 , for @xmath231 , be the permutations @xmath232 let @xmath233 be the permutations @xmath234 + we denote with @xmath235 the list @xmath236 then @xmath237 is the derivation @xmath196 { { \pi_1.b_1,\ldots,\pi_n.b_n\seqsym \pi_0.b_0 } } { \deduce{{\delta_1'\seqsym d_1}}{\langle \vec \pi(1)\rangle.\pi_1 } & \ldots & \deduce{{\delta_m'\seqsym d_m}}{\langle \vec \pi(m)\rangle.\pi_m } & \deduce{{d_1,\ldots , d_m,\delta_{m+1}'\seqsym \pi_0.b_0}}{\langle \vec \pi(m+1)\rangle.\pi ' } } \ ] ] 3 . suppose @xmath182 ends with @xmath110 : @xmath238 { { \sigma ; b_1,\ldots , b_n \seqsym \nabla_\iota x.b } } { \deduce{{\sigma ; b_1,\ldots , b_n \seqsym b[a / x]}}{\pi_1}}\ ] ] where @xmath239 let @xmath240 be a nominal constant such that @xmath241 and @xmath242 . such a constant exists since @xmath243 is finite and @xmath244 is a finite permutation . thus @xmath245 = \pi_0.b_0[d / x].$ ] then @xmath221 is the derivation : @xmath238 { { \sigma ; \pi_1.b_1 , \ldots , \pi_n.b_n \seqsym \pi_0.(\nabla x.b ) } } { \deduce{{\sigma ; \pi_1.b_1 , \ldots , \pi_n.b_n \seqsym \pi_0.b[d / x ] } } { \langle\pi_0.(a~d ) , \dots,\pi_n\rangle.\pi_1}}\ ] ] 4 . suppose @xmath182 ends with @xmath109 : this case is analogous to previous one . 5 . suppose @xmath182 ends with @xmath246 : @xmath247 { { b_1,\ldots , b_j,\ldots , b_n\seqsym b_0 } } { \deduce { { b_1,\ldots , b_j , b_j\ldots , b_n\seqsym b_0 } } { \pi ' } } \ ] ] then @xmath221 is @xmath247 { { \pi_1.b_1,\ldots,\pi_{j}.b_j,\ldots,\pi_n.b_n\seqsym \pi_0.b_0 } } { \deduce { { \pi_1.b_1,\ldots,\pi_{j}.b_j,\pi_j.b_j\ldots,\pi_n.b_n\seqsym \pi_0.b_0 } } { \langle \pi_1,\ldots,\pi_j,\pi_j,\ldots,\pi_n\rangle.\pi ' } } \ ] ] 6 . if @xmath182 ends with any other rule and has premise derivations @xmath248 , then @xmath237 ends with the same rule and has premise derivations @xmath249 @xmath199 @xmath250 [ def : res ] let @xmath182 be a proof of @xmath251 and let @xmath252 be a list of nominal constants such that @xmath253 we define a derivation @xmath254 of @xmath255 , \ldots , b_n[a_n / x ] \seqsym b_0[a_0/x]},$ ] as follows : 1 . suppose @xmath182 is @xmath120 { { \sigma , x ; b_1,\ldots , b_n \seqsym b_0 } } { \pi.b_j = \pi'.b_0 } \enspace .\ ] ] let @xmath240 be a nominal constant which is not in the support of @xmath256 and @xmath257 , and @xmath258 and @xmath259 . then @xmath260 is @xmath120 { { \sigma ; b_1[a_1/x ] , \ldots , b_n[a_n / x ] \seqsym b_0[a_0/x ] } } { \pi.(a_j~d).b_1[a_1/x ] = \pi'.(a_0~d).b_0[a_0/x]}\ ] ] 2 . suppose @xmath182 ends with @xmath195 : @xmath196 { { \sigma , x ; b_1,\ldots , b_n \seqsym b_0 } } { \deduce{{\sigma , x ; \delta_1 \seqsym d_1}}{\pi_1 } & \ldots & \deduce{{\sigma , x ; \delta_m \seqsym d_m}}{\pi_m } & \deduce{{\sigma , x ; d_1,\ldots , d_m,\delta_{m+1 } \seqsym b_0}}{\pi ' } } \ ] ] where @xmath225 is a partition of @xmath226 suppose that for each @xmath227 , @xmath228 for some index @xmath229 let @xmath261 be a list of nominal constants such that @xmath262 let @xmath263 , for @xmath231 be the list @xmath264 and let @xmath265 be the list @xmath266 let @xmath235 be the list @xmath267,\ldots , b_{ik_i}[a_{ik_i}/x]\ ] ] and let @xmath104 be the list @xmath268,\ldots , d_{m}[d_m / x ] , \delta_{m+1}'.\ ] ] then @xmath269 is the derivation @xmath196 { { \sigma ; b_1[a_1/x],\ldots , b_n[a_n / x ] \seqsym b_0[a_0/x ] } } { \deduce{{\sigma ; \delta_1 ' \seqsym d_1[d_1/x]}}{r(x , f(1),\pi_1 ) } & \ldots & \deduce{{\sigma ; \delta_m ' \seqsym d_m[a_m / x]}}{r(x , f(m),\pi_m ) } & \deduce{{\sigma ; \gamma \seqsym b_0[a_0/x]}}{r(x , f(m+1),\pi ' ) } } \ ] ] 3 . suppose @xmath182 is @xmath238 { { \sigma , x ; b_1 , \ldots , b_n \seqsym \nabla y.b } } { \deduce{{\sigma , x ; b_1,\ldots , b_n \seqsym b[c / y]}}{\pi_1 } } \enspace .\ ] ] if @xmath270 then @xmath269 is @xmath238 { { \sigma , x ; b_1 , \ldots , b_n \seqsym \nabla y.b } } { \deduce{{\sigma , x ; b_1,\ldots , b_n \seqsym b[c / y]}}{r(x,\vec a , \pi_1 ) } } \enspace .\ ] ] if @xmath271 , then we swap @xmath272 with a fresh constant . let @xmath240 be a nominal constant not in the support of @xmath273 $ ] . we apply the swapping @xmath274 to the conclusion of the end sequent of @xmath275 according to the construction in definition [ def : perm ] to get a proof @xmath276 of @xmath277}.$ ] the derivation @xmath260 is constructed as follows : @xmath238 { { \sigma ; b_1[a_1/x],\ldots , b_n[a_n / x ] \seqsym \nabla y.b[a_0/x ] } } { \deduce{{\sigma ; b_1[a_1/x],\ldots , b_n[a_n / x ] \seqsym b[a_0/x , d / y]}}{r(x,\vec a,\pi_2)}}\ ] ] 4 . if @xmath182 ends with @xmath109 apply the same construction as in the previous case . suppose @xmath182 ends with @xmath112 @xmath208 { { \sigma , x ; b_1,\ldots , b_n \seqsym \forall y.b } } { \deduce{{\sigma , x , h ; b_1,\ldots , b_n \seqsym b[h\,\vec c / y]}}{\pi_1 } } \enspace .\ ] ] let @xmath278 $ ] where @xmath214 is a variable not in @xmath279 apply the construction in definition [ def : subst ] to get the proof @xmath202 of @xmath280}\ ] ] then @xmath269 is @xmath208 { { \sigma ; b_1[a_1/x],\ldots , b_n[a_n / x ] \seqsym \forall y.b[a_0/x ] } } { \deduce{{\sigma , h ' ; b_1[a_1/x],\ldots , b_n[a_n / x ] \seqsym b[a_0/x , ( h'\,\vec c a_0)/y]}}{r(x,\vec a , \pi\theta ) } } \enspace .\ ] ] 6 . if @xmath182 ends with @xmath111 , apply the same construction as in the previous case . suppose @xmath182 ends with @xmath281 : @xmath282 { { \sigma , x ; b_1,\ldots , b_n \seqsym \exists y.b } } { \deduce{{\sigma , x ; b_1,\ldots , b_n \seqsym b[t / y]}}{\pi_1 } } \enspace .\ ] ] if @xmath283)$ ] then @xmath269 is @xmath282 { { \sigma ; b_1[a_1/x ] , \ldots , b_n[a_n / x ] \seqsym \exists y.b[a_0/x ] } } { \deduce{{\sigma ; b_1[a_1/x ] , \ldots , b_n[a_n / x ] \seqsym b[a_0/x , t / y]}}{r(x,\vec a,\pi_1 ) } } \enspace .\ ] ] if @xmath284 $ ] , we exchange it with a fresh constant . let @xmath285 be a nominal constant distinct from @xmath286 and not in the support of @xmath287.$ ] then @xmath288)[a_0/x ] = b[(a_0~d).t / y , a_0/x].$ ] we first apply the construction in definition [ def : perm ] to @xmath275 to get a derivation @xmath276 of @xmath289}.$ ] the derivation @xmath269 is thus @xmath282 { { \sigma ; b_1[a_1/x],\ldots , b[a_n / x ] \seqsym \exists y.b[a_0/x ] } } { \deduce{{\sigma ; b_1[a_1/x ] , \ldots , b_n[a_n / x ] \seqsym b[(a_0~d).t / y , a_0/x]}}{r(x,\vec a,\pi_2 ) } } \enspace .\ ] ] 8 . suppose @xmath182 ends with @xmath130 : @xmath128 { { \sigma , x ; s = t , b_2 , \ldots , b_n \seqsym b_0 } } { \left\ { \raisebox{-1.5ex } { \deduce{{(\sigma , x)\theta ; b_2\theta,\ldots , b_n\theta \seqsym b_0\theta}}{\pi_\theta } } \right\}_\theta } \ ] ] where each @xmath80 is a unifier of @xmath290 and @xmath291 we need to show that for each unifier of @xmath292 , \lambda a_1\lambda \vec c.t[a_1/x])$ ] there is a corresponding unifier for @xmath131 and @xmath293 we can assume without loss of generality that @xmath4 is not in the domain of @xmath81 . + we first show the case where @xmath4 is not free in @xmath81 . it is clear that in this case @xmath81 is a unifier of @xmath131 and @xmath132 . therefore we apply the procedure recursively to the premise derivation @xmath294 , to get the derivation @xmath295 of @xmath296)\rho , \ldots , ( b_n[a_n / x])\rho \seqsym ( b_0[a_0/x])\rho}.\ ] ] in the other case , where @xmath4 is free in the range of @xmath81 , we show that it can be reduced to the previous case . first we define a substitution @xmath204 to be the substitution @xmath81 where @xmath4 is replaced by a new variable @xmath297 which is not free in @xmath81 . clearly @xmath204 is also a unifier of @xmath298 $ ] and @xmath299.$ ] moreover , it is more general than @xmath81 , since @xmath300 \circ \rho'.$ ] therefore we can apply the construction in the previous case to get a derivation @xmath301 and apply the substitution @xmath302 $ ] to to this derivation , using the procedure in definition [ def : subst ] , to get a derivation of @xmath296)\rho , \ldots , ( b_n[a_n / x])\rho \seqsym ( b_0[a_0/x])\rho.}\ ] ] the derivation @xmath260 is then constructed as follows @xmath128 { { \sigma ; s[a_1/x ] = t[a_1/x ] , \ldots , b_n[a_n / x ] \seqsym b_0[a_0/x ] } } { \left\ { \raisebox{-1.5ex } { \deduce{{\sigma\rho ; ( b_2[a_2/x])\rho , \ldots , ( b_n[a_n / x])\rho \seqsym ( b_0[a_0/x])\rho } } { \pi_\rho ' } } \right\}_{\rho } } \ ] ] where each @xmath303 is constructed as explained above . if @xmath182 ends with @xmath246 : @xmath247 { { b_1,\ldots , b_j , \ldots , b_n\seqsym b_0 } } { \deduce { { b_1,\ldots , b_j , b_j , \ldots , b_n\seqsym b_0 } } { \pi ' } } \ ] ] then @xmath304 is @xmath247 { { b_1[a_1/x],\ldots , b_j[a_j / x ] , \ldots , b_n[a_n / x]\seqsym b_0[a_0/x ] } } { \deduce { { b_1[a_1/x],\ldots , b_j[a_j / x ] , b_j[a_j / x ] , \ldots , b_n[a_n / x]\seqsym b_0[a_0/x ] } } { r(x , ( a_0,\dots , a_j , a_j,\dots , a_n ) , \pi ' ) } } \ ] ] 10 . if @xmath182 ends with any other rule and has premise derivations @xmath275,@xmath305 , @xmath306 , then @xmath260 ends with the same rule and has premise derivations @xmath307 , @xmath305 , @xmath308 [ lm : weak drv ] for any derivation @xmath182 of @xmath201 and any multiset of @xmath65-formulas @xmath189 , @xmath190 is a derivation of @xmath309 and @xmath310 [ lm : subst drv ] for any derivation @xmath182 of @xmath201 and any @xmath65-substitution @xmath80 , @xmath202 is a derivation of @xmath203 and @xmath311 [ lm : perm drv ] for any derivation @xmath182 of @xmath312 and permutations @xmath220 , @xmath237 is a derivation of @xmath313 and @xmath314 [ lm : res drv ] for any derivation @xmath182 of @xmath315 and any list of nominal constants @xmath316 such that @xmath317 @xmath260 is a derivation of @xmath255,\ldots , b_n[a_n / x ] \seqsym b_0[a_0/x]}$ ] and @xmath318 [ lm : subst]_substitutions . _ let @xmath182 be a proof of @xmath201 and let @xmath80 be a @xmath65-substitution . then there exists a proof @xmath319 of @xmath203 such that @xmath320 follows immediately from lemma [ lm : subst drv ] . [ lm : perm ] _ permutations . _ let @xmath182 be a proof of @xmath321 then there exists a proof @xmath319 of @xmath222 such that @xmath320 follows immediately from lemma [ lm : perm drv ] . [ lm : res]_restricted name substitutions . _ let @xmath182 be a proof of @xmath322 then there exists a proof of @xmath319 of @xmath255 , \ldots , b_n[a_n / x ] \seqsym b_0[a_0/x]},$ ] where @xmath323 for each @xmath324 such that @xmath320 follows immediately from lemma [ lm : res drv ] . the next two lemmas are crucial to the cut - elimination proof : they allow one to reintroduce the symmetry between @xmath325 and @xmath112 , and dually , between @xmath111 and @xmath281 rules . [ lm : supp1]_support extension . _ let @xmath182 be a proof of @xmath326}$ ] where @xmath327 , @xmath328 and @xmath113 is not free in @xmath104 and @xmath5 . let @xmath329 be a list of nominal constants not in the support of @xmath5 . then there exists a proof @xmath319 of @xmath330}$ ] where @xmath331 suppose @xmath114 is the list of constants @xmath332 . let @xmath333 be a list of distinct variables not appearing in @xmath334 . we first apply the substitution @xmath335 $ ] to the sequent @xmath336}.$ ] by lemma [ lm : subst ] , there is a proof @xmath275 of @xmath337}\ ] ] the derivation @xmath319 is then obtained by repeatedly applying lemma [ lm : res ] to @xmath275 to change @xmath338 into @xmath114 . [ lm : supp2]_support extension . _ let @xmath182 be a proof of @xmath339 , \gamma \seqsym c}$ ] where @xmath327 , @xmath328 and @xmath113 is not free in @xmath104 , @xmath5 and @xmath105 . let @xmath329 be a list of nominal constants not in the support of @xmath5 . then there exists a proof @xmath319 of @xmath340 , \gamma \seqsym c}$ ] where @xmath331 use the same construction as in the proof of lemma [ lm : supp1 ] . we define a _ reduction _ relation between derivations , following closely the reduction relation in @xcite . for simplicity of presentation , we shall omit the signatures in the sequents in the following reduction of cuts when the signatures are not changed by the reduction or when it is clear from context which signatures should be assigned to the sequents . the redex is always a derivation @xmath341 ending with the multicut rule @xmath342{{\sigma ; \delta_1,\ldots,\delta_n,\gamma \seqsym c } } { \deduce{{\sigma ; \delta_1 \seqsym b_1 } } { \pi_1 } & \cdots & \deduce{{\sigma ; \delta_n \seqsym b_n } } { \pi_n } & \deduce{{\sigma ; b_1,\ldots , b_n,\gamma \seqsym c } } { \pi } } \enspace .\ ] ] we refer to the formulas @xmath343 produced by the @xmath344 as _ cut formulas_. if @xmath345 , @xmath341 reduces to the premise derivation @xmath182 . for @xmath346 we specify the reduction relation based on the last rule of the premise derivations . if the rightmost premise derivation @xmath182 ends with a left rule acting on a cut formula @xmath67 , then the last rule of @xmath347 and the last rule of @xmath182 together determine the reduction rules that apply . we classify these rules according to the following criteria : we call the rule an _ essential _ case when @xmath347 ends with a right rule ; if it ends with a left rule , it is a _ left - commutative _ case ; if @xmath347 ends with the @xmath348 rule , then we have an _ axiom _ case ; a _ multicut _ case arises when it ends with the @xmath344 rule . when @xmath182 does not end with a left rule acting on a cut formula , then its last rule is alone sufficient to determine the reduction rules that apply . if @xmath182 ends in a rule acting on a formula other than a cut formula , then we call this a _ right - commutative _ case . structural _ case results when @xmath182 ends with a contraction or weakening on a cut formula . if @xmath182 ends with the @xmath348 rule , this is also an axiom case ; similarly a multicut case arises if @xmath182 ends in the @xmath344 rule . for simplicity of presentation , we always show @xmath349 . _ @xmath350 : if @xmath275 and @xmath182 are @xmath351{{\delta_1\seqsym b_1 ' \land b_1 '' } } { \deduce{{\delta_1\seqsym b_1 ' } } { \pi_1 ' } & \deduce{{\delta_1\seqsym b_1 '' } } { \pi_1 '' } } \qquad\qquad\qquad \infer[{\land{\cal l}}]{{b_1 ' \land b_1'',b_2,\ldots , b_n,\gamma\seqsym c } } { \deduce{{b_1',b_2,\ldots , b_n,\gamma\seqsym c } } { \pi ' } } \enspace , \ ] ] then @xmath341 reduces to @xmath342{{\delta_1,\ldots,\delta_n,\gamma\seqsym c } } { \deduce{{\delta_1\seqsym b_1 ' } } { \pi_1 ' } & \deduce{{\delta_2\seqsym b_2 } } { \pi_2 } & \cdots & \deduce{{\delta_n\seqsym b_n } } { \pi_n } & \deduce{{b_1',b_2,\ldots , b_n,\gamma\seqsym c } } { \pi ' } } \enspace .\ ] ] the case for the other @xmath352 rule is symmetric . @xmath353 : if @xmath275 and @xmath182 are @xmath354{{\delta_1\seqsym b_1 ' \lor b_1 '' } } { \deduce{{\delta_1\seqsym b_1 ' } } { \pi_1 ' } } \qquad\qquad\!\!\ ! \infer[{\lor{\cal l}}]{{b_1 ' \lor b_1'',b_2,\ldots , b_n,\gamma\seqsym c } } { \deduce{{b_1',b_2,\ldots , b_n,\gamma\seqsym c } } { \pi ' } & \deduce{{b_1'',b_2,\ldots , b_n,\gamma\seqsym c } } { \pi '' } } \enspace , \ ] ] then @xmath341 reduces to @xmath342{{\delta_1,\ldots,\delta_n,\gamma\seqsym c } } { \deduce{{\delta_1\seqsym b_1 ' } } { \pi_1 ' } & \deduce{{\delta_2\seqsym b_2 } } { \pi_2 } & \cdots & \deduce{{\delta_n\seqsym b_n } } { \pi_n } & \deduce{{b_1',b_2,\ldots , b_n,\gamma\seqsym c } } { \pi ' } } \enspace .\ ] ] the case for the other @xmath355 rule is symmetric . @xmath356 : suppose @xmath275 and @xmath182 are @xmath357{{\delta_1\seqsym b_1 ' { \supset}b_1 '' } } { \deduce{{b_1',\delta_1\seqsym b_1 '' } } { \pi_1 ' } } \qquad\qquad\!\ ! \infer[{{\supset}{\cal l}}]{{b_1 ' { \supset}b_1'',b_2,\ldots , b_n,\gamma\seqsym c } } { \deduce{{b_2,\ldots , b_n,\gamma\seqsym b_1 ' } } { \pi ' } & \deduce{{b_1'',b_2,\ldots , b_n,\gamma\seqsym c } } { \pi '' } } \enspace .\ ] ] let @xmath358 be @xmath342{{\delta_1,\ldots,\delta_n,\gamma\seqsym b_1 '' } } { \infer[{mc}]{{\delta_2,\ldots,\delta_n,\gamma\seqsym b_1 ' } } { \left\{\raisebox{-1.5ex}{\deduce{{\delta_i\seqsym b_i } } { \pi_i}}\right\}_{i \in \{2 .. n\ } } & \raisebox{-2.5ex}{\deduce{{b_2,\ldots , b_n,\gamma\seqsym b_1 ' } } { \pi ' } } } & \deduce{{b_1',\delta_1\seqsym b_1 '' } } { \pi_1 ' } } \enspace .\ ] ] then @xmath341 reduces to @xmath359{\makebox[\infwidthi ] { } } { \infer[{mc}]{{\delta_1,\ldots,\delta_n,\gamma , \delta_2,\ldots,\delta_n,\gamma\seqsym c } } { \raisebox{-2.5ex}{\deduce{{\ldots\seqsym b_1 '' } } { \xi_1 } } & \left\{\raisebox{-1.5ex}{\deduce{{\delta_i\seqsym b_i } } { \pi_i}}\right\}_{i \in \{2 .. n\ } } & \raisebox{-2.5ex}{\deduce{{b_1'',\{b_i\}_{i \in \{2 .. n\}},\gamma\seqsym c } } { \pi '' } } } } } \enspace .\ ] ] we use the double horizontal lines to indicate that the relevant inference rule ( in this case , @xmath246 ) may need to be applied zero or more times . @xmath360 : suppose @xmath275 and @xmath182 are @xmath208 { { \sigma ; \delta_1 \seqsym \forall x.b_1 ' } } { \deduce{{\sigma , h ; \delta_1 \seqsym b_1'[(h\,\vec{c})/x ] } } { \pi_1 ' } } \qquad\qquad\qquad \infer[{\forall{\cal l } } ] { { \sigma ; \forall x.b_1',b_2,\ldots , b_n,\gamma \seqsym c } } { \deduce{{\sigma ; b_1'[t / x],b_2,\ldots , b_n,\gamma \seqsym c } } { \pi ' } } \enspace , \ ] ] where @xmath361 let @xmath362 ) \setminus supp(b_1').$ ] apply lemma [ lm : supp1 ] to get a derivation @xmath363 of @xmath364}.$ ] the derivation @xmath341 reduces to @xmath342 { { \sigma ; \delta_1,\ldots,\delta_n,\gamma \seqsym c } } { \raisebox{-2.5ex } { \deduce{{\sigma ; \delta_1 \seqsym b_1'[t / x ] } } { \pi_1''[\lambda\vec{c}\vec d.t / h ' ] } } & \left\{\raisebox{-1.5ex } { \deduce{{\sigma ; \delta_i \seqsym b_i } } { \pi_i}}\right\}_{i \in \{2 .. n\ } } & \raisebox{-2.5ex}{\deduce{{\ldots\seqsym c } } { \pi ' } } } \enspace .\ ] ] @xmath365 : suppose @xmath275 and @xmath182 are @xmath282 { { \sigma ; \delta_1 \seqsym \exists x.b_1 ' } } { \deduce{{\sigma ; \delta_1 \seqsym b_1'[t / x ] } } { \pi_1 ' } } \qquad\qquad\qquad \infer[{\exists{\cal l } } ] { { \sigma ; \exists x.b_1',b_2,\ldots , b_n,\gamma \seqsym c } } { \deduce{{\sigma , h ; b_1'[(h\,\vec{c})/x],b_2,\ldots , b_n , \gamma \seqsym c } } { \pi ' } } \enspace , \ ] ] where @xmath361 let @xmath362)\setminus supp(b_1').$ ] apply lemma [ lm : supp2 ] to @xmath319 to get a derivation @xmath366 of @xmath367}.$ ] then @xmath341 reduces to @xmath342{{\sigma ; \delta_1,\ldots,\delta_n,\gamma \seqsym c } } { \deduce{{\sigma ; \delta_1 \seqsym b_1'[t / x]}}{\pi_1 ' } & \ldots & \deduce{{\sigma ; b_1'[t / x ] , b_2,\dots,\gamma \seqsym c } } { \pi''[\lambda \vec{c}\vec d.t / h ' ] } } \enspace .\ ] ] @xmath368 : suppose @xmath275 and @xmath182 are @xmath238 { { \delta_1\seqsym \nabla x.b_1 ' } } { \deduce{{\delta_1\seqsym b_1'[a / x ] } } { \pi_1 ' } } \qquad\qquad\qquad \infer[{\nabla{\cal l}}]{{\nabla x.b_1',\ldots , b_n,\gamma\seqsym c } } { \deduce{{b_1'[b / x],\ldots , b_n , \gamma\seqsym c } } { \pi ' } } \enspace .\ ] ] apply the construction in definition [ def : perm ] to to @xmath369 to swap @xmath42 with @xmath43 to get a derivation @xmath363 of @xmath370}.$ ] @xmath341 reduces to @xmath342 { { \delta_1,\ldots,\delta_n,\gamma\seqsym c } } { \deduce{{\delta_1\seqsym b_1'[b / x]}}{\pi_1 '' } & \ldots & \deduce{{b_1'[b / x ] , \dots , b_n , \gamma\seqsym c } } { \pi ' } } \enspace .\ ] ] @xmath371 suppose @xmath275 is @xmath372 { { \delta_1\seqsym nat~z } } { } $ ] and @xmath182 is @xmath194 { { nat~z , b_2 , \ldots , b_n , \gamma\seqsym c } } { \deduce{{\seqsym d~z}}{\pi ' } & \deduce{{d~j\seqsym d~(s\,j)}}{\pi '' } & \deduce{{d~z , b_2 , \ldots , b_n , \gamma\seqsym c}}{\pi ' '' } } \enspace .\ ] ] then @xmath341 reduces to @xmath342 { { \delta_1,\delta_2,\ldots,\delta_n,\gamma\seqsym c } } { \raisebox{-1.5ex}{\deduce{{\delta_1\seqsym d~z}}{w(\delta_1,\pi ' ) } } & \left\{\raisebox{-1.5ex } { \deduce{{\delta_i\seqsym b_i}}{\pi_i } } \right\}_{i\in\{2\dots n\ } } & \raisebox{-1.5ex}{\deduce{{d~z , b_2 , \ldots , b_n , \gamma\seqsym c}}{\pi ' '' } } } \ ] ] @xmath371 suppose @xmath275 is @xmath373 { { \delta_1\seqsym nat~(s\,i ) } } { \deduce{{\delta\seqsym nat\,i}}{\pi_1'}}\ ] ] and @xmath182 is @xmath194 { { nat~(s\,i ) , b_2 , \ldots , b_n , \gamma\seqsym c } } { \deduce{{\seqsym d\,z}}{\pi ' } & \deduce{{d\,j\seqsym d\,(s\,j)}}{\pi '' } & \deduce{{d\,(s\,i ) , b_2 , \ldots , b_n , \gamma\seqsym c}}{\pi ' '' } } \ ] ] let @xmath358 be @xmath374 { { \delta_1\seqsym d~i } } { \deduce{{\delta_1\seqsym nat\,i}}{\pi_1 ' } & \infer[{nat{\cal l } } ] { { nat\,i\seqsym d\,i } } { \deduce{{\seqsym d\,z}}{\pi ' } & \deduce{{d\,j\seqsym d\,(s\,j)}}{\pi '' } & \infer[id_\pi]{{d\,i\seqsym d\,i } } { } } } \ ] ] suppose @xmath375 we apply the procedures in definition [ def : subst ] and definition [ def : res ] to @xmath366 to obtain the derivation @xmath376 of @xmath377 let @xmath378 be @xmath374 { { \delta_1\seqsym d\,(s\,i ) } } { \deduce{{\delta_1\seqsym d\,i}}{\xi_1 } & \deduce{{d\,i\seqsym d\,(s\,i)}}{\pi^{\bullet}[\lambda \vec c.i / h ] } } \ ] ] then @xmath341 reduces to @xmath374 { { \delta_1,\ldots,\delta_n,\gamma\seqsym c } } { \deduce{{\delta_1\seqsym d\,(s\,i)}}{\xi_2 } & \deduce{{\delta_2\seqsym b_2}}{\pi_2 } & \ldots & \deduce{{\delta_n\seqsym b_n}}{\pi_n } & \deduce{{d\,(s\,i ) , b_2,\dots , b_n,\gamma\seqsym c}}{\pi ' '' } } \ ] ] @xmath379 if @xmath275 and @xmath182 are @xmath380 { { \sigma ; \delta_1 \seqsym t = t } } { } \qquad \infer[{{\rm eq}{\cal l } } ] { { \sigma ; t = t , \gamma \seqsym c } } { \left\ { \raisebox{-1.5ex } { \deduce{{\sigma\theta ; \gamma\theta \seqsym c\theta}}{\pi_\theta } } \right\}_\theta } \ ] ] then @xmath341 reduces to @xmath196 { { \sigma ; \delta_1,\ldots,\delta_n,\gamma \seqsym c } } { \deduce{{\sigma ; \delta_2 \seqsym b_2}}{\pi_2 } & \ldots & \deduce{{\sigma ; \delta_n \seqsym b_n}}{\pi_n } & \deduce{{\sigma ; \delta_1,b_2,\ldots , b_n,\gamma \seqsym c}}{w(\delta_1,\pi_\epsilon ) } } \ ] ] where @xmath381 is the empty substitution . @xmath382 : suppose @xmath275 and @xmath182 are @xmath383 { { \delta_1\seqsym p\,\bar{t } } } { \deduce{{\delta_1\seqsym b[\vec{t}/\vec x]}}{\pi_1 ' } } \qquad \qquad \infer[{\hbox{\sl def}{\cal l}}]{{p\,\vec{t},b_2,\dots,\gamma\seqsym c } } { \deduce { { b[\vec{t}/\vec x],b_2 , \dots,\gamma\seqsym c } } { \pi ' } } \enspace .\ ] ] then @xmath341 reduces to @xmath342 { { \delta_1,\dots,\delta_n,\gamma\seqsym c } } { \deduce{{\delta_1\seqsym b[\vec{t}/\vec x ] } } { \pi_1 ' } & \deduce{{\delta_2\seqsym b_2}}{\pi_2 } & \ldots & \deduce{{\delta_n\seqsym b_n}}{\pi_n } & \deduce{{b[\vec t/\vec x ] , \dots , \gamma\seqsym c}}{\pi ' } } \enspace .\ ] ] _ _ @xmath384 : suppose @xmath182 ends with a left rule other than @xmath246 acting on @xmath385 and @xmath275 is @xmath386 { { \delta_1\seqsym b_1 } } { \left\ { \raisebox{-1.5ex } { \deduce{{\delta_1^i\seqsym b_1 } } { \pi_1^i } } \right\ } } \enspace , \ ] ] where @xmath387 is any left rule except @xmath388 , @xmath130 , or @xmath193 . then @xmath341 reduces to @xmath386 { { \delta_1,\delta_2,\ldots,\delta_n,\gamma\seqsym c } } { \deduce{\makebox[\infwidthi ] { } } { \left\{\raisebox{-3.5ex } { \infer[{mc } ] { { \delta_1^i,\delta_2,\ldots,\delta_n,\gamma\seqsym c } } { \raisebox{-2.5ex } { \deduce{{\delta_1^i\seqsym b_1 } } { \pi_1^i } } & \left\{\raisebox{-1.5ex}{\deduce{{\delta_j\seqsym b_j } } { \pi_j}}\right\}_{j \in \{2 .. n\ } } & \raisebox{-2.5ex}{\deduce{{b_1,\ldots , b_n,\gamma\seqsym c } } { \pi}}}}\right\}\makebox[\infwidthii ] { } } } \enspace .\ ] ] @xmath389 : suppose @xmath182 ends with a left rule other than @xmath246 acting on @xmath385 and @xmath275 is @xmath124 { { d_1 ' { \supset}d_1'',\delta_1'\seqsym b_1 } } { \deduce{{\delta_1'\seqsym d_1 ' } } { \pi_1 ' } & \deduce{{d_1'',\delta_1'\seqsym b_1 } } { \pi_1 '' } } \enspace .\ ] ] let @xmath358 be @xmath342 { { d_1'',\delta_1',\delta_2,\ldots,\delta_n,\gamma\seqsym c } } { \deduce{{d_1'',\delta_1'\seqsym b_1 } } { \pi_1 '' } & \deduce{{\delta_2\seqsym b_2 } } { \pi_2 } & \cdots & \deduce{{\delta_n\seqsym b_n } } { \pi_n } & \deduce{{b_1,\ldots , b_n,\gamma\seqsym c } } { \pi } } \enspace .\ ] ] then @xmath341 reduces to @xmath124 { { d_1 ' { \supset}d_1'',\delta_1',\delta_2,\ldots,\delta_n,\gamma\seqsym c } } { \deduce{{\delta_1',\delta_2,\ldots,\delta_n,\gamma\seqsym d_1 ' } } { w(\delta_2\cup \dots \cup \delta_n\cup \gamma , \pi_1 ' ) } & \deduce{{d_1'',\delta_1',\delta_2,\ldots,\delta_n,\gamma\seqsym c } } { \xi_1 } } \enspace .\ ] ] @xmath390 suppose @xmath182 ends with a left rule other than @xmath246 acting on @xmath385 and @xmath275 is @xmath194 { { nat\,i , \delta_1'\seqsym b_1 } } { \deduce{{\seqsym d_1\,z}}{\pi_1 ^ 1 } & \deduce{{d_1\,j\seqsym d_1(s\,j)}}{\pi_1 ^ 2 } & \deduce{{d_1 i , \delta_1'\seqsym b_1}}{\pi_1 ^ 3 } } \ ] ] let @xmath358 be @xmath196 { { d_1 i , \delta_1 ' , \delta_2 , \ldots , \delta_n , \gamma\seqsym c } } { \deduce{{d_1 i , \delta_1'\seqsym b_1}}{\pi_1 ^ 3 } & \deduce{{\delta_2\seqsym b_2}}{\pi_2 } & \ldots & \deduce{{\delta_n\seqsym b_n}}{\pi_n } & \deduce{{b_1,\ldots , b_n\seqsym c}}{\pi } } \ ] ] then @xmath341 reduces to @xmath194 { { nat\,i , \delta_1 ' , \delta_2 , \ldots,\delta_n,\gamma\seqsym c } } { \deduce{{\seqsym d_1 z}}{\pi_1 ^ 1 } & \deduce{{d_1 j\seqsym d_1 ( s\,j)}}{\pi_1 ^ 2 } & \deduce{{d_1 i , \delta_1 ' , \delta_2,\ldots,\delta_2,\gamma\seqsym c}}{\xi_1 } } \ ] ] @xmath391 if @xmath182 ends with a left rule other than @xmath246 acting on @xmath385 and @xmath275 is @xmath128 { { s = t , \delta_1'\seqsym b_1 } } { \left\ { \raisebox{-1.5ex } { \deduce{{\delta_1'\theta\seqsym b_1\theta}}{\pi^\theta } } \right\}_\theta } \ ] ] then @xmath341 reduces to @xmath128 { { \qquad \qquad \qquad \quad s = t,\delta_1',\delta_2,\ldots,\delta_n,\gamma\seqsym c \qquad \qquad \qquad \qquad \qquad \qquad } } { \left\ { \raisebox{-1.5ex } { \infer[mc ] { { \delta_1'\theta , \delta_2\theta,\ldots , \delta_n\theta , \gamma\theta\seqsym c\theta } } { \deduce{{\delta_1'\theta\seqsym b_1\theta}}{\pi^\theta } & \deduce{{\delta_2\theta\seqsym b_2\theta}}{\pi_2\theta } & \ldots & \deduce{{\delta_n\theta\seqsym b_n\theta}}{\pi_n\theta } & \deduce{{b_1\theta,\ldots , b_n\theta,\gamma\theta\seqsym c\theta}}{\pi\theta } } } \right\}_\theta } \ ] ] _ _ @xmath392 : suppose @xmath182 is @xmath393 { { b_1,\ldots , b_n,\gamma\seqsym c } } { \left\{\raisebox{-1.5ex } { \deduce{{b_1,\ldots , b_n,\gamma^i\seqsym c } } { \pi^i}}\right\ } } \enspace , \ ] ] where @xmath394 is any left rule other than @xmath388 , @xmath130 , or @xmath193 ( but including @xmath246 ) acting on a formula other than @xmath395 . the derivation @xmath341 reduces to @xmath393{{\delta_1,\ldots,\delta_n,\gamma\seqsym c } } { \deduce{\makebox[\infwidthi ] { } } { \left\{\raisebox{-2.45ex } { \infer[{mc}]{{\delta_1,\ldots,\delta_n,\gamma^i\seqsym c } } { \deduce{{\delta_1\seqsym b_1 } } { \pi_1 } & \cdots & \deduce{{\delta_n\seqsym b_n } } { \pi_n } & \deduce{{b_1,\ldots , b_n,\gamma^i\seqsym c } } { \pi^i}}}\right\}\makebox[\infwidthii ] { } } } \enspace , \ ] ] @xmath396 : suppose @xmath182 is @xmath124{{b_1,\ldots , b_n , d ' { \supset}d'',\gamma'\seqsym c } } { \deduce{{b_1,\ldots , b_n,\gamma'\seqsym d ' } } { \pi ' } & \deduce{{b_1,\ldots , b_n , d'',\gamma'\seqsym c } } { \pi '' } } \enspace .\ ] ] let @xmath358 be @xmath342{{\delta_1,\ldots,\delta_n,\gamma'\seqsym d ' } } { \deduce{{\delta_1\seqsym b_1 } } { \pi_1 } & \cdots & \deduce{{\delta_n\seqsym b_n } } { \pi_n } & \deduce{{b_1,\ldots , b_n,\gamma'\seqsym d ' } } { \pi'}}\ ] ] and @xmath378 be @xmath342{{\delta_1,\ldots,\delta_n , d'',\gamma'\seqsym c } } { \deduce{{\delta_1\seqsym b_1 } } { \pi_1 } & \cdots & \deduce{{\delta_n\seqsym b_n } } { \pi_n } & \deduce{{b_1,\ldots , b_n , d'',\gamma'\seqsym c } } { \pi '' } } \enspace .\ ] ] then @xmath341 reduces to @xmath124{{\delta_1,\ldots,\delta_n , d ' { \supset}d'',\gamma'\seqsym c } } { \deduce{{\delta_1,\ldots,\delta_n,\gamma'\seqsym d ' } } { \xi_1 } & \deduce{{\delta_1,\ldots,\delta_n , d'',\gamma'\seqsym c } } { \xi_2 } } \enspace .\ ] ] @xmath397 suppose @xmath182 is @xmath194 { { b_1,\ldots , b_n , nat\,i , \gamma'\seqsym c } } { \deduce{{\seqsym d\,z}}{\pi ' } & \deduce{{d\,j\seqsym d\,(s\,j)}}{\pi '' } & \deduce{{b_1,\ldots , b_n , d\,i,\gamma'\seqsym c}}{\pi ' '' } } \ ] ] let @xmath358 be @xmath398 { { \delta_1,\ldots,\delta_n , d\,i,\gamma'\seqsym c } } { \deduce{{\delta_1\seqsym b_1}}{\pi_1 } & \ldots & \deduce{{\delta_n\seqsym b_n}}{\pi_n } & \deduce{{b_1,\ldots , b_n , d\,i , \gamma'\seqsym c}}{\pi ' '' } } \ ] ] then @xmath341 reduces to @xmath194 { { \delta_1,\ldots,\delta_n , nat\,i,\gamma'\seqsym c } } { \deduce{{\seqsym d\,z}}{\pi ' } & \deduce{{d\,j\seqsym d\,(s\,j)}}{\pi '' } & \deduce{{\delta_1,\ldots,\delta_n , d\,i,\gamma'\seqsym c}}{\xi_1 } } \ ] ] @xmath399 : if @xmath182 is @xmath128{{b_1,\ldots , b_n,{s = t } , \gamma'\seqsym c } } { \left\{\raisebox{-1.5ex } { \deduce{{b_1\rho,\ldots , b_n\rho,\gamma'\rho\seqsym c\rho } } { \pi^{\rho}}}\right\ } } \enspace , \ ] ] then @xmath341 reduces to @xmath128 { { \delta_1,\ldots,\delta_n , s = t,\gamma'\seqsym c } } { \deduce{\makebox[\infwidthi ] { } } { \left\{\raisebox{-3.5ex}{\infer[{mc } ] { { \delta_1\rho,\ldots , \delta_n\rho,\gamma'\rho\seqsym c\rho } } { \left\{\raisebox{-1.5ex } { \deduce{{\delta_i\rho\seqsym b_i\rho } } { \pi_i\rho } } \right\}_{i \in \{1 .. n\ } } & \raisebox{-2.5ex } { \deduce{{b_i\rho , \ldots , \gamma'\rho\seqsym c\rho } } { \pi^{\rho } } } } } \right\}\makebox[\infwidthii ] { } } } \enspace .\ ] ] @xmath400 : if @xmath182 is @xmath401 { { b_1,\ldots , b_n,\gamma\seqsym c } } { \left\{\raisebox{-1.5ex } { \deduce{{b_1,\ldots , b_n,\gamma^i\seqsym c^i } } { \pi^i}}\right\ } } \enspace , \ ] ] where @xmath402 is any right rule , then @xmath341 reduces to @xmath401 { { \delta_1,\ldots,\delta_n,\gamma\seqsym c } } { \deduce{\makebox[\infwidthi ] { } } { \left\{\raisebox{-2.45ex } { \infer[{mc}]{{\delta_1,\ldots,\delta_n,\gamma^i\seqsym c^i } } { \deduce { { \delta_1\seqsym b_1 } } { \pi_1 } & \cdots & \deduce{{\delta_n\seqsym b_n } } { \pi_n ' } & \deduce{{b_1,\ldots , b_n,\gamma^i\seqsym c^i } } { \pi^i}}}\right\}\makebox[\infwidthii ] { } } } \enspace .\ ] ] _ _ @xmath403 : if @xmath182 ends with a left rule other than @xmath246 acting on @xmath385 and @xmath275 ends with a multicut and reduces to @xmath369 , then @xmath341 reduces to @xmath342{{\delta_1,\ldots,\delta_n,\gamma\seqsym c } } { \deduce{{\delta_1\seqsym b_1 } } { \pi_1 ' } & \deduce{{\delta_2\seqsym b_2 } } { \pi_2 } & \cdots & \deduce{{\delta_n\seqsym b_n } } { \pi_n } & \deduce{{b_1,\ldots , b_n,\gamma\seqsym c } } { \pi } } \enspace .\ ] ] @xmath404 : suppose @xmath182 is @xmath342{{b_1,\ldots , b_n,\gamma^1,\ldots,\gamma^m,\gamma'\seqsym c } } { \left\{\raisebox{-1.5ex}{\deduce{{\{b_i\}_{i \in i^j},\gamma^j\seqsym d^j } } { \pi^j}}\right\}_{j \in \{1 .. m\ } } & \raisebox{-2.5ex}{\deduce{{\{d^j\}_{j \in \{1 .. m\}},\{b_i\}_{i \in i'},\gamma'\seqsym c } } { \pi ' } } } \enspace , \ ] ] where @xmath405 partition the formulas @xmath406 among the premise derivations @xmath275 , , @xmath407,@xmath319 . for @xmath408 let @xmath409 be @xmath342{{\{\delta_i\}_{i \in i^j},\gamma^j\seqsym d^j } } { \left\{\raisebox{-1.5ex}{\deduce{{\delta_i\seqsym b_i } } { \pi_i}}\right\}_{i \in i^j } & \raisebox{-2.5ex}{\deduce{{\{b_i\}_{i \in i^j},\gamma^j\seqsym d^j } } { \pi^j } } } \enspace .\ ] ] then @xmath341 reduces to @xmath342{{\delta_1,\ldots,\delta_n,\gamma^1,\ldots\gamma^m,\gamma'\seqsym c } } { \left\{\raisebox{-1.5ex}{\deduce{{\ldots\seqsym d^j } } { \xi^j}}\right\}_{j \in \{1 .. m\ } } & \left\{\raisebox{-1.5ex}{\deduce{{\delta_i\seqsym b_i } } { \pi_i}}\right\}_{i \in i ' } & \raisebox{-2.5ex}{\deduce{{\ldots\seqsym c } } { \pi ' } } } \enspace .\ ] ] _ _ @xmath410 : if @xmath182 is @xmath247{{b_1,b_2,\ldots , b_n,\gamma\seqsym c } } { \deduce{{b_1,b_1,b_2,\ldots , b_n,\gamma\seqsym c } } { \pi ' } } \enspace , \ ] ] then @xmath341 reduces to @xmath411{\makebox[\infwidthi ] { } } { \infer[{mc}]{{\delta_1,\delta_1,\delta_2,\ldots,\delta_n,\delta_n,\gamma\seqsym c } } { \raisebox{-2.5ex}{\deduce{{\delta_1\seqsym b_1 } } { \pi_1 } } & \left\{\raisebox{-1.5ex}{\deduce{{\delta_i\seqsym b_i } } { \pi_i}}\right\}_{i \in \{1 .. n\ } } & \raisebox{-2.5ex}{\deduce{{b_1,b_1,b_2,\ldots , b_n,\gamma\seqsym c } } { \pi ' } } } } } \enspace .\ ] ] _ _ @xmath412 : suppose @xmath182 ends with either @xmath193 or @xmath130 on @xmath385 and @xmath275 ends with the @xmath223 rule : @xmath120 { { \delta_1',b\seqsym b_1 } } { \pi_1.b = \pi_2.b_1}\ ] ] then it is the case that @xmath413 apply the construction in definition [ def : perm ] to @xmath182 to get a derivation @xmath319 of @xmath414 the derivation @xmath341 reduces to @xmath342{{b , \delta_1 ' , \delta_2,\ldots,\delta_n,\gamma\seqsym c } } { \deduce{{\delta_2\seqsym b_2 } } { \pi_2 } & \cdots & \deduce{{\delta_n\seqsym b_n } } { \pi_n } & \deduce{{b,\delta_1 ' , b_2,\ldots , b_n,\gamma\seqsym c } } { w(\delta_1',\pi ' ) } } \enspace .\ ] ] @xmath415 : if @xmath182 ends with the @xmath223 rule with a matching formula in @xmath104 , i.e. , there exists @xmath416 such that @xmath417 for some permutations @xmath37 and @xmath38 , then then @xmath341 reduces to @xmath120 { { \delta_1,\ldots,\delta_n,\gamma\seqsym c}}{}\ ] ] if @xmath182 ends with the @xmath223 rule but @xmath105 does not match any formula in @xmath104 , then @xmath105 must match one of the cut formulas , say @xmath385 , i.e. , there exists permutations @xmath418 and @xmath419 such that @xmath420 . that is , @xmath421 in this case , we first apply the permutation @xmath422 to @xmath275 according to the construction in definition [ def : perm ] to get a derivation @xmath369 of @xmath423 . @xmath341 then reduces to @xmath424 an inspection of the rules of the logic and this definition will reveal that every derivation ending with a multicut has a reduct . because we use a multiset as the left side of the sequent , there may be ambiguity as to whether a formula occurring on the left side of the rightmost premise to a multicut rule is in fact a cut formula , and if so , which of the left premises corresponds to it . as a result , several of the reduction rules may apply , and so a derivation may have multiple reducts . we now define two properties of derivations : normalizability and reducibility . each of these properties implies that the derivation can be reduced to a cut - free derivation of the same end - sequent . in the following , substitutions mean @xmath65-substitutions for some signature @xmath279 the definitions are similar to those by mcdowell and miller @xcite . however , since the cut reduction in our case involves several transformations of derivations , other than substitutions and weakening , we need to build this transformations into the definitions of normalizability and reducibility . a _ height - preserving _ ( hp ) transformation @xmath425 is a finite sequence of transformations @xmath426 where each @xmath427 is one of the transformations described in definition [ def : weak ] , definition [ def : subst ] , definition [ def : perm ] and definition [ def : res ] . the number @xmath428 is the _ order _ of @xmath425 . the application of @xmath425 to @xmath182 is defined as follows : @xmath429 note that a height - preserving transformation may not be defined for all derivations , and that it may be the identity transformation ( i.e. , it does nothing ) . height - preserving transformations are ranged over by @xmath430 and @xmath431 [ lm : height ] let @xmath425 be a height - preserving transformation . for any derivation @xmath182 , if @xmath432 is defined , then @xmath433 [ def : norm ] we define the set of _ normalizable _ derivations to be the smallest set that satisfies the following conditions : 1 . if a derivation @xmath182 ends with a multicut , then it is normalizable if for every height - preserving transformation @xmath425 such that @xmath432 is defined , there is a normalizable reduct of @xmath432 . if a derivation ends with any rule other than a multicut , then it is normalizable if the premise derivations are normalizable . these clauses assert that a given derivation is normalizable provided certain ( perhaps infinitely many ) other derivations are normalizable . if we call these other derivations the predecessors of the given derivation , then a derivation is normalizable if and only if the tree of the derivation and its successive predecessors is well - founded . in this case , the well - founded tree is called the _ normalization _ of the derivation . the set of normalizable derivations is not empty ; the cut - free proofs , for instance , are normalizable . since a normalization is well - founded , it has an associated induction principle : for any property @xmath434 of derivations , if for every derivation @xmath182 in the normalization , @xmath434 holds for every predecessor of @xmath182 implies that @xmath434 holds for @xmath182 , then @xmath434 holds for every derivation in the normalization . [ lm : norm - cut - free ] if there is a normalizable derivation of a sequent , then there is a cut - free derivation of the sequent . let @xmath182 be a normalizable derivation of the sequent @xmath435 . we show by induction on the normalization of @xmath182 that there is a cut - free derivation of @xmath435 . 1 . if @xmath182 ends with a multicut , then any of its reducts is one of its predecessors and so is normalizable . one of its reduct , via the empty transformation , is also a derivation of @xmath435 , so by the induction hypothesis this sequent has a cut - free derivation . 2 . suppose @xmath182 ends with a rule other than multicut . since we are given that @xmath182 is normalizable , by definition the premise derivations are normalizable . these premise derivations are the predecessors of @xmath182 , so by the induction hypothesis there are cut - free derivations of the premises . thus there is a cut - free derivation of @xmath435 . the next four lemmas are also proved by induction on the normalization of derivations . [ lm : norm subst ] if @xmath182 is a normalizable derivation , then for any substitution @xmath80 such that @xmath202 is defined , @xmath202 is normalizable . [ lm : norm weak ] if @xmath182 is normalizable , then for any multiset of formulas @xmath189 , if @xmath190 is defined , then @xmath190 is normalizable . [ lm : norm perm ] if @xmath182 is normalizable , then for any permutations @xmath436 such that @xmath237 is defined , @xmath237 is normalizable . [ lm : norm res ] if @xmath182 is normalizable , then for any nominal constants @xmath437 such that @xmath260 is defined , @xmath260 is normalizable . [ lm : norm hpt ] if @xmath182 is normalizable , then for any height - preserving transformation @xmath425 such that @xmath432 is defined , @xmath432 is normalizable . [ def : level drv ] the level of a sequent @xmath438 is the level of @xmath105 . the level of a derivation @xmath182 is the level of its root sequent . the definition of reducibility for derivations is done by induction on the level of derivations : in defining the reducibility of level-@xmath439 derivations , we assume that the reducibility of derivations of level @xmath440 , for all @xmath441 is already defined . in the following definition , when we apply a transformation @xmath425 to a derivation @xmath182 of @xmath442 we use the notation @xmath443 to denote the formula in the root sequent of @xmath432 that results from applying the transformation to @xmath67 . [ def : reducibility ] _ reducibility . _ for any @xmath439 , we define the set of _ reducible _ @xmath439-level derivations to be the smallest set of @xmath439-level derivations that satisfies the following conditions : 1 . if a derivation @xmath182 ends with a multicut then it is reducible if for every height - preserving transformation @xmath425 such that @xmath432 is defined , there is a reducible reduct of @xmath444 2 . suppose the derivation ends with the implication right rule @xmath357 { { \gamma\seqsym b{\supset}c } } { \deduce{{b,\gamma\seqsym c}}{\pi}}\ ] ] then the derivation is reducible if @xmath182 is reducible and for every height - preserving transformation @xmath425 such that @xmath432 is defined , multiset of formulas @xmath189 and reducible derivation @xmath319 of @xmath445 , where @xmath446 , the derivation @xmath196 { { \delta , \gamma'\seqsym c ' } } { \deduce{{\delta\seqsym b'}}{\pi ' } & \deduce{{b',\gamma'\seqsym c'}}{{\cal t}(\pi ) } } \ ] ] is reducible . if the derivation ends with the implication left rule or the @xmath171 rule , then it is reducible if the right premise derivation is reducible and the other premise derivations are normalizable . 4 . if the derivation ends with any other rule , then it is reducible if the premise derivations are reducible . these clauses assert that a given derivation is reducible provided certain other derivations are reducible . if we call these other derivations the predecessors of the given derivation , then a derivation is reducible only if the tree of the derivation and its successive predecessors is well founded . in this case , the well founded tree is called the _ reduction _ of the derivation . [ lm : reducible implies norm ] if a derivation is reducible , then it is normalizable . by induction on the reduction of the derivation . [ lm : reducible hpt ] if a derivation @xmath182 is reducible , then for any height - preserving @xmath425 such that @xmath432 is defined , @xmath432 is reducible . by induction on the reduction of @xmath182 and lemma [ lm : norm hpt ] . in the following , when we mention @xmath432 we assume implicitly that it is defined . we shall also use the notation @xmath447 to denote @xmath448 , that is the application of the transformation to the formula @xmath449 similarly , the multiset @xmath450 will be written @xmath451 we drop the subscript @xmath425 if it is clear from context which transformation we refer to . [ lm : reducibility ] for any derivation @xmath182 of @xmath452 and reducible derivations @xmath453 of @xmath454 where @xmath455 , and for any transformations @xmath456 such that @xmath457 is defined and @xmath458 , the derivation @xmath341 @xmath342 { { \sigma ' ; { \underline{\delta_1}}_{{{\cal t}}_1},\ldots , { \underline{\delta_n}}_{{{\cal t}}_n},{\underline{\gamma}}_{{{\cal t } } } \seqsym { \underline{c}}_{{{\cal t } } } } } { \deduce{{\sigma ' ; { \underline{\delta_1}}_{{{\cal t}}_1 } \seqsym { \underline{b_1}}_{{{\cal t}}}}}{{{\cal t}}_1(\pi_1 ) } & \ldots & \deduce{{\sigma ' ; { \underline{\delta_n}}_{{{\cal t}}_n } \seqsym { \underline{b_n}}_{{{\cal t}}}}}{{{\cal t}}_n(\pi_n ) } & \deduce{{\sigma ' ; { \underline{b_1}}_{{{\cal t}}},\ldots,{\underline{b_n}}_{{{\cal t } } } , { \underline{\gamma}}_{{{\cal t } } } \seqsym { \underline{c}}_{{{\cal t}}}}}{{{\cal t}}(\pi ) } } \ ] ] is reducible . the proof is by induction on @xmath185 with subordinate induction on @xmath428 and on the reductions of @xmath459 since the proof does not depend on the order of the inductions on reductions , when we need to distinguish of one the @xmath347 s we shall refer to it as @xmath275 without loss of generality . we need to show that for every @xmath460 , the derivation every reduct of @xmath461 is reducible . if @xmath345 then @xmath461 reduces to @xmath462 since reducibility is preserved by height - preserving transformation , it suffices to consider the case where @xmath463 and @xmath460 are the identity transformation , that is , we need only to show that @xmath182 is reducible . this is proved by case analysis on the last rule of @xmath464 for each case , the results follow from the outer induction hypothesis and definition [ def : reducibility ] . the case with @xmath465 requires that height - preserving transformations do not increase the height of the derivations ( see lemma [ lm : height ] ) . in the cases for @xmath388 and @xmath193 we need the additional information that reducibility implies normalizability ( see lemma [ lm : reducible implies norm ] ) . for @xmath346 , we analyze all possible reductions that apply to @xmath461 and show that every reduct of @xmath466 is reducible . we suppose that @xmath461 is of the following form : @xmath196 { { { \underline{\delta_1}}_{{{\cal f}}_1 } , \ldots , { \underline{\delta_n}}_{{{\cal f}}_n } , { \underline{\gamma}}_{{{\cal f}}}\seqsym { \underline{c}}_{{{\cal f } } } } } { \deduce{{{\underline{\delta_1}}_{{{\cal f}}_1}\seqsym { \underline{c_1}}_{{{\cal f}}_1}}}{{{\cal f}}_1(\pi_1 ) } & \ldots & \deduce{{{\underline{\delta_n}}_{{{\cal f}}_n}\seqsym { \underline{c_n}}_{{{\cal f}}_n}}}{{{\cal f}}_n(\pi_n ) } & \deduce{{{\underline{b_1}}_{{{\cal f}}},{\underline{b_n}}_{{{\cal f } } } , { \underline{\gamma}}_{{{\cal f}}}\seqsym { \underline{c}}_{{{\cal f}}}}}{{{\cal f}}(\pi ) } } \ ] ] where @xmath467 in several cases below , we often omit the subscripts @xmath468 or @xmath469 when it is clear from context which transformations we refer to . we also often switch between @xmath470 and @xmath471 to make the inference figures more readable . most cases follow immediately from the inductive hypothesis and definition [ def : reducibility ] and lemma [ lm : reducible implies norm ] , lemma [ lm : reducible hpt ] and lemma [ lm : height ] . we show here the interesting cases . @xmath356 : suppose @xmath275 and @xmath182 are @xmath357 { { \delta_1\seqsym b_1'{\supset}b_1 '' } } { \deduce{{\delta_1,b_1'\seqsym b_1''}}{\pi_1 ' } } \qquad \qquad \infer[{{\supset}{\cal l } } ] { { b_1'{\supset}b_1'',b_2,\dots , b_n,\gamma\seqsym c } } { \deduce{{b_2,\dots,\gamma\seqsym b_1'}}{\pi ' } & \deduce{{b_1'',b_2,\dots,\gamma\seqsym c}}{\pi '' } } \enspace .\ ] ] let @xmath358 be the derivation @xmath342 { { \underline{\delta_2},\dots,\underline{\delta_n},\underline{\gamma}\seqsym \underline{b_1 ' } } } { \deduce{{\underline { \delta_2}\seqsym \underline{b_2}}}{{{\cal f}}_2(\pi_2 ) } & \ldots & \deduce{{\underline{\delta_n}\seqsym \underline{b_n}}}{{{\cal f}}_n(\pi_n ) } & \deduce{{\underline{b_2},\dots,\underline{b_n } , \underline \gamma\seqsym \underline{b_1 ' } } } { { { \cal f}}_n(\pi ' ) } } \ ] ] then @xmath358 is reducible by induction hypothesis since @xmath468 and @xmath469 preserve reducibility ( lemma [ lm : reducible hpt ] ) and do not increase the height of derivations ( lemma [ lm : height ] ) . since we are given that @xmath275 is reducible , by definition [ def : reducibility ] , the derivation @xmath378 @xmath342 { { { \underline{\delta_1 } } , \ldots , { \underline{\delta_n } } , { \underline{\gamma}}\seqsym { \underline{b_1 '' } } } } { \deduce{{{\underline{\delta_2 } } , \ldots , { \underline{\delta_n } } , { \underline{\gamma}}\seqsym { \underline{b_1'}}}}{\xi_1 } & \deduce{{{\underline{b_1 ' } } , { \underline{\delta_1}}\seqsym { \underline{b_1''}}}}{{{\cal f}}_1(\pi_1 ' ) } } \ ] ] is reducible as well . therefore , the reduct of @xmath461 @xmath472{\makebox[\infwidthi ] { } } { \infer[{mc}]{{{\underline{\delta_1}},\ldots,{\underline{\delta_n } } , { \underline{\gamma } } , { \underline{\delta_2}},\ldots,{\underline{\delta_n}},{\underline{\gamma}}\seqsym { \underline{c } } } } { \raisebox{-2.5ex}{\deduce{{\ldots\seqsym { \underline{b_1 '' } } } } { \xi_2 } } & \left\{\raisebox{-1.5ex}{\deduce{{{\underline{\delta_i}}\seqsym { \underline{b_i } } } } { { { \cal f}}_i(\pi_i)}}\right\}_{i \in \{2 .. n\ } } & \raisebox{-2.5ex } { \deduce{{{\underline{b_1''}},\{{\underline{b_i}}\}_{i \in \{2 .. n\}},{\underline{\gamma}}\seqsym { \underline{c } } } } { { { \cal f}}(\pi '' ) } } } } } \enspace .\ ] ] is reducible by the outer induction hypothesis and definition [ def : reducibility ] . @xmath473 suppose @xmath275 and @xmath182 are @xmath208 { { \sigma ; \delta_1 \seqsym \forall x.b } } { \deduce{{\sigma , h ; \delta_1 \seqsym b[h\,\vec c / x]}}{\pi_1 ' } } \qquad \infer[{\forall{\cal l } } ] { { \sigma ; \forall x.b , b_2 , \ldots , b_n,\gamma \seqsym c } } { \deduce{{\sigma ; b[t / x ] , b_2,\ldots , b_n,\gamma \seqsym c}}{\pi ' } } \ ] ] applying the transformation @xmath474 to @xmath275 ( and similarly , @xmath468 to @xmath182 ) might require several transformation be done on the premise of the derivation , e.g. , to avoid clashes of nominal constants , etc . , so let us suppose that @xmath475 and @xmath476 are of the following shapes : @xmath208 { { \sigma ' ; { \underline{\delta_1 } } \seqsym \forall x.d } } { \deduce{{\sigma ' , h ; \delta_1 \seqsym d[h'\,\vec d / x]}}{{{\cal g}}_1(\pi_1 ' ) } } \qquad \infer[{\forall{\cal l } } ] { { \sigma ' ; \forall x.d , { \underline{b_2 } } , \ldots , { \underline{b_n } } , { \underline{\gamma } } \seqsym { \underline{c } } } } { \deduce{{\sigma ' ; d[s / x ] , { \underline{b_2}},\ldots , { \underline{b_n } } , { \underline{\gamma } } \seqsym { \underline{c } } } } { { { \cal g}}(\pi ' ) } } \ ] ] where @xmath477 and @xmath478 = { \underline{b[t / x]}}.$ ] if the support of @xmath478 $ ] is larger than @xmath479 , then the reduction rule for @xmath360 requires further transformations be applied to @xmath480 , i.e. , as is described in lemma [ lm : supp1 ] . so let us suppose that this transformation is applied , resulting in a derivation @xmath481}}{{{\cal g}}_1'(\pi_1 ' ) } \enspace .\ ] ] then @xmath461 reduces to @xmath196 { { \sigma ' ; { \underline{\delta_1}},\ldots,{\underline{\delta_n } } , { \underline{\gamma } } \seqsym { \underline{c } } } } { \deduce{{\sigma ' ; { \underline{\delta_1 } } \seqsym d[s / x ] } } { { { \cal g}}_1'(\pi_1')[\lambda \vec e.s / f ] } & \deduce{{{\underline{\delta_2}}\seqsym { \underline{b_2}}}}{{{\cal f}}_2(\pi_2 ) } & \ldots & \deduce{{{\underline{\delta_2}}\seqsym { \underline{b_2}}}}{{{\cal f}}_n(\pi_n ) } & \deduce{{\sigma ' ; d[s / x ] , \ldots , { \underline{\gamma } } \seqsym { \underline{c}}}}{{{\cal g}}(\pi ' ) } } \ ] ] which is reducible by the outer induction hypothesis . @xmath482 suppose @xmath275 and @xmath182 are @xmath373 { { \delta_1\seqsym nat\,m } } { \deduce{{\delta_1\seqsym nat\,m}}{\pi_1 ' } } \qquad \infer[{nat{\cal l } } ] { { nat\,(s\,i ) , b_2 , \ldots , b_n,\gamma\seqsym c } } { \deduce{{\seqsym d\,z}}{\pi ' } & \deduce{{d\,j\seqsym d\,(s\,j)}}{\pi '' } & \deduce{{d\,(s\,m),b_2,\ldots , b_n,\gamma\seqsym c}}{\pi ' '' } } \ ] ] then @xmath475 and @xmath476 are @xmath373 { { { \underline{\delta_1}}\seqsym nat\,i } } { \deduce{{{\underline{\delta_1}}\seqsym nat\,i}}{{{\cal f}}_1(\pi_1 ' ) } } \qquad \infer[{nat{\cal l } } ] { { nat\,(s\,i ) , { \underline{b_2 } } , \ldots , { \underline{b_n}},{\underline{\gamma}}\seqsym { \underline{c } } } } { \deduce{{\seqsym d\,z}}{\pi ' } & \deduce{{d\,j\seqsym d\,(s\,j)}}{\pi '' } & \deduce{{d\,(s\,i),{\underline{b_2}},\ldots,{\underline{b_n}},{\underline{\gamma}}\seqsym { \underline{c}}}}{{{\cal f}}(\pi ' '' ) } } \ ] ] note that the derivations @xmath319 and @xmath366 are not affected by the transformation @xmath468 since @xmath483 is a closed term with no occurrences of nominal constants and @xmath440 in @xmath366 is a new eigenvariable . let @xmath358 be the derivation @xmath196 { { { \underline{\delta_1}}\seqsym d\,i } } { \deduce{{{\underline{\delta_1}}\seqsym nat\,i}}{{{\cal f}}_1(\pi_1 ' ) } & \infer[{nat{\cal l } } ] { { nat\,i\seqsym d\,i } } { \deduce{{\seqsym d\,z}}{\pi ' } & \deduce{{d\,j\seqsym d\,(s\,j)}}{\pi '' } & \infer[id_\pi ] { { d\,i\seqsym d\,i } } { } } } \enspace .\ ] ] since the height of the right premise is no larger than @xmath185 , and @xmath369 is a predecessor of @xmath275 , @xmath358 is reducible by induction on the reduction of @xmath484 let @xmath162 be the support of @xmath485 we construct the derivation @xmath376 of @xmath486 from @xmath366 using the procedures described in definition [ def : subst ] and definition [ def : res ] . let @xmath378 be @xmath487 { { { \underline{\delta_1}}\seqsym d\,(s\ , i ) } } { \deduce{{{\underline{\delta_1}}\seqsym d\,i}}{\xi_1 } & \deduce{{d\,i\seqsym d\,(s\,i)}}{\pi^{\bullet}[\lambda \vec c.i / h ] } } \ ] ] since @xmath488 ) \leq ht ( \pi'')$ ] , by the outer induction hypothesis , @xmath378 is also reducible . therefore the reduct of @xmath461 @xmath196 { { { \underline{\delta_1 } } , \ldots , { \underline{\delta_2}},{\underline{\gamma}}\seqsym { \underline{c } } } } { \deduce{{{\underline{\delta_1}}\seqsym d\,(s\,i)}}{\xi_2 } & \deduce{{{\underline{\delta_2}}\seqsym { \underline{b_2}}}}{{{\cal f}}_2(\pi_2 ) } & \ldots & \deduce{{{\underline{\delta_n}}\seqsym { \underline{b_n } } } } { { { \cal f}}_n(\pi_n ) } & \deduce{{d\,(s\,i ) , { \underline{b_2 } } , \ldots , { \underline{\gamma}}\seqsym { \underline{c}}}}{{{\cal f}}(\pi ' '' ) } } \ ] ] is reducible by the outer induction hypothesis . @xmath391 suppose @xmath275 is @xmath128 { { s = t , \delta_1\seqsym b_1 } } { \left\ { \raisebox{-1.5ex}{\deduce{{\delta_1\theta\seqsym b_1\theta}}{\pi^\theta } } \right\}_\theta } \ ] ] then @xmath475 is @xmath489 { { { \underline{s } } = { \underline{t } } , { \underline{\delta_1}}\seqsym { \underline{b_1 } } } } { \left\ { \raisebox{-1.5ex}{\deduce{{{\underline{\delta_1}}\theta\seqsym { \underline{b_1}}\theta}}{\pi^{\bullet\rho } } } \right\}_\rho } \ ] ] where each @xmath490 is obtained from some @xmath491 by the transformations described in definition [ def : weak ] , definition [ def : subst ] , definition [ def : perm ] and definition [ def : res ] . we denote with @xmath492 the substitution @xmath80 such that @xmath490 is constructed out of @xmath493 thus we can write each @xmath490 as the derivation @xmath494 for some transformation @xmath495 the reduct of @xmath461 @xmath128 { { \qquad \qquad \qquad \quad { \underline{s}}= { \underline{t}},{\underline{\delta_1'}},{\underline{\delta_2}},\ldots,{\underline{\delta_n}},{\underline{\gamma}}\seqsym { \underline{c } } \qquad \qquad \qquad \qquad \qquad \qquad \qquad\qquad } } { \left\ { \raisebox{-1.5ex } { \infer[mc ] { { { \underline{\delta_1'}}\rho , { \underline{\delta_2}}\rho,\ldots , { \underline{\delta_n}}\rho , { \underline{\gamma}}\rho\seqsym { \underline{c } } \rho } } { \deduce{{{\underline{\delta_1'}}\rho\seqsym { \underline{b_1}}\rho}}{{{\cal f}}_\rho(\pi^{f(\rho ) } ) } & \deduce{{{\underline{\delta_2}}\rho\seqsym { \underline{b_2}}\rho}}{{{\cal f}}_2(\pi_2)\rho } & \ldots & \deduce{{{\underline{\delta_n}}\rho\seqsym { \underline{b_n}}\rho}}{{{\cal f}}_n(\pi_n)\rho } & \deduce{{{\underline{b_1}}\rho,\ldots,{\underline{b_n}}\rho,{\underline{\gamma}}\rho\seqsym { \underline{c}}\rho}}{{{\cal f}}(\pi)\rho } } } \right\}_\rho } \ ] ] each premise derivation of the above derivation is reducible by the induction hypothesis on the reduction of @xmath275 , since each @xmath496 is a predecessor of @xmath484 the reduct of @xmath461 is therefore reducible by definition [ def : reducibility ] . @xmath497 suppose @xmath182 is @xmath357 { { b_1 , \ldots , b_n , \gamma\seqsym { c_1 { \supset}c_2 } } } { \deduce{{b_1 , \ldots , b_n , \gamma , c_1 \seqsym c_2}}{{{\cal f}}(\pi ' ) } } \ ] ] then @xmath498 @xmath357 { { { \underline{b_1 } } , \ldots , { \underline{b_n } } , { \underline{\gamma}}\seqsym { \underline{c_1 { \supset}c_2 } } } } { \deduce{{{\underline{b_1 } } , \ldots , { \underline{b_n } } , { \underline{\gamma } } , { \underline{c_1 } } \seqsym { \underline{c_2}}}}{{{\cal f}}(\pi ' ) } } \ ] ] let @xmath358 be @xmath499 { { { \underline{\delta_1 } } , \ldots , { \underline{\delta_n } } , { \underline{c_1}}\seqsym { \underline{c_2 } } } } { \deduce{{{\underline{\delta_1}}\seqsym { \underline{b_1}}}}{{{\cal f}}_1(\pi_1 ) } & \ldots & \deduce{{{\underline{\delta_n}}\seqsym { \underline{b_n}}}}{{{\cal f}}_n(\pi_n ) } & \deduce{{{\underline{b_1}},\ldots,{\underline{b_1}},{\underline{\gamma } } , { \underline{c_1}}\seqsym { \underline{c_2}}}}{{{\cal f}}(\pi ' ) } } \ ] ] which is reducible by the outer induction hypothesis . let @xmath378 be the derivation @xmath357 { { { \underline{\delta_1 } } , \ldots , { \underline{\delta_n } } , { \underline{\gamma}}\seqsym { \underline{c_1{\supset}c_2 } } } } { \deduce{{{\underline{\delta_1 } } , \ldots , { \underline{\delta_n } } , { \underline{\gamma } } , { \underline{c_1}}\seqsym { \underline{c_2}}}}{\xi_1 } } \enspace , \ ] ] which is the reduct of @xmath500 to show that @xmath378 is reducible , we need to show that for any @xmath501 , and for any derivation @xmath366 of @xmath502 where @xmath503 , the derivation @xmath504 @xmath196 { { \delta , { \underline{\delta_1}}_{{{\cal g}}_1 } , \ldots , { \underline{\delta_n}}_{{{\cal g}}_n } , { \underline{\gamma}}_{{{\cal g}}}\seqsym { \underline{c_2}}_{{{\cal g } } } } } { \deduce{{\delta\seqsym d}}{\pi '' } & \deduce{{d , { \underline{\delta_1}}_{{{\cal g}}_1 } , \ldots , { \underline{\delta_n}}_{{{\cal g}}_n } , { \underline{\gamma}}_{{{\cal g}}}\seqsym { \underline{c_2}}_{{{\cal g } } } } } { { { \cal t}}''(\xi_2 ) } } \ ] ] is reducible . here the transformations @xmath505 and @xmath506 are transformations associated with the premise derivations in @xmath507 @xmath504 is reducible if for any transformation @xmath508 , every reduct of the derivation @xmath509 is reducible . the reduct of @xmath509 in this case is : @xmath342 { { { \underline{\delta } } , { \underline{\delta_1 } } , \ldots , { \underline{\delta_n } } , { \underline{\gamma}}\seqsym { \underline{c_2 } } } } { \deduce{{{\underline{\delta}}\seqsym { \underline{d}}}}{{{\cal h}}'(\pi '' ) } & \deduce{{{\underline{\delta_1}}\seqsym { \underline{b_1}}}}{{{\cal h}}_1(\pi_1 ) } & \ldots & \deduce{{{\underline{\delta_n}}\seqsym { \underline{b_n}}}}{{{\cal h}}_n(\pi_n ) } & \deduce{{{\underline{d } } , { \underline{b_1 } } , \ldots , { \underline{b_n } } , { \underline{\gamma}}\seqsym { \underline{c_2}}}}{{{\cal h}}''(\pi ' ) } } \ ] ] where @xmath510 and @xmath511 are transformations applied to the premises of @xmath512 and @xmath513 is the transformation applied to the left premise of @xmath514 this derivation is reducible by the outer induction hypothesis . [ cor : reducibility ] every derivation is reducible . this result follows immediately from lemma [ lm : reducibility ] with @xmath515 [ thm : cut elim ] the cut rule is admissible in @xmath33 . follows immediately from corollary [ cor : reducibility ] , lemma [ lm : reducible implies norm ] and lemma [ lm : norm - cut - free ] . the logic @xmath33 is consistent , i.e. , it is not the case that both @xmath165 and @xmath516 are provable . @xmath36 { { \sigma ; { \sigma \triangleright b},\gamma \seqsym { \sigma \triangleright b } } } { } \qquad \infer[cut ] { { \sigma ; \delta,\gamma \seqsym { { \cal c } } } } { { \sigma ; \delta \seqsym { { \cal b } } } \qquad { \sigma ; { { \cal b}},\gamma \seqsym { { \cal c}}}}\ ] ] @xmath517 { { \sigma ; { \sigma \triangleright b \land c},\gamma \seqsym { { \cal d } } } } { { \sigma ; { \sigma \triangleright b } , { \sigma \triangleright c } , \gamma \seqsym { { \cal d } } } } \qquad \infer[{\land{\cal r } } ] { { \sigma ; \gamma \seqsym { \sigma \triangleright b \land c } } } { { \sigma ; \gamma \seqsym { \sigma \triangleright b } } \qquad { \sigma ; \gamma \seqsym { \sigma \triangleright c } } } \ ] ] @xmath123 { { \sigma ; { \sigma \triangleright b \lor c},\gamma \seqsym { { \cal d } } } } { { \sigma ; { \sigma \triangleright b},\gamma \seqsym { { \cal d } } } \qquad { \sigma ; { \sigma \triangleright c},\gamma \seqsym { { \cal d } } } } \qquad \infer[{\lor{\cal r } } ] { { \sigma ; \gamma \seqsym { \sigma \triangleright b \lor c } } } { { \sigma ; \gamma \seqsym { \sigma \triangleright b}}}\ ] ] @xmath121 { { \sigma ; { \sigma \triangleright \bot},\gamma \seqsym { { \cal b } } } } { } \qquad \infer[{\lor{\cal r}}]{{\sigma ; \gamma \seqsym { \sigma \triangleright b \lor c } } } { { \sigma ; \gamma \seqsym { \sigma \triangleright c}}}\ ] ] @xmath124{{\sigma ; { \sigma \triangleright b { \supset}c},\gamma \seqsym { { \cal d } } } } { { \sigma ; \gamma \seqsym { \sigma \triangleright b } } \qquad { \sigma ; { \sigma \triangleright c},\gamma \seqsym { { \cal d } } } } \qquad \infer[{{\supset}{\cal r}}]{{\sigma ; \gamma \seqsym { \sigma \triangleright b { \supset}c } } } { { \sigma ; { \sigma \triangleright b},\gamma \seqsym { \sigma \triangleright c}}}\ ] ] @xmath125 { { \sigma ; { \sigma \triangleright \forall_\gamma x.b},\gamma \seqsym { { \cal c } } } } { { \sigma , \sigma \vdash t : \gamma } \qquad { \sigma ; { \sigma \triangleright b[t / x]},\gamma \seqsym { { \cal c } } } } \qquad \infer[{\forall{\cal r } } ] { { \sigma ; \gamma \seqsym { \sigma \triangleright \forall x.b } } } { { \sigma , h ; \gamma \seqsym { \sigma \triangleright b[(h~\sigma)/x]}}}\ ] ] @xmath518{{\sigma ; { \sigma \triangleright \exists x.b},\gamma \seqsym { { \cal c } } } } { { \sigma , h ; { \sigma \triangleright b[(h~\sigma)/x]},\gamma \seqsym { { \cal c } } } } \qquad \infer[{\exists{\cal r } } ] { { \sigma ; \gamma \seqsym { \sigma \triangleright \exists_\gamma x.b } } } { { \sigma , \sigma \vdash t : \gamma } \qquad { \sigma ; \gamma \seqsym { \sigma \triangleright b[t / x]}}}\ ] ] @xmath519 { { \sigma ; { \sigma \triangleright \nabla x\ b},\gamma \seqsym { { \cal c } } } } { { \sigma ; { ( \sigma , y ) \triangleright b[y / x ] } , \gamma \seqsym { { \cal c } } } } \qquad \infer[{\nabla{\cal r } } ] { { \sigma ; \gamma \seqsym { \sigma \triangleright \nabla x\ b } } } { { \sigma ; \gamma \seqsym { ( \sigma , y ) \triangleright b[y / x ] } } } \ ] ] @xmath247{{\sigma ; { { \cal b}},\gamma \seqsym { { \cal c } } } } { { \sigma ; { { \cal b}},{{\cal b}},\gamma \seqsym { { \cal c } } } } \qquad \infer[{\hbox{\sl w}{\cal l } } ] { { \sigma ; { { \cal b } } , \gamma \seqsym { { \cal c } } } } { { \sigma ; \gamma \seqsym { { \cal c } } } } \qquad \infer[{\top{\cal r}}]{{\sigma ; \gamma \seqsym { \sigma \triangleright \top}}}{}\ ] ] we now show that the formulation of @xmath48 is equivalent to @xmath50 extended with the axiom schemes of name permutations and weakening : @xmath520 where @xmath4 is not free in @xmath5 in the second scheme . sequents in @xmath50 are expressions of the form @xmath521 @xmath65 is the _ signature _ of the sequent , @xmath66 is a list of variables locally scoped over @xmath67 , and is referred to as _ local signature_. the expression @xmath522 is called a _ local judgment _ , or _ judgment _ for short . in @xcite , local judgments are considered equal modulo renaming of their local signatures , e.g. , @xmath523 is equal to @xmath524 local judgments are ranged over by scripted capital letters , e.g. , @xmath525 , @xmath526 , etc . for the purpose of proving the correspondence with @xmath48 , however , we will make this renaming step explicit , by including the rules : @xmath527 { { { \vec x \triangleright b } , \gamma\seqsym { { \cal c } } } } { { { \vec y \triangleright b ' } , \gamma\seqsym { { \cal c } } } } \qquad \infer[{\alpha_{\cal l } } , \lambda \vec x.b \equiv_\alpha \lambda \vec y.b ' ] { { \gamma\seqsym { \vec x \triangleright b } } } { { \gamma\seqsym { \vec y \triangleright b'}}}\ ] ] the inference rules of @xmath50 are given in figure [ fig : folnb ] . we now consider the correspondence between @xmath48 with @xmath50 extended with the following axiom schemes : @xmath528 @xmath529 we can equivalently state these two axioms as the following inference rules : @xmath530 { { { ( \vec x , a , b,\vec y ) \triangleright b } , \gamma\seqsym { { \cal c } } } } { { { ( \vec x , b , a,\vec y ) \triangleright b } , \gamma\seqsym { { \cal c } } } } \qquad \infer[{p{\cal l } } ] { { \gamma\seqsym { ( \vec x , a , b,\vec y ) \triangleright b } } } { { \gamma\seqsym { ( \vec x , b , a,\vec y ) \triangleright b}}}\ ] ] @xmath531 { { { ( \vec x\vec y ) \triangleright b } , \gamma\seqsym { { \cal c } } } } { { { ( \vec x , a,\vec y ) \triangleright b } , \gamma\seqsym { { \cal c } } } } \qquad \infer[{ss{\cal r } } , a \not \in \{\vec x , \vec y\ } ] { { \gamma\seqsym { ( \vec x\vec y ) \triangleright b } } } { { \gamma\seqsym { ( \vec x , a,\vec y ) \triangleright b } } } \ ] ] @xmath532 { { { ( \vec x , a,\vec y ) \triangleright b } , \gamma\seqsym { { \cal c } } } } { { { ( \vec x\vec y ) \triangleright b } , \gamma\seqsym { { \cal c } } } } \qquad \infer[{ss{\cal r } } , a \not \in supp(b ) ] { { \gamma\seqsym { ( \vec x , a,\vec y ) \triangleright b } } } { { \gamma\seqsym { ( \vec x\vec y ) \triangleright b } } } \ ] ] implicit in the above rules is the assumption that variables in local signatures are considered as special constants , much like the nominal constants in @xmath48 . the support of @xmath5 , within a local signature @xmath533 , is defined similarly as it is in @xmath48 : it is the set @xmath534 the logical system with the inference rules in figure [ fig : folnb ] together with @xmath535 , @xmath536 , @xmath537 , @xmath538 , @xmath539 , @xmath540 , @xmath541 and @xmath542 is referred to as @xmath543 . in relating @xmath48 and @xmath543 , we map the local signatures to nominal constants , and vice versa . in the following , given a formula @xmath5 , we assume a particular enumeration of the nominal constants appearing in @xmath5 based the left - to - right order of their appearance in @xmath5 . * suppose @xmath182 ends with @xmath551 @xmath120 { { \gamma ' , b_i\seqsym b_0 } } { \pi.b_i = \pi'.b_0}\ ] ] the permutations @xmath37 and @xmath38 can be imitated by a series of renaming ( @xmath535 and @xmath536 rules ) . the derivation @xmath319 is therefore constructed by applying a series of @xmath552 , @xmath553 , followed by the @xmath187 rule . * suppose @xmath182 ends with @xmath554 in this case we suppose that @xmath555 @xmath357 { { b_1,\ldots , b_n\seqsym c { \supset}d } } { \deduce{{b_1,\ldots , b_n , c\seqsym d}}{\pi_1}}\ ] ] by induction hypothesis we have a derivation @xmath276 of @xmath556 we first have to weaken the signatures @xmath437 and @xmath557 to @xmath558 before applying the introduction rule for @xmath56 . that is , @xmath319 is the derivation @xmath357 { { { \vec c_1 \triangleright b_1,\ldots , { \vec c_n \triangleright b_n}}\seqsym { \vec c_0 \triangleright c { \supset}d } } } { \infer [ * ] { { { \vec c_1 \triangleright b_1,\ldots , { \vec c_n \triangleright b_n } } , { \vec c_0 \triangleright c}\seqsym { \vec c_0 \triangleright d } } } { \infer [ ] { \makebox[\infwidthi ] { } } { \deduce{{{\vec c_1 \triangleright b_1,\ldots , { \vec c_n \triangleright b_n } } , { \vec a \triangleright c}\seqsym { \vec b \triangleright d } } } { \pi_2 } } } } \ ] ] here the star ` * ' denotes a series of applications of @xmath541 , @xmath542 , @xmath537 and @xmath559 * suppose @xmath182 is @xmath282 { { b_1,\ldots , b_n\seqsym \exists x.c } } { \deduce{{b_1,\ldots , b_n\seqsym c[t / x]}}{\pi_1}}\ ] ] it is possible that @xmath22 contains new constants that are not in the support of @xmath560 suppose @xmath557 is an enumeration of the support of @xmath561 $ ] . the derivation @xmath319 is constructed as follows @xmath562 { { { \vec c_1 \triangleright b_1 } , \ldots , { \vec c_n \triangleright b_n } \seqsym { \vec c_0 \triangleright \exists x.c } } } { \infer [ ] { \makebox[\infwidthi ] { } } { \infer[{\exists{\cal r } } ] { { { \vec c_1 \triangleright b_1 } , \ldots , { \vec c_n \triangleright b_n } \seqsym { \vec d \triangleright \exists x.c } } } { \deduce { { { \vec c_1 \triangleright b_1 } , \ldots , { \vec c_n \triangleright b_n } \seqsym { \vec d \triangleright c[t / x ] } } } { \pi_2 } } } } \ ] ] where @xmath276 is obtained from induction hypothesis applied to @xmath275 , and the rule ` * ' denotes a series of applications of @xmath540 ( for introducing new constants ) and @xmath538 ( for rearranging the order of the local signature ) . * for other cases , the construction of @xmath319 follows the same pattern as in the previous cases , i.e. , by induction hypothesis , followed by some rearranging , extension , or weakening of local signatures . suppose @xmath182 is a derivation of @xmath563 we construct a derivation @xmath319 of @xmath312 by induction on @xmath185 . we show here the interesting cases ; the other cases follow immediately from induction hypothesis : * if @xmath182 ends with @xmath187 , @xmath564 , or @xmath565 then @xmath319 ends with the same rule . * suppose @xmath182 is @xmath566 { { { \vec c_1 \triangleright b_1 } , \ldots , { \vec c_n \triangleright b_n}\seqsym { \vec c_0 \triangleright b_0 } } } { \deduce{{{\vec c_1 \triangleright b_1 } , \ldots , { \vec c_n \triangleright b_n } \seqsym { \vec d \triangleright b } } } { \pi_1 } } \ ] ] by induction hypothesis , there is a derivation @xmath276 of @xmath567 to get @xmath319 apply the procedure in definition [ def : perm ] to @xmath276 to rename @xmath5 to @xmath257 . * suppose @xmath182 is @xmath208 { { { \vec c_1 \triangleright b_1 } , \ldots , { \vec c_n \triangleright b_n}\seqsym { \vec c_0 \triangleright \forall x.c } } } { \deduce { { { \vec c_1 \triangleright b_1 } , \ldots , { \vec c_n \triangleright b_n}\seqsym { \vec c_0 \triangleright c[(h\,\vec c_0)/x ] } } } { \pi_1 } } \ ] ] by induction hypothesis , there is a derivation @xmath276 of @xmath568}}.$ ] suppose @xmath569 then @xmath319 is @xmath208 { { b_1,\ldots , b_n\seqsym \forall x.c } } { \deduce{{b_1,\ldots , b_n\seqsym c[h'\,\vec d / x]}}{\pi_2[\lambda \vec c_0.h'\,\vec d / h ] } } \ ] ] * if @xmath182 ends with @xmath111 , apply the same construction as in the previous case . [ thm : lg equal folnb ] let @xmath27 be a formula which contains no occurrences of nominal constants . then @xmath27 is provable in @xmath50 extended with the axiom schemes @xmath32 and @xmath570 if and only if @xmath27 is provable in @xmath49 d. miller and c. palamidessi . foundational aspects of syntax . in p. degano , r. gorrieri , a. marchetti - spaccamela , and p. wegner , editors , _ acm computing surveys symposium on theoretical computer science : a perspective _ , volume 31 . acm , september 1999 . p. schroeder - heister . cut - elimination in logics with definitional reflection . in d. pearce and h. wansing , editors , _ nonclassical logics and information processing _ , volume 619 of _ lncs _ , pages 146171 . springer , 1992 .

this paper presents a cut - elimination proof for the logic @xmath0 , which is an extension of a proof system for encoding generic judgments , the logic @xmath1 of miller and tiu , with an induction principle . the logic @xmath0 , just as @xmath1 , features extensions of first - order intuitionistic logic with fixed points and a `` generic quantifier '' , @xmath2 , which is used to reason about the dynamics of bindings in object systems encoded in the logic . a previous attempt to extend @xmath1 with an induction principle has been unsuccessful in modeling some behaviours of bindings in inductive specifications . it turns out that this problem can be solved by relaxing some restrictions on @xmath2 , in particular by adding the axiom @xmath3 , where @xmath4 is not free in @xmath5 . we show that by adopting the equivariance principle , the presentation of the extended logic can be much simplified . this paper contains the technical proofs for the results stated in @xcite ; readers are encouraged to consult @xcite for motivations and examples for @xmath6

the paradigm of random walks with correlated traps is relevant to many interdisciplinary problems ( random search strategies , exciton trapping by polymer chains , ligand binding to receptors on cell surfaces , foraging patterns , etc . ) , but it is also interesting from the more general perspective of the dynamics of complex systems with correlated disorder @xcite . many previous works concerned trapping kinetics for regularly distributed traps @xcite , some specific types of finite - range correlations and clusters @xcite , as well as traps distributed with critical ( long - ranged ) positional correlations @xcite . in this paper we consider random walks on a host lattice with spectral dimension @xmath3 ( where diffusion is compact @xcite ) with correlated imperfect traps forming a proper fractal sublattice . for such a model , provided that the initial coordinates of random walks are chosen randomly to be at or within a fixed distance of a trap , we found that the survival probability @xmath0 first decays according to stretched exponential kinetics @xmath13 , followed by a transition to power law decay @xmath9 at longer time scales , with the same exponent @xmath14 for both regimes . the crossover time @xmath15 between the two stages is related to the traps absorption rate @xmath16 by @xmath17 and shrinks to zero in the limit of perfect traps @xmath18 . thus , for strongly absorbing traps the stretched exponential regime is practically absent and @xmath0 decays according to a power law for all time scales ( except very short ones ) . on the other hand , for weakly absorbing traps ( small @xmath16 ) , both regimes are distinctly present . the stretched exponential regime is relatively short , lasting only while @xmath0 has not dropped too far ( roughly , less than ten per cent ) from its initial value . however , the absolute duration of the regime @xmath17 may be significant for a sufficiently small absorption rate @xmath16 . a particularly interesting instance of this case is when absorption is controlled by a thermally activated reaction , @xmath19 . for a high activation energy @xmath20 , the duration of the stretched exponential regime may hold over several orders of magnitude in time . we will show that the two - stage long - tailed kinetics of imperfect correlated traps are consistent with known results for regular spatial trap distributions @xcite , and are different from the most - studied model of perfect uncorrelated traps distributed randomly with relative concentration @xmath21 . for that model , a good approximation for short time scales is given by the rosenstock mean - field expression @xcite @xmath22 where @xmath23 is the number of distinct sites visited by a walker . let @xmath24 denote the root - mean - square displacement of the walker @xmath25 , where @xmath8 is the dimension of the random walk . one recovers stretched exponential relaxation of @xmath0 from ( [ ros ] ) whenever exploration is _ compact _ , i.e. all sites within radius @xmath24 of the origin are visited with equal probability . in that case @xmath26 where @xmath2 and @xmath27 are the space fractal and spectral dimensions , respectively . then it follows from ( [ ros ] ) and ( [ visits ] ) that the survival probability has the stretched exponential form @xmath28 with @xmath29 and stretching exponent @xmath30 . at long time scales , instead of the power law kinetics we found for fractally correlated traps , the models with _ uncorrelated _ perfect traps predict stretched exponential decay . this time , it is due to a more subtle mechanism related to rare spatial fluctuations in the trap distribution . due to the presence of arbitrarily large trap - free regions where the walker can survive for a long time , the survival probability decays slower than exponentially , following a stretched exponential function with stretching exponent @xmath31 for euclidean host lattices @xcite and @xmath32 for fractals @xcite . the same asymptotic behavior was also found to hold for randomly distributed _ imperfect _ traps @xcite . while some authors argue that stretched exponential long - time decay due to this mechanism occurs only when @xmath0 drops to extremely low values and thus is not practically observable , more recent simulations show that the mechanism may manifest itself in an experimentally observable range of values of @xmath0 @xcite . thus , one will observe that the survival kinetics of uncorrelated and fractally correlated traps are qualitatively different , for both short and long time scales . nevertheless , we shall show that the asymptotic behavior of the survival probability @xmath0 for the model with fractally correlated traps can be accounted for by simple heuristic arguments based on the concept of compact exploration , i.e. in a manner not so different from that outlined above for the case of uncorrelated traps . the layout of the paper is as follows . in section 2 we formulate the model above in detail and use heuristic arguments to predict that initial relaxation for weakly absorbing traps takes a stretched exponential form . in sections 3 through 6 we verify this prediction for a number of specific systems and discuss relevant simulation techniques , still focusing on the initial relaxation stage in systems with weak traps . the crossover to power law relaxation for longer time scales and strongly absorbing traps are discussed in section 7 . some concluding remarks appear in section 8 . consider a classical particle taking discrete steps at a constant hopping rate @xmath1 between nearest - neighbor sites of a host lattice @xmath33 of dimension @xmath2 , with static imperfect traps interspersed according to a sublattice @xmath34 of dimension @xmath35 . we characterize the trapping imperfection by a finite _ absorption rate _ @xmath5 , at which the survival probability decays on each trap . this induces the following master equation for @xmath36 , the probability that the particle is at site @xmath37 at time @xmath38 : @xmath39 where @xmath40 is the set of immediate neighbors of @xmath37 . we shall assume that the _ coordination number _ , i.e. the number of neighbors @xmath41 , is the same for all sites @xmath42 of the host lattice , except those on the boundary . in this case , the master equation takes the form @xmath43 the problem is then to find the survival probability @xmath44 the two characteristic parameters of the problem are the ratio @xmath16 of the absorption and hopping rates , and the the probability @xmath45 that a walker arriving at a trap will be absorbed : @xmath46 the above expression for @xmath45 can be obtained as the probability that absorption occurs before the particle can hop to a neighboring site @xmath47 , where the exponential term gives the ( poissonian ) probability that the particle neither leaves the trap nor is absorbed over the interval @xmath48 , and @xmath49 is the probability that absorption occurs in the interval @xmath50 . the limits of weak and strong absorption correspond to @xmath51 and @xmath52 , @xmath53 , respectively . in the simulation we approximate the process described by ( [ master22 ] ) by averaging over discrete random walks with time increment @xmath54 and jumping probability @xmath55 . upon arrival at a trap site , the particle is , in its next step , either annihilated with probability @xmath45 , or jumps to one of its @xmath56 neighboring sites , each with probability @xmath57 . in some special cases the master equation ( [ master22 ] ) is amenable to analytic treatment , particularly in the continuous limit @xcite . however for the general case we can glean some insight by using qualitative mean - field arguments , briefly mentioned in the previous section . first , summing over @xmath37 one obtains from ( [ master ] ) @xmath58 where @xmath59 is the probability that the particle survived up to moment @xmath38 , and occupies a trap at that moment . it may be instructive ( particularly for the purpose of simulation ) to define the probabilities @xmath0 and @xmath59 explicitly for an ensemble of @xmath60 particles , @xmath61 where @xmath62 is the number of particles that survived up to moment @xmath38 , and @xmath63 the number of survivors which at that moment occupy a trap . the above expression for @xmath59 can be also presented as @xmath64 therefore @xmath59 can be written in the form @xmath65 where the function @xmath66 has the meaning of the conditional probability that a particle that survived up to moment @xmath38 occupies a trap . from ( [ exact ] ) and ( [ factorization ] ) one obtains for the survival probability @xmath0 the equation @xmath67 which is still exact but of little help unless one knows the function @xmath68 in an explicit form , or its relation to @xmath0 . definition ( [ p_exact ] ) suggests a straightforward way to evaluate @xmath68 in a simulation for any absorption rate ; we shall briefly discuss the results of such evaluation at the end of section 7 . but in fact , for a system with weakly absorbing traps @xmath69 , we can get a simple analytical approximation of @xmath68 by speculating that at sufficiently small time scales one can neglect the effects of absorption on the occupation of traps : @xmath70 where @xmath71 is the number of particles occupying traps when the absorption rate is zero . in other words , the approximation @xmath72 is the probability of occupying a site on @xmath73 in a system with absorption turned off . an explicit form of the function @xmath72 is easy to construct for lattices of spectral dimension @xmath74 using a compact exploration argument . we first suggest that @xmath75 where @xmath76 and @xmath77 are the average numbers of distinct sites visited by the particle , in @xmath78 and @xmath79 respectively , up to time @xmath38 in a system with @xmath80 . if @xmath81 ( or @xmath82 ) then we ensure that diffusive exploration is compact , and therefore @xmath77 and @xmath76 approximate the average number of sites in @xmath73 and @xmath33 within a radius @xmath83 , i.e. the root - mean - square displacement of the particle on a trap - free lattice . this gives @xmath84 substitution of ( [ aux111 ] ) into ( [ p1 ] ) gives for @xmath72 an asymptotic power law @xmath85 then the corresponding solution of ( [ f ] ) @xmath86 has stretched exponential form @xmath87 with @xmath88 , @xmath89 the empirical constant @xmath90 in ( [ f3 ] ) remains undefined , but is expected to be of order of one . the experimental evidence of such a relaxation would be a linear dependence of @xmath91 versus @xmath92 , with slope @xmath10 . in order to facilitate comparison with discrete time simulation , it is convenient to express @xmath16 in terms of the absorption probability @xmath45 . rearranging ( [ gamma ] ) , we obtain @xmath93 . by also taking into account that the time unit in our simulation is @xmath94 , we may then write ( [ f3 ] ) as @xmath95 the empirical constants @xmath90 in eqs . ( [ f3 ] ) and ( [ f3 m ] ) differ by a factor of @xmath96 . although compact exploration only holds for euclidean lattices when @xmath97 , we shall see in section 6 that our approach still produces a reasonable approximation when @xmath98 . on the other hand , many types of fractals satisfy the condition of @xmath99 . in particular , for random walks on a critical percolation cluster , it holds for any dimension of the embedding lattice @xcite . it is not _ a priori _ clear how to extend the above reasoning to the case where @xmath100 , and exploration is no longer compact . we leave this as an open question , and shall not discuss it below . one expects that the above mechanism of stretched exponential relaxation is limited in both short and long time scales . on one hand , the mean - field - like expression ( [ p1 ] ) and scaling relations ( [ aux111 ] ) presuppose that the walker has performed many steps , @xmath101 ; the simulations below give an empirical lower bound of between 10 and 100 steps . on the other hand , the above estimation of @xmath68 assumes that the visiting frequency of trapping sites is not affected by annihilation , which implies that @xmath0 must be close to one . from ( [ f3 ] ) one estimates the upper bound to be @xmath102 . thus the validity domain of the proposed mechanism for an infinite system is expected to be @xmath103 for a finite system of size @xmath104 , the upper bound is a minimum of @xmath17 and @xmath105 . we stress again that while the interval ( [ validity ] ) corresponds only to the initial stage of relaxation , the absolute duration of this stage for weakly absorbing traps ( @xmath106 ) may be significant . for times much larger than @xmath107 , the approximation ( [ p0 ] ) for @xmath68 ceases to be valid , and simulation shows that stretched exponential relaxation is replaced by power law decay @xmath9 with the same @xmath10 given by ( [ alpha ] ) . we shall postpone detailed discussion of this regime until section 7 . the previous argument relies on the assumption that traps are weakly absorbing . for strong absorption rates @xmath108 , including the limit of perfect traps @xmath18 , the validity interval ( [ validity ] ) of the stretched exponential regime is inconsistent and , as we shall see , the approximation ( [ p_approx ] ) for @xmath68 is not valid at any time . in this case , as we discuss in section 7 , the stretched exponential regime is absent and @xmath0 , after a short transient period , follows the same power law as for weakly absorbing traps at long time scales . another restriction on our heuristic argument for the stretched exponential decay of @xmath0 is that the initial location of the walker must be on , or within a fixed distance of , a trap . otherwise ( e.g. if an initial site is chosen randomly on the host lattice ) the second asymptotic relation in ( [ aux111 ] ) may be invalid . if the initial distance between the particle and a trap is @xmath109 ( in lattice spacing units ) , then one will expect that the above reasoning starts to work only after the time needed for a walker to reach a trap , @xmath110 . in this case , instead of ( [ validity ] ) one expects a validity interval with a higher lower bound , namely @xmath111hence for the validity interval to be significant we require that @xmath112 , which again can only be a meaningful condition when absorption is weak , @xmath113 . note that mathematically our model is similar to the class of defect - diffusion models of dipole relaxation , which assume that dipole reorientation ( relaxation ) is triggered by mobile defects @xcite . in this case the rate equation for the fraction of surviving dipoles @xmath0 has the form ( [ f ] ) , where @xmath68 is now the diffusive current of defects . if the latter is characterized by power law decay like ( [ p0 ] ) , then this model is formally equivalent to ours . the two models do , however , employ very different mechanisms to argue power law decay of @xmath68 . in our model it is due to fractal spatial correlations of traps , whereas in defect - diffusion models it is due to dispersive transport of defects ( in which case @xmath10 is typically temperature - dependent ) . the difference is also reflected in the fact that the validity range of defect - diffusion models is not restricted by the initial time interval and , when relevant , is capable of describing a much larger section of the relaxation function than the model we discuss here . as a final note for this section , we assumed above that the exponent @xmath114 in ( [ p0 ] ) is less than one . for the special case where @xmath115 , the above reasoning leads , instead of to stretched exponential decay , to a power law @xmath116 with an empirical constant @xmath90 . as a simple test of the qualitative argument outlined in the previous section , consider the limiting setting when the host lattice is one - dimensional , diffusion is regular , and the trapping sublattice consists of a single trap site : @xmath117 for a weakly absorbing trap we expect the initial relaxation of the survival probability to follow a stretched exponential law with @xmath118 . if the trap site is at position @xmath119 , then the master equation ( [ master22 ] ) has the form @xmath120 with the initial condition @xmath121 that the particle starts at the initial site @xmath109 , which may coincide with the trap if @xmath122 is finite . ( [ exact ] ) takes the form @xmath123 , and the survival probability is determined by the probability to be on the trap site @xmath124 : @xmath125 this problem is exactly solvable in the continuous limit ( see @xcite and references therein ) . we outline the solution in the appendix and show that over interval ( [ validity_m ] ) , which in this case becomes @xmath126 the survival probability has the approximate form @xmath127 with @xmath128 . this is consistent with prediction ( [ alpha ] ) : over interval ( [ validity2 ] ) for small @xmath16 , expression ( [ analytic ] ) is a good approximation of the the stretched exponential function with @xmath129 , @xmath130 for random walks on a line with a single trap for different values of the absorption probability @xmath45 . for all curves , the initial position of the walker coincides with the position of the trap . solid lines show the simulation results ( averaged over about @xmath131 trajectories ) and dashed lines show corresponding stretched exponential curves according to eq . ( [ ser2 ] ) with @xmath132.,height=234 ] for random walks on a line with absorption probability @xmath133 and different initial positions @xmath109 . a single trap is located at the origin , @xmath119.,height=234 ] we found the result to be in good agreement with numerical simulation . a comparison of ( [ ser2 ] ) with a numerical experiment for different ( small ) values of the absorption probability @xmath45 and initial position @xmath134 is presented in fig . 1 . according to ( [ ser2 ] ) , a log - log plot of the function @xmath135 , i.e. the plot of @xmath91 versus @xmath136 , must appear to be a straight line with slope given by the exponent @xmath129 . our simulation confirms this prediction , and shows an increase in the duration of its validity domain for smaller @xmath45 and @xmath16 in a way that is consistent with ( [ validity2 ] ) . all curves show deviation from stretched exponential relaxation for small @xmath38 when @xmath137 is of order ten . deviation for large @xmath38 is easily noticeable for @xmath138 and @xmath139 , but is beyond the experiment s time range for the curves with @xmath133 and @xmath140 . 2 shows similar plots for the same value @xmath133 and different initial positions @xmath141 of the particle . in those cases , the stretched exponential regime emerges after a transient time which increases quadratically in @xmath109 ; this is in agreement with ( [ validity2 ] ) . similar results hold for the mathematically equivalent problem of a two - dimensional host lattice with traps on a one - dimensional line : @xmath142 in this case , equation ( [ alpha ] ) predicts stretched exponential relaxation with @xmath129 , and numerical simulation confirms this over the time interval ( [ validity2 ] ) . the simulations also show agreement with theoretical predictions for other types of simple ( euclidean ) trap configurations embedded in one- and two - dimensional host lattices . in the sections to follow , we consider host and trap lattices with nontrivial fractal dimensions . we shall now extend the previous example to have traps distributed along a constructive analogue of the cantor set . we shall refer to the host lattice , together with the traps , as the cantor lattice . in order to simplify the enumeration of trapping sites , in this and the following sections we shall associate hopping sites with lattice cells , rather than the cells end points . unlike the cantor set , which is defined `` from the outside , in '' by starting from the ( uncountable ) interval @xmath143 $ ] and recursively removing the middle third of every resulting subinterval , we employ an `` inside - out '' ( and countable ) iterative construction ; see fig . 3 . the first generation @xmath144 consists of three consecutive cells , the first and last of which are traps . then we define the @xmath145-th generation recursively to be two copies of @xmath146 flanking a trap - free @xmath146-sized block . hence each @xmath147 has size @xmath148 and contains @xmath149 traps . one refers to each generation @xmath147 as a _ finite _ cantor lattice , and the limit @xmath150 as _ the _ cantor lattice , characterized by the following dimensions : @xmath151 according to the heuristic argument in section 2 , the survival probability for weakly absorbing traps is expected to be characterized by stretched exponential decay ( [ f3 ] ) with the exponent @xmath152 we verified this prediction with numerical simulations of @xmath131 random walks , each starting from a random trap cell , on the finite cantor lattice @xmath153 , which consists of over @xmath154 cells . in fact one could go so far as to dynamically generate the cantor lattice , but averaging over initial conditions for a sufficiently large but finite lattice like @xmath153 is simpler , and the size effects are negligible . ( we found empirically that for random walks of about @xmath155 steps , finite - size effects become noticeable only for lattices smaller than @xmath156 . ) in lieu of explicitly storing the locations of traps , which would be infeasible , we will exploit a property of the natural left - right enumeration ( see fig . 3 ) of any given generation . let @xmath37 be the label of a cell in @xmath157 , and @xmath158 its ternary representation , i.e. the unique sequence of @xmath159 such that @xmath160 . the convenience of this representation is that traps can be identified immediately : a site with a ternary address @xmath161 is a trap if and only if @xmath162 for all @xmath163 ; see fig . 3 . in other words , the subset @xmath73 of trap sites on @xmath157 is exactly @xmath164 the relevance of a ternary enumeration is made apparent and natural when one considers the recursive composition of the cantor lattice : every generation consists of left , right , and middle parts of equal length , and only the latter is guaranteed to be trap - free . then one may read the ternary label @xmath165 as a sequence of choices : @xmath166 correspond to the left , middle , and right @xmath167-sized blocks within @xmath157 , and so on for each successive digit . for example , in the @xmath168 lattice , the cell labeled @xmath169 is in the left @xmath170 block , since @xmath171 . that specific @xmath170 block is made of three @xmath144 blocks , and @xmath172 indicates that we are in the middle one . finally , within that specific @xmath144 block the cell is on the right , so @xmath173 . on the other hand , the cell has the decimal label @xmath174 , which is precisely the decimal representation of the ternary number @xmath169 . for @xmath131 random walks on a @xmath153 cantor lattice , for several values of the absorption probability @xmath45 . solid lines show the simulation results , dashed lines show the corresponding stretched exponential curves according to eq . ( [ f3 m ] ) with @xmath175 , as given by ( [ alpha3 ] ) , the coordination number @xmath176 , and the empirical parameter @xmath177 . each trajectory begins from a randomly selected trap.,height=234 ] simulation results for @xmath153 are shown in fig . after a transient of about ten steps , the initial relaxation of the survival probability closely follows stretched exponential kinetics ( [ f3 ] ) with the exponent @xmath10 given by ( [ alpha3 ] ) . one observes that the upper time bound for this behavior increases as the absorption probability @xmath45 decreases , in a way consistent with prediction ( [ validity ] ) . as @xmath38 exceeds the interval of validity , the transition to slower ( power law ) relaxation occurs , which in fig . 4 is visible for @xmath178 . for @xmath140 and @xmath179 the transition is beyond the simulation s time scope . we postpone discussion of the long - time power - law relaxation regime and systems with strong absorption until section 7 . if the initial site were not a trap , but instead chosen to be at a given distance @xmath180 away from a ( randomly chosen ) trap , then the relaxation curves would have a form similar to that in fig . 2 , i.e. approaching stretched exponential form at long times scales , with a transition time increasing with @xmath180 . for our next example , we will consider random walks on a constructive variant of the ( fractal ) sierpinski gasket , with traps located on a one - dimensional subset , namely the bottom `` edge '' ; see fig . this case differs from the two previous examples , in that diffusion on the host lattice is anomalous , with @xmath181 . here the set of relevant dimensions is @xcite @xmath182 we expect for the initial relaxation of the survival probability to have the stretched exponential form ( [ f3 ] ) with the exponent @xmath183 as long as the starting point of each random walk is on or near a trap and absorption is weak @xmath184 . numerical simulation confirms this prediction with time bounds similar to those for the two previous cases ; see fig . below we discuss some technical details of the simulation , which for sierpinski lattices has some peculiarities of its own . for random walks on the sierpinski lattice @xmath153 with a one - dimensional sublattice of traps ( see fig . 5 ) for different values of the absorption probability @xmath45 . solid lines show the simulation results ( averaged over about @xmath131 trajectories ) , dashed lines show the corresponding stretched exponential curves according to ( [ f3 m ] ) with @xmath185 , as given by ( [ alpha5 ] ) , the coordination number of traps @xmath186 , and the empirical constant @xmath187 . initial sites are chosen randomly from the trap lattice @xmath73.,height=234 ] for the same reasons as the cantor lattice , it is convenient to generate our sierpinski lattice using a recursive `` inside - out '' construction . this induces a natural ternary enumeration of the lattice cells , depicted in fig . 5 . as in the previous section , the hopping sites of the walk are the cells of the lattice , which this time are the `` elementary '' triangles of every generation @xmath157 . the lattice of the first generation @xmath144 consists of three triangles whose positions are labeled @xmath188 ( left ) , @xmath189 ( top ) , and @xmath190 ( right ) . the lattice of the second generation @xmath170 consists of three @xmath144 blocks , whose three positions `` left '' , `` top '' , and `` right '' are again denoted @xmath188 , @xmath189 , and @xmath190 respectively . the three @xmath170 blocks compose in a similar manner to make the third generation lattice @xmath168 , and the process may be repeated to any desirable order . hence we may once again use the ternary addressing scheme wholesale , and the traps are exactly those sites without any digits equal to 1 . indeed the only difference , from the perspective of simulation , between this and the preceding section is that the set of neighbors for each cell has changed . as in the preceding section , the simulation was carried out on the lattice @xmath153 , consisting of @xmath191 cells , which we found to be large enough for finite - size effects to be negligible . as we foreshadowed , in order to simulate random walks on a sierpinski lattice of generation @xmath192 , one needs an algorithm for determining the neighbors of a given cell . suppose the cell has label @xmath193 two of its neighbors ( both for an apex cell ) must belong to the same @xmath144-block as @xmath15 , meaning the neighbors labels @xmath194 and @xmath195 differ from @xmath15 only by the final digit : @xmath196 for example , the cell at @xmath169 in the @xmath168 lattice has two neighbors from the same @xmath144-block , with labels @xmath197 and @xmath198 ; see fig . 5 . however , the algorithm for finding the label @xmath199 of the third neighbor is more involved @xcite . first , for a given cell with the label ( [ label ] ) , one checks whether @xmath200 . if the condition is satisfied , i.e the label has the form @xmath201 then the cell and its third neighbor belong to different @xmath144 blocks but to the same @xmath170 block . in this case the label of the third neighbor @xmath199 has the same first @xmath202 digits as @xmath15 , while the last two digits replace each other : @xmath203 for example , for the cell with @xmath204 , the third neighbor is labeled @xmath205 ; see fig . 5 . now , if the condition above were not satisfied , i.e. @xmath206 , then one must check whether @xmath207 , i.e. @xmath208 in this case the cell and its third neighbor belong to different @xmath144 and @xmath170 blocks , but to the same @xmath168 block . then the label of the third neighbor @xmath199 has the same first @xmath209 digits as the label @xmath15 ( [ label3 ] ) , while the last three digits @xmath210 are replaced by @xmath211 , @xmath212 for example , in @xmath168 the cell with @xmath213 has its third neighbor labeled @xmath214 ; see fig . 5 . one will by now see the pattern of this method . proceed as above until meeting the condition that @xmath215 , in which case @xmath216 then the label of the third neighbor has the form @xmath217 of course , if all the digits @xmath218 are equal , then @xmath15 labels an apex cell , and there is no third neighbor . for example , in @xmath219 of the sierpinski lattice the cell with @xmath220 has two neighbors with labels given by ( [ label0 ] ) , namely @xmath221 and @xmath222 , and the third neighbor with the label @xmath223 given by ( [ label5 ] ) ( with @xmath224 ) . the construction , enumeration , and nearest - neighbor algorithms outlined in this section can be readily extended to sierpinski lattices of higher dimensions . for our final example , we consider random walks on a two - dimensional euclidean lattice with traps forming a sierpinski gasket ; see fig . 7 . in this case , @xmath225 and according to sec . 2 the stretching exponent should be @xmath226 one might worry more about the validity of this prediction than for our other examples because , strictly speaking , a random walk in two dimensions is not compact : the average number of distinct visited sites @xmath76 increases in 2d as @xmath227 @xcite , rather than linearly , as anticipated by the compact - exploration ansatz ( [ aux111 ] ) . yet one may expect that the slowly varying logarithmic factor can be approximated without much error by a constant , so that the prediction ( [ alpha6 ] ) may still be justified . numerical simulation supports this optimism : the slope of @xmath0 in the double - logarithmic scale is found to be in good agreement with ( [ f3 m ] ) and ( [ alpha6 ] ) ; see fig . similarly to the previous two sections , in our simulation we construct the 2d lattice with the embedded sierpinski gasket by composing higher generations from lower ones recursively , as shown in fig . simulation results presented in fig . 8 are for the lattice @xmath153 , with periodic boundary conditions . simulations for lattices larger than @xmath156 and with other types of boundary conditions show very similar results , indicating that finite - size effects are negligible . for random walks on a 2d lattice with traps on the sierpinski gasket @xmath153 ( see fig . 7 ) for different values of the absorption probability @xmath45 . solid lines show the simulation results ( averaged over about @xmath131 trajectories ) , dashed lines show the corresponding stretched exponential curves according to ( [ f3 m ] ) with @xmath228 , as given by ( [ alpha6 ] ) , the coordination number @xmath229 , and the empirical constant @xmath230 . initial sites are chosen randomly from among the trap sites.,height=230 ] cells are enumerated by pairs of cartesian coordinates @xmath231 expressed in binary . for a lattice @xmath232 of generation @xmath192 , binary coordinates have @xmath192 digits @xmath233 where each digit @xmath234 has a value of either zero or one ; see fig . the binary enumeration is convenient because traps ( i.e. cells belonging to the sierpinski gasket ) can be identified as those , and only those , cells whose binary addresses satisfy the condition that the sum of digits in every position does not exceed one , @xmath235 ( in other words , the bitwise ` and ' operation applied to @xmath236 and @xmath237 is zero if and only if the site is a trap . ) for example , in the lattice @xmath168 ( see fig . 7 ) for the trapping cell with the binary address @xmath238 , we have @xmath239 for @xmath240 , and therefore the condition ( [ trap_condition ] ) is satisfied . on the other hand , for the non - trapping cell with binary address @xmath241 , @xmath242 the condition ( [ trap_condition ] ) is not satisfied because @xmath243 . this method can also be used to model random walks on the sierpinski gasket , as a perhaps simpler alternative to the method described in the previous section . for random walks on the sierpinski lattice @xmath244 with a one - dimensional sublattice of traps ( see fig . 5 ) for several values of the absorption probability @xmath45 . solid lines show simulation results ( averaged over between @xmath155 and @xmath245 walks ) and dashed lines show the corresponding power law relaxation functions given by eq . ( [ power ] ) , @xmath246 , with @xmath185 and empirical constant @xmath247 . , height=230 ] for all the example systems discussed above , simulation shows that for weakly absorbing traps ( @xmath248 ) the initial stretched exponential kinetics are replaced at longer time scales @xmath249 by algebraic decay with the same exponent @xmath10 as for the stretched exponential regime , and with a prefactor proportional to the inverse absorption probability , @xmath250 in this case , double - logarithmic plots of @xmath0 at long times become straight lines with slope @xmath251 . as the absorption rate increases , the crossover time @xmath15 decreases , and the stretched exponential regime becomes less visible . for strongly absorbing ( @xmath52 ) and perfect ( @xmath252 , @xmath253 ) traps , @xmath0 follows power law kinetics for all times , except very short ones . 9 illustrates this behavior for random walks on the sierpinski lattice described in section 5 , considering generation @xmath244 with initial coordinates chosen randomly from the sites adjacent to a trap . for other systems the results are similar ( but curiously , the graph produced by the cantor lattice has some small but noticeable ripples , even after averaging over a great many walks ) . 9 reveals that the long - time behavior of the survival probability @xmath0 is the same for imperfect ( @xmath254 ) and perfect ( @xmath253 ) traps . this is a well - known result for the case of a _ single _ trap ( see appendix and @xcite ) , and we see that it persists for a network of correlated traps as well . it suggests that on a long time scale , regardless whether traps are perfect , the kinetics of absorption are controlled not by occupation statistics ( as we assumed in section 2 when evaluating @xmath68 , the probability to occupy a trap ) , but rather by first passage time ( fpt ) statistics . while for perfect traps the relevance of the fpt statistics is obvious , for imperfect traps it can be understood intuitively by speculating that the main contribution to the survival probability at long time - scales comes from particles performing long excursions in large trap - free regions . the duration of such an excursion is essentially the time of first return to the absorbing lattice , and assumed to be much larger than the time the particle spends after returning to a trap - rich region . we show below that this picture leads naturally to the long - time asymptotic behavior of ( [ power ] ) . in standard fpt problems , one seeks to evaluate the fpt distribution @xmath255 , or its moments . the former is is the probability density that a particle , starting from the origin , will hit a specific target point located at the distance @xmath256 from the origin at time @xmath38 . for our needs we generalize the problem , replacing the point - like target by an extended network - like target , as follows : _ random walks are performed on a lattice @xmath33 of dimension @xmath2 with targets forming a proper sublattice @xmath257 of dimension @xmath4 . at @xmath258 , an initial position is chosen randomly on @xmath73 . find the distribution function @xmath259 that a particle will return to @xmath73 ( not necessarily the initial position ) for the first time at time @xmath260 . _ let @xmath72 be a probability density for a particle to occupy the target sublattice @xmath73 at time @xmath38 , provided it was on @xmath73 at @xmath258 . ( as in section 2 , the subscript @xmath188 indicates that @xmath72 is evaluated with absorption turned off ; the targets in the above fpt problem are not traps , perfect or otherwise . ) clearly , the fpt probability distribution @xmath259 and the occupation probability distribution @xmath72 are related in the same way as for a point - like target @xcite , @xmath261 here the first term on the right reflects the initial condition of being on @xmath73 at @xmath258 , and the second term says that to occupy @xmath73 at time @xmath38 the particle must hit it for the first time at some moment @xmath262 and then return to @xmath73 after time @xmath263 . the corresponding equation for laplace transforms ( which we denote by tildes ) reads @xmath264 , so that @xmath265 as was shown in section 2 , the compact exploration argument gives for @xmath72 the asymptotic scaling ( [ p0 ] ) , @xmath266 . then according to the tauberian theorem @xcite , @xmath267 for small @xmath268 , and from ( [ f_laplace ] ) one obtains @xmath269 . in the long time domain this corresponds to power law decay of the fpt distribution , @xmath270 for a single point - like target @xmath271 , @xmath272 , and ( [ fpt ] ) takes the form @xmath273 . this specific form of the result ( [ fpt ] ) was obtained and tested in @xcite . of surviving until , and occupying a trap at , time @xmath38 ( see eq . [ p_exact ] ) . it is evaluated over @xmath274 walks on the 2d lattice with imperfect traps on the sierpinski gasket @xmath153 ( see fig . 7 ) for several values of the absorption probability @xmath45 . solid lines show simulation results , the dashed line shows the dependence ( [ p0 ] ) , @xmath275 , and dash - dotted lines represent the ansatz @xmath276 . , height=234 ] with the fpt distribution found , we now return to the trapping problem . we identify the target network with the trap sublattice and appeal to the above reasoning that on long time - scales the time of absorption is approximately that of the time of first return on the trap sublattice , which is distributed according to ( [ fpt ] ) . the survival probability @xmath0 is the probability that first passage occurs at @xmath277 , so one immediately recovers the power law ( [ power ] ) , @xmath278 although it is not formally recovered in this derivation , the prefactor @xmath279 which we found experimentally ; see ( [ power ] ) is intuitively to be expected in ( [ power2 ] ) . one may alternatively seek to account for algebraic asymptotic decay @xmath280 in terms of the conditional probability @xmath68 of occupying a trap ( [ p_exact ] ) . namely , this behavior can be formally derived as a solution of equation ( [ f ] ) , @xmath281 , when @xmath68 has the form @xmath282 . indeed , numerical simulation shows that at large time scales there is a crossover from @xmath266 to @xmath283 ; see fig . we find it difficult , however , to justify or interpret the above ansatz for @xmath68 theoretically , and numerically the function @xmath68 is hard to evaluate ; it quickly becomes very small and , unlike @xmath0 , fluctuates wildly even when evaluated over a very large number of walks . stretched exponential and power laws are the two most commonly observed heavy - tailed distributions in disordered and complex systems , and for this reason are often believed to originate from very general mechanisms . in particular , power law kinetics are often a signature of continuous - time random walk processes @xcite characterized by a broad distribution of transition rates or waiting times , and there are several models @xcite still competing as generic explanations for the origin of stretched exponential kinetics . from this perspective , the emergence of both these distributions within the same conceptually simple model studied in this paper is perhaps remarkable . we studied the survival probability @xmath0 of random walks in the presence of fractally correlated traps . for imperfect , weakly absorbing traps , the initial relaxation of @xmath0 is stretched exponential , followed by power law decay , with both regimes characterized by the same exponent @xmath10 . the regime of stretched exponential relaxation is shorter for strongly absorbing traps , but may hold over several orders of magnitude in time for weakly absorbing traps . both regimes may be accounted for by arguments based on the concept of compact exploration , applied to evaluate pertinent occupational and first - passage time distributions , for stretched exponential and power law regimes respectively . we illustrated and verified theoretical predictions with monte carlo simulations for regular host and trap lattices , but we also expect these results for random fractal lattices like critical percolation clusters @xcite and multidimensional potential landscape structures relevant to complex systems with correlated disorder @xcite . in the latter case , imperfect correlated traps may correspond to deep potential valley regions separated from the relaxation pathway by a potential ridge . theoretical arguments employed in the paper imply that the host and trap lattices are infinite , and in the simulation we tried to minimize the effects of boundary conditions . the enumeration algorithms employed in this paper allow one to simulate random walks on very large fractal structures . all results presented are for fractals of generation @xmath284 , each consisting of at least @xmath154 units , which we found to be sufficiently large to neglect finite - size effects . for the time scale considered ( @xmath285 ) , we found empirically that finite size effects become noticeable only for much smaller structures of generation @xmath286 . while specific forms of finite size effects depend on boundary conditions , we found as a general trend that they make the survival probability @xmath0 decay faster at long times than in an infinite system . this is intuitively clear , since in an infinite fractal system a particle finds itself , as time progresses , in larger and larger trap - free regions , whereas in a finite system the maximum size of a trap - free region is fixed . we thank g. buck , j. schnick , s. shea , and j. parodi for discussions and interest , and the anonymous referees for their insightful comments and suggestions . in this appendix we outline a method for the evaluation of the survival probability @xmath0 for a random walk on a one - dimensional lattice with an imperfect trap located at the origin . the problem is described by the master equation ( [ master2 ] ) . using the standard laplace and discrete fourier transforms , one can readily find from that equation the exact expression for the laplace transform of @xmath0 , whose exact inverse however is unknown . more analytic progress can be achieved by considering , instead of the exact master equation ( [ master2 ] ) , its continuous limit version @xmath287 for the probability density @xmath288 . this equation follows from ( [ master2 ] ) after replacements @xmath289 , @xmath290 , @xmath291 , and taking the limits @xmath292 and @xmath293 with finite @xmath294 let @xmath295 be a free - diffusion propagator , that is , a solution of the trap - free diffusion equation ( eq . ( [ a_master ] ) with @xmath296 ) with the initial condition @xmath297 . then the solution of eq . ( [ a_master ] ) with initial condition @xmath298 can be expressed as follows @xcite : @xmath299 this expression is easy to interpret : @xmath288 is smaller than the free - diffusion propagator @xmath300 by the contribution from the particles , which were captured by the trap at an earlier time @xmath301 and , had they not been captured , would have diffused to the point @xmath236 at time @xmath38 . the negative contribution of such particles is given by the second term in the right - hand - side of ( [ a_sol1 ] ) . in the laplace space , ( [ a_sol1 ] ) reads @xmath302 where the tilde denotes laplace transforms . from here one finds @xmath303 and substituting this expression back to ( [ a_sol2 ] ) one gets @xmath304 for the laplace transform of the survival probability this yields @xmath305 substituting the laplace transform of the propagator ( [ a_propagator ] ) @xmath306 one finds @xmath307 . \label{a_image0 } \end{aligned}\ ] ] one expects this result , obtained in the continuous limit , to be a reasonable approximation for a discrete lattice as well . for that case , taking into account ( [ a_d ] ) , we can re - write ( [ a_image0 ] ) using notation from the main text , @xmath308 , \label{a_image } \end{aligned}\ ] ] where @xmath309 is the initial position in lattice spacing units , and @xmath310 is the dimensionless parameter characterizing the absorption strength . while this laplace transform enjoys an exact closed - form inverse , see eq . ( [ a_exact ] ) below , the asymptotic forms of @xmath0 can be derived from that of ( [ a_image ] ) . consider first the case of weak absorption , @xmath311 then for the domain @xmath312 ( [ a_image ] ) can be approximated as @xmath313 interval ( [ a_domain1 ] ) corresponds to the time domain @xmath314 for which the inversion of ( [ a_approx1 ] ) gives @xmath315 this is a good short - time approximation of the stretched exponential function ( [ ser2 ] ) . on the other hand , for @xmath316 ( [ a_image ] ) is reduced to @xmath317 in the time domain this corresponds to a power law asymptotics @xmath318 for @xmath319 . if absorption is not small , @xmath108 , the conditions ( [ a_domain2 ] ) and @xmath320 become inconsistent , and the stretched exponential regime is absent . in this case , for @xmath321 one obtains from ( [ a_exact ] ) instead of ( [ a_qq ] ) , the approximation @xmath322 in the time domain this corresponds to @xmath323 for @xmath324 . in particular , for the limit of a perfect trap @xmath18 , @xmath325 the exact inverse of ( [ a_image ] ) has the form @xcite : @xmath326 ( in the limit of a perfect trap @xmath18 only the second term survives in this expression . ) the above asymptotic formula may alternatively be obtained directly from ( [ a_exact ] ) using asymptotic relations @xmath327 for @xmath328 and @xmath329 for @xmath330 .

we consider the survival probability @xmath0 of a random walk with a constant hopping rate @xmath1 on a host lattice of fractal dimension @xmath2 and spectral dimension @xmath3 , with spatially correlated traps . the traps form a sublattice with fractal dimension @xmath4 and are characterized by the absorption rate @xmath5 which may be finite ( imperfect traps ) or infinite ( perfect traps ) . initial coordinates are chosen randomly at or within a fixed distance of a trap . for weakly absorbing traps ( @xmath6 ) , we find that @xmath0 can be closely approximated by a stretched exponential function over the initial stage of relaxation , with stretching exponent @xmath7 , where @xmath8 is the random walk dimension of the host lattice . at the end of this initial stage there occurs a crossover to power law kinetics @xmath9 with the same exponent @xmath10 as for the stretched exponential regime . for strong absorption @xmath11 , including the limit of perfect traps @xmath12 , the stretched exponential regime is absent and the decay of @xmath0 follows , after a short transient , the aforementioned power law for all times .

motivated by the seemingly intrinsic nature of the glassy behavior in such crystalline spin glasses , we explore a heisenberg model on a magnetic lattice realized in scgo and qs - ferrites . the magnetic lattice of interest is a triangular network of bi - pyramids that are formed by two corner - sharing tetrahedra and are connected by linking triangles ( see fig . 1a , d ) . here we consider a simple nearest neighbor spin interaction hamiltonian @xmath2 . classically , any spin configuration in which each tetrahedron and linking triangle has a total zero spin is a ground state . there are an infinite number of energetically equivalent configurations . an important subset of these states is the set of states in which the spins in each bi - pyramid are collinear@xcite . it is well established that collinear configurations are commonly favored in frustrated magnets , which , as we will show later is also the case for the model at hand . note , in passing , that in most cases co - linearity of spin configurations is global over the entire lattice , while here the collinear direction is not global . henceforth , we will refer to such states simply as locally collinear ( lc ) states . lc states can be conveniently explored as a simple problem of two degrees of freedom : tri - color ( representing the three types of spins for the @xmath3 configuration for the afm linking triangle ) and binary sign ( representing the parallel ( + ) or anti parallel ( - ) direction of each spin within a collinear bi - pyramid of given color ( fig . the triangular network of the bi - pyramids forces the tri - color to order long range in a @xmath4 structure as shown by circles in blue , green and red in fig . there are 18 possible sign configuration per bi - pyramid , and the sign degrees of freedom are constrained to have the same sign for each linking triangle that connects each three neighboring bi - pyramids @xcite . spin liquid candidate systems , such as kagome and pyrochlore , where any spin configurations with zero spin triangles and zero spin tetrahedra , respectively , are ground states , have local zero - energy modes , involving a a finite number of spins , and thus their ground state degeneracy is extensive and scales with the volume . below , however , we give an entropic argument using a tiling approach to the absence of local - zero modes in our model . absence of such modes greatly enhances the dynamical barrier to transitions between lc states , and facilitates freezing . we map each sign state into a hexagon tile as shown in fig . the six corners of the hexagon tile represent the six spins forming the upper and the lower triangles of the bi - pyramid . the tiles are chosen to have the exact matching and enumeration properties of the sign representation ( fig . 1b ) , spins on the boundary are associated with black and blue colors according to their sign . the middle spin of the bi - pyramid does not interact with other bi - pyramids , and thus we are free to choose the color of the center of the hexagon so as to create the simplest patterns that preserve the topology of the network of positive and negative spins on the boundary . even within the subset of the lc states ( sign states ) , there are numerous ways of covering the entire lattice . with the hexagon representation , the problem of counting the number of sign states in the system becomes a tiling problem . our tiling problem seems new and bears a remote visual resemblance with a two colored piecewise herringbone tiling . in order to investigate how the degeneracy increases with the size of the system , we first identified numerically all possible sign states with varying the number of column and rows of bi - pyramids , and thus varying the size of the system . as shown in fig . 2b , for a given column the number of the possible sign states , @xmath5 , increases with the number of rows in a slower rate than exponentially , which indicates that @xmath5 does not scale with the volume ( area in this quasi - two - dimensional case ) of the system . surprisingly , as shown in the inset , @xmath5 seems to scale with the number of bi - pyramids on the boundary , i.e. , the perimeter of the system . this behavior is corroborated by studying the number of states using transfer matrix methods , we find , numerically , that the largest eigenvalue of the transfer matrix corresponding to a strip of k rows , seems to decrease with k , up to 11 rows , involving 77 spins per unit length . this scaling starkly contrasts with the volume scaling of @xmath5 of the kagome and pyrochlore systems in which local zero energy modes exist . this non - extensive scaling may be viewed as a consequence of the absence of local zero energy modes . instead , the smallest unit of zero energy modes scales with the linear dimension of the system , as it involves bi - pyramids along a line , as shown in fig . 3a . the hexagon representation for the sign state of each bi - pyramid allows us to establish bounds on the number of collinear states @xmath6 for a system of volume @xmath7 and circumference @xmath8 . we find that @xmath9 , where @xmath10 are constants . in particular , for @xmath11 , we have : @xmath12 , concluding that ( up to a possible logarithmic correction ) the number of states is extensive in the boundary length . the lower bound is easy to establish : for a given boundary length @xmath8 , we can construct explicitly a number of states which scales as @xmath13 . one way of doing so is by starting from one of the long - range structures as shown in fig . these structures support straight quasi one - dimensional modes that change the state of the bi - pyramid along them . for a square sample of side @xmath8 , we can put up to @xmath8 independent parallel modes of this type , which supplies us with the lower bound . to show the upper bound we recast the sign states as a tiling problem . we use the hexagonal tiles depicted in fig . 1c to obtain a representation of the system as a network of lines . the resultant network may be considered as a fully packed network of rectilinear stripes of alternating color on a lattice , made of straight lines , @xmath14 degree turns ( `` elbows '' ) and junctions as shown in fig . 3b . the elements generating this network yield the following properties : @xmath15 . lines can not terminate , and @xmath16 . there are no closed loops . property @xmath15 can be verified by inspection of possible termination points , and ruling each of them out . to prove property @xmath16 , assume the contrary and consider a closed loop of black color , inside there must be loops of smaller and smaller sizes . since the colors alternate , we must have an enclosed simply connected region that is entirely black or entirely blue . since we do not have an entirely blue or entirely black hexagon in our disposal , such a region must be of limited thickness , therefore the inner region must be made of lines with termination points . by property @xmath15 , such termination points are not allowed . to proceed , we define a `` laminar region '' as a region where no junctions are present . property @xmath17 : in a laminar region , by definition , the lines are parallel ; moreover , for each of the lines parallel to a chosen reference line ( not necessarily a straight one ) , the thickness at any point along it can be deterministically inferred if the thickness at any other point is known . property @xmath17 is established by classifying all possible elbow points that do not involve a junction ( fig . 3b ) . properties @xmath15 and @xmath16 imply that each line must go through the boundary . property @xmath17 , shows that in a laminar region , the thickness degree of freedom of each pattern can be pushed to the boundary , moreover , any elbow must be reflected at two points on boundary of the sample , a detailed study of these properties shows that for a laminar region we can systematically reconstruct the internal state given the boundary of the region . next , we consider junctions to show that @xmath18 for any network where @xmath19 is the number of junctions . by properties @xmath15 , @xmath16 , the network is a graph with the only possible termination points on the boundary , and no closed circles : it is thus a forest ( disjoint union of trees ) , with leaves only on the boundary . an elementary fact of graph theory @xcite is that the number of nodes in a full binary tree can not exceed the number of leaves , therefore @xmath18 . we can have at most @xmath20 locations for placing junctions in the sample . there are a finite number of possible junction elements . once the locations and nature of the junctions have been established , the sample excluding the junction is a laminar region by definition , with effective boundary length proportional to @xmath21 , following observation @xmath17 , the state is determined by it s boundary . summing over possible numbers of junctions we have @xmath22 , which yields the aforementioned upper bound . thus we have proved that the configurational entropy of collinear states scales with its perimeter rather than its volume . let us now turn to the energetics of the sign states . among the myriad of the sign states , there are six long range ordered states where three types of sign bi - pyramids are arranged in a @xmath4 structure , one of which is the @xmath23 state shown in fig . 1d . once a sign state is constructed over the entire lattice , the corresponding lc state is constructed by imposing the color ordering . from the lc states , one can generate non - collinear coplanar bi - pyramid states ( henceforth coplanar states ) by collectively rotating each pair of antiparallel spins in each tetrahedron @xcite . in the mean - field level , the collective motions do not cost any energy , leading to an energy landscape with infinitely large flat bottom formed by collinear and coplanar state and thus to low temperature spin liquid behaviors@xcite . to investigate what happens when quantum fluctuations are taken into account , we have calculated the energy cost of the quantum fluctuations , within the harmonic ( holstein - primakoff ) approximation around numerous classical spin configurations of minimal energy , with up to @xmath24 bi - pyramids per sample . this is done by carrying out numerically a symplectic transformation to diagonalize the resultant bosonic hamiltonians in real space for each state , without assuming long - range order . an example of the procedure is shown in fig . 4 for @xmath25 bi - pyramids with several different cdifferent lc states as local minima . since the long range ordered sign state is special , we considered the lc states near the @xmath4 @xmath23 state that are connected with each other through coplanar states . 4 shows the results ; the degeneracy between the collinear and the coplanar states is also lifted , making the lc states local minima and creating energy barriers by the coplanar states . the degeneracy among the lc states is also lifted ; the @xmath4 sign state has a lower energy than the other sign states , making the @xmath4 long range ordered state a global minimum and the other lc states local minima . explicit enumeration shows that there are 6 possible @xmath4 sign states , giving 36 possible @xmath4 spin states when combined with the @xmath26 possible color configurations . thus quantum fluctuations lift the mean field ground state degeneracy to form @xmath27 global minima of the long range ordered lc states and numerous local minima of other lc states , the number of which scales with the perimeter of the system . since there are no local spin reorientations which connect between the mean field minima , the energy barriers between different states are huge . as a result , upon cooling , the system gets trapped in one of the local minima of collinear bi - pyramids without a long - range order . the spin freezing explicitly breaks the @xmath28 invariance of our heisenberg hamiltonian . as a result of this symmetry breaking , and the finite spin stiffness for deforming the aperiodic static antiferromagnetic spin texture , its thermodynamics at low temperatures will be governed by low - energy hydrodynamic halperin - saslow modes@xcite . such modes are linearly dispersive and lead to a @xmath29 behavior for a quasi - two - dimensional system such as ours . in conventional spin glasses where dilute magnetic ions are embedded in a nonmagnetic metal , there is also a linear in @xmath30 contribution to the specific heat due to localized two - level systems@xcite , which dominates its thermodynamics as observed experimentally@xcite . in our system , however , such a linear contribution is negligible at low doping ( see also ref . @xcite ) , leading to a @xmath31 behavior at low temperatures , consistent with the experimentally observed behavior in scgo@xcite . the concept of a rugged energy landscape was originally proposed to explain freezing phenomenon found in classic spin glass in which dilute magnetic ions in a nonmagnetic metal interact via long range rkky interactions that change with distance between the magnetic ions and even change in sign @xcite . the random magnetic interactions induce frustration , which leads to many states of nearly identical energy and a rugged energy landscape . since then , it has also been suggested to be responsible for other quenching processes that are ubiquitous in nature , ranging from gelation @xcite to metallic glass @xcite , to protein folding @xcite . in such systems , precise mapping of the complex energy landscape as a function of configurations and thus the microscopic mechanism for the freezing phenomena has been challenging . the triangular network of bi - pyramids , on the other hand , does not possess the problem of randomness , and thus provides a unique opportunity to microscopically determine the rugged energy landscape and study the mechanism of the spin freezing , as shown in this work . an important ingredient in our treatment was the tiling based proof that no classical local zero modes are allowed in the system . we remark that the relevance of tiling as model systems for glassy behavior has been extensively studied for glasses in the context of kcms@xcite . as dynamics is usually allowed only when vacancies in the system are present@xcite , a system that is highly packed ( or fully packed as in our discussion here ) will be `` stuck '' in a configuration for a very long time . sub - extensive entropy appears in some kcm models . for example the ising plaquette model on the square lattice exhibits a non - extensive entropy at low temperatures @xcite , in this model glassines is present on the classical level , and @xmath28 symmetry is broken on the hamiltonian level . the system is gapped , rendering trivial low temperature thermodynamics . in the context of spin - liquids , a checkerboard model was studied@xcite , where in a valance bond solid ( vbs ) phase , bond configurations are stripe - like , and carry entropy that is extensive in boundary length . however , we note that there , individual spins have no static moment ( in fact , spin configurational entropy is extensive in volume in that model ) . a vbs state was also suggested on a structurally different but related lattice to ours , a \{111 } slice of pyrochlore@xcite , which has also a volume scaling entropy . a nematic phase of pseudo - spin was explored in s=1 kagome antiferromagnet with a strong single ion anisotropy @xcite which is realized without a static spin moment . finally , we note that similar exotic entropy scaling has been of great interest in other branches of physics , from cosmology , where the entropy of black holes has been argued by bekenstein and hawking to scale as the boundary area @xcite , to more recently , in many body quantum mechanical model systems at zero temperature . for example , boundary extensive ground state degeneracy is a feature of some supersymmetric lattice models@xcite . a related phenomena is the scaling of entanglement entropy with the boundary area for free scalar fields @xcite , that obtains logarithmic corrections when a fermi surface is present @xcite . in our case the perimeter scaling entropy is due to the fact that the local minima of the energy landscape are not separated by local spin rotations ( which would typically result in an extensive entropy ) , but rather are connected with each other by a continuous extended collective rotations of spins , which are sensitive to the states on the boundary . it would be interesting to see if other physical systems possess similar properties . s.h.l . and i.k . were supported by the division of materials sciences and engineering , basic energy sciences ( bes ) , us department of energy ( de - 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[ scgolatt ] below . one bi - pyramid consists of two tetrahedrons that share a corner with each other . for a collinear bi - pyramid spin configuration , each spin can be assigned a binary sign , representing a parallel ( + ) or anti - parallel ( - ) direction . to satisfy their antiferromagnetic constraints , each collinear tetrahedron must have two plus and two minus , leading to total zero spin . there are 18 possible sign configurations per each bi - pyramid ( 13 ) , as shown in fig . when the @xmath4 color order is imposed on a sign state , a collinear bi - pyramid state is obtained . [ s2]a illustrates an example with a sign state that does not have any long range order , yielding a collinear bi - pyramid state without any long range order . long range ordered collinear bi - pyramid state from the 1 - 6 - 8 sign state . , title="fig:",width=288 ] long range ordered collinear bi - pyramid state from the 1 - 6 - 8 sign state . , title="fig:",width=288 ] fig . [ s2]b , [ s2cont]c , and [ s2cont]d show three cases with @xmath4 long range ordered sign states that yield three @xmath4 long range ordered collinear bi - pyramid states . sign flip operation on the three long range ordered states yields another set of three @xmath4 collinear bi - pyramid states of the 10 - 15 - 17 , 11 - 13 - 18 , and 12 - 14 - 16 sign states . long range ordered collinear bi - pyramid state from the 2 - 4 - 9 sign state . ( d ) a @xmath4 long range ordered collinear bi - pyramid state from the 3 - 5 - 7 sign state . , title="fig:",width=288 ] long range ordered collinear bi - pyramid state from the 2 - 4 - 9 sign state . ( d ) a @xmath4 long range ordered collinear bi - pyramid state from the 3 - 5 - 7 sign state . , title="fig:",width=288 ] starting from a classical spin configuration we can rotate simultaneously a subset of the spins , while remaining in the classically degenerate manifold satisfying the zero total spin on the tetrahedra and triangles . presence of modes involving a finite amount of spins , suggests that at low temperatures the system may be in a liquid state due to quantum tunneling . however , in the bi - pyramid lattice we do not find such modes . the modes involving motion of a minimal amount of spins are most easily seen in the ordered states . examples for these quasi 1d modes are the `` ladder '' and `` spaghetti '' modes depicted in the figures below . ) . the mode involves the simultaneous rotation of the spins marked in green . , width=288 ] structure can support partial modes . left panel : 3 straight `` spaghetti '' modes traversing an ordered state in the vertical direction . right panel : simultaneous rotation of @xmath32 of the spins in one kagome layer . , title="fig:",width=288 ] structure can support partial modes . left panel : 3 straight `` spaghetti '' modes traversing an ordered state in the vertical direction . right panel : simultaneous rotation of @xmath32 of the spins in one kagome layer . , title="fig:",width=288 ] in this section we will use an alternative representation of the state in terms of a planar , fully packed pipe network . in this representation , the sign states are represented by the hexagonal tiles depicted in the figure [ tiles ] . the rules for tiling are simple : colors must match . a typical state is shown in the figure [ typicalnetwork ] , and we can think of it as a fully packed diagram of pipes . we note that straight segments of pipes may have only two possible thicknesses , which we refer to as thick or thin , and we set as 1 or 2 , in appropriate units . the ordered , 618 state is depicted in [ 618pipes ] . the same state after putting in a full @xmath33 rotation along a parallel and diagonal spaghetti ( see depiction of `` spaghetti '' modes in fig . [ mode examples ] ) , respectively , is shown in the next two figures . state represented by tiles . middle panel : the @xmath23 state with a full @xmath33 parallel spaghetti . right panel : the @xmath23 state with a full @xmath33 diagonal spaghetti . , title="fig:",width=192 ] state represented by tiles . middle panel : the @xmath23 state with a full @xmath33 parallel spaghetti . right panel : the @xmath23 state with a full @xmath33 diagonal spaghetti . , title="fig:",width=192 ] state represented by tiles . middle panel : the @xmath23 state with a full @xmath33 parallel spaghetti . right panel : the @xmath23 state with a full @xmath33 diagonal spaghetti . , title="fig:",width=172 ] * property 1 * : it is impossible to terminate a line . _ proof of property 1 : _ this property can be verified by inspection of possible termination points , and ruling each of them out . * property 2 * : there can be no closed loops in a pipe diagram . _ proof of property 2 : _ to prove this , assume the contrary and consider a closed loop of black color , inside there must be loops of smaller and smaller sizes . since the colors alternate , we must have inside an enclosed simply connected region which is entirely black or entirely blue . as we do nt have a an all blue or all black hexagon , therefore the inner region must be made of lines with termination points . by property 1 , this can not happen . * definition : * _ `` laminar region''_. a laminar region is a region where lines do not split ( i.e. there are no junctions ) . property 3 : the network in a laminar region is a set of black and blue pipes ( not necessarily straight ) , which are parallel to each other . moreover , the thickness of each pipes is determined by the thickness at any given point along the pipe . _ proof of property 3 : _ if no junctions are present , since the lines are fully packed and can not cross or join , the arrangement is of parallel curves . we now inspect the thickness of the curves . since the thickness is constant along straight segments , we have to consider elbows ( corners ) . there is only a finite number of possible elbows , which can be checked explicitly . we find that at a corner without a junction , the thickness of a pipe is switched deterministically from thin to thick and vise versa . thus , given the thickness of the pipe , and the set of points where it has changed direction , the thickness is determined all along the pipe . an example of a corner is in figure [ corner ] . a simple example of laminar regions are the 1 - 6 - 8 states , as well as it s variants involving a spaghetti exhibited in the figures above . in a laminar region , property 3 leads to the following immediate estimate of the number of possible states in a region bound by a curve of length @xmath8 . choosing a reference black curve of length @xmath8 , we can specify the thicknesses of @xmath5 of its parallel pipes . the area @xmath34 of such a configuration is @xmath35 , and the number of possibilities involved is bounded above by @xmath36 : at each point on the reference curve we can either stay straight or make a @xmath37 degree turn left or right , in addition , we have @xmath38 choices of the thicknesses of the parallel curves . remark : this , of course is a large over - estimate , since the reference curve can not self intersect , moreover , it s shape is highly restricted by properties 1,2 , which applies to it as well as all the parallel curves . finally , we show that given the configuration of tiles a boundary layer of a thickness 3 hexagons of a laminar region , the state can be completely determined . to see this , note that to establish the state we have to determine the locations of the elbows , as well as the thickness of the lines . the thicknesses are always determined at the boundary . for each internal elbow , it s image will appear on the boundary of the region in at least two points ( as it can not disappear going from one line to the next ) . if we determine a layer of the boundary thick enough to detect all possible elbows , which can be done with a boundary of thickness 3 , we can continue to determine the internal state using the information on the thickness of the lines at the boundary . by the arguments above , the number of laminar regions scales as the boundary length . to complete our bound on the total number of states , we must include junctions . there are several kinds of junctions . most involve a narrow and wide lines meeting . in addition there is a triple narrow junction type ( fig . [ nnnjunction ] ) . all junctions of valence larger than 3 may be viewed as joining of two valence 3 junctions . we now bound the number @xmath19 of possible junctions in the sample . this can be done by a simple argument : by properties 1,2 , the system is a set of lines with no closed cycles , and termination points only on the boundary . such a system may be viewed as a graph consisting of a disjoint union of trees with leaves on the boundary . since the number of vertices on a full tree is always smaller then the number of trees we have that : @xmath39 for each configuration of junctions , we can consider the system consisting of the boundary and a small region around the junctions as a laminar region . the effective boundary degrees of freedom are less than @xmath40 , where the constant @xmath41 describes the `` thickening '' needed to observe elbows , as before . therefore we have at most : @xmath42 where @xmath43 are binomial coefficients , and @xmath44 . we can easily bound the last equation using @xmath45 and get : @xmath46 for a typical @xmath47 , and we have : @xmath48 in this section we show how locally colinear states may be counted using a transfer matrix method . while we get fairly quickly an upper bound on the number of states , it is extensive in system volume . however , numerics shows that this seems like a large over - estimate : the numerical behavior is quite peculiar and is consistent with a boundary entropy . let us consider a description of this problem as follows . we consider a square array of the bi - pyramids . it may be viewed as alternating two zig - zag columns which are shifted with respect to each other vertically . we enumerate the possible signs a long the zig - zag columns as follows . we consider adding another column in two steps : we first add sites at even levels and then the sites at odd level as depicted in the fig [ twosteps ] . in this procedure all the sites in the first stage can be added independently of each other , and after it is complete the second stage shares this property . we are now left with the task of transferring this into a formula . for a bi - pyramids in the first move , we note that the signs @xmath49 and in general @xmath50 , where @xmath51 is an integer , remain unchanged . the sites which might change after the move are of the form @xmath52 . each such pair only depends on the states of @xmath53 . let us denote by @xmath54 the number of possible sign states with the signs on the left @xmath53 and sites on the right : @xmath55 . we can view this as a linear transformation @xmath56 on @xmath57 @xmath58 for a column of @xmath59 bi - pyramids , there are @xmath60 signs on the border which participate in the counting . in the first stage we can combine all the @xmath61 moves to @xmath62 next we note that the transformation governing the added bi - pyramids in step 2 , is described in the same , albeit shifted by two sites . in addition , it involves adding boundary bi - pyramids , which require special counting . we can summarize this as : @xmath63 for @xmath64 columns , the number of states may now be computed as @xmath65 where @xmath66 specify boundary conditions on the left and on the right . now , the number of states in a large region , @xmath67 scales as @xmath68 , where @xmath69 is the largest eigenvalue of @xmath70 . next we consider the eigenvalues of @xmath71 and @xmath72 separately . assuming @xmath59 is large , these are determined by @xmath61 . the matrix @xmath61 is special , and it s eigenvalues can be determined analytically . to do so we first determine the invariant subspaces of this matrix . @xmath61 is a @xmath73 matrix , acting on 4 ising spins . it turns out more convenient to write it using two double spins , in basis 4 by assigning : @xmath74 we now find the cycles of the matrix : @xmath75 interestingly , the 6 largest eigenvalues are @xmath76 , each doubly degenerate . here the largest eigenvalue is @xmath77 is the golden ratio . by the norm inequality : @xmath78 , we immediately conclude that the largest eigenvalue of @xmath72 and of @xmath71 scale as @xmath79 . from this we have a rough estimate that the number of states scales at most as @xmath80 . the largest eigenvalues of transfer matrices can be computed numerically up to several rows . here are the two largest eigenvalues of the transfer matrix @xmath81 up to 11 rows : @xmath82 @xmath83 it is very clear that , at least up to 11 rows , the largest eigenvalue goes down . this means , that for a long enough strip , the number of states of 11 rows , will be smaller than , say the number of states of 3 rows . this reflects the highly constrained nature of the system : for certain sign configuration of the 3 rows , there can be no way to continue adding rows in a consistent way up to 11 rows . the decreasing nature the eigenvalues give us strong evidence that , in fact , the number of states should scale as the boundary length , without an extra @xmath84 . for one bi - pyramid , a coplanar state can be generated from a collinear state by rotating the 7 spins in several collective ways , four of which with a set of parameters @xmath85 are shown in fig . [ s3]a . for the entire triangular lattice of bi - pyramids , because of the color and ferro - sign bond constraints , the three angles with the same magnitude are sufficient to generate a long range ordered coplanar state . [ s3]b shows one orthogonal bi - pyramid spin state , as an example , generated from the 1 - 6 - 8 sign state by rotating the spins with ( 45,-45,-45 ) . if the spins are rotated with ( 90,-90,-90 ) , then the bi - pyramids become collinear again , as shown as an example in fig . [ s3]c that corresponds to the 7 - 3 - 5 sign state . the collinear spin states and their resulting coplanar states are continuously connected with each other in the spin - energy phase space . the connections among the sign states are listed in the table below . [ cols="<,^,>",options="header " , ] the initial and final long - range - ordered sign states connected by the global spin zero - energy excitations illustrated in fig . in order to explore the energy landscape generated by quantum fluctuations , we next concentrated on evaluating the spin wave energy for non uniform systems . the calculations have been done in the framework of the holstein - primakoff representation . we consider the hamiltonian : @xmath86 first we pick a classical spin configuration which is a local energy minimum . in the next step we take each classical spin direction , and replace by an operator as : @xmath87 where @xmath88 are boson creation / annihilation operators , and @xmath89 are any couple of unit vectors which combine into an orthogonal frame with the classical direction @xmath90 . at this point , in order to get a tractable theory , the square roots are expanded to lowest order in @xmath91 . since it is assumed that the spin is at a classical minimum , this procedure yields a quadratic hamiltonian in the bosonic operators . uniform ( long range ordered states ) are studied , as usual , by rewriting the hamiltonian in momentum space . the number of degrees of freedom , is then determined simply by the unit size . we then compute the zero point energy of the resulting @xmath92 modes , and sum them over the brillouin zone . state by a global reorientation as described in fig [ s3 ] . ( d ) total spin wave energy as function of rotation angle . maximum is obtained for orthogonal configurations and minimum for collinear . we also show the energy obtained for the stripe state consisting of alternating rows of 8 and 17 type of sign states ( see fig . [ stripes ] and [ ladderinstripe]),width=441 ] to consider states which are not translationally invariant requires a real space treatment . to diagonalize the hamiltonian , on a lattice with @xmath94 bi - pyramids , we get a @xmath95 quadratic boson hamiltonian . to compute the state of the system , we wrote a procedure affecting the symplectic transformations needed to diagonalize the hamiltonian numerically . we then computed the spin wave energy for the hamiltonian for various system sizes ( as large as @xmath96 bi - pyramids , involving a total of 4032 spins ) .

* when magnetic moments ( spins ) are regularly arranged in a geometry of a triangular motif , the spins may not satisfy simultaneously their interactions with their neighbors . this phenomenon , called frustration , leads to numerous energetically equivalent magnetic states ( ground states ) , which results in exotic states such as spin liquid and spin ice . here we report an alternative situation : a system that , classically , is to be a liquid in the clean limit freezes into a glassy state induced by quantum fluctuations . the case in point is a frustrated magnet in which spins are arranged in a triangular network of bi - pyramids . when taking into account quantum corrections , the classical degeneracy is broken into a set of local minima in a rugged energy landscape , which are separated by large energy barriers , over a finite number of degenerate , periodic , ground states . the appearance of large barriers is due to the absence of local zero - energy modes that are typical in spin - liquid candidate systems . we establish this by mapping the set of local energy minima states into a tiling with colored hexagonal tiles . we show that the system exhibits a large number of aperiodic tessellations . the configuration entropy of the local minima is extremely sensitive to boundary conditions , scaling with the boundary length rather than its volume . the low temperature thermodynamics is also discussed to compare it with other glassy materials . * , it is well known , since the classical work of pauling on ice @xcite , that certain systems can exhibit an extensive number of energetically equivalent ground states , leading to finite entropy at low temperatures . in a spin ice , states are separated by local single ionic energy barriers , and the spins freeze into one of the equivalent states at low enough temperatures @xcite . in pyrochlore with large spins , locally confined zero energy motions of spins are possible , which can lead to a classical spin liquid state @xcite . when quantum effects are taken into account , for small spins , such systems may settle into a super - position of states , forming a quantum spin - liquid@xcite , as suggested by anderson @xcite . a closely related but distinct type of systems is glassy systems . one example is amorphous alloys in which the atoms are arranged in a disordered way @xcite . another is spin glass systems in which low concentration of magnetic impurities interact via random long - range interactions @xcite . in such systems randomness ( or quenching ) is the driving force for the freezing phenomena . the randomness , however , makes it difficult to fully understand the complex physics of the freezing phenomenon . interesting effective models for glassy behavior without disorder have been presented in order to understand various types of glasses . for example , glassy behavior has been explored in systems with long - range interactions @xcite and in models with hard - core classical constraints and stochastic dynamics known as kinetically constrained models@xcite ( kcm ) , as well as in certain quantum plaquette and quantum dimer models@xcite . here we show that a spin glass can actually arise from simple nearest neighbor heisenberg interactions at low temperatures due to quantum effects . moreover , this behavior may be present in real materials , and may provide a framework to understand the unconventional@xcite glassy behaviors found in classes of frustrated magnets such as @xmath0 ( scgo(p)),@xcite and qs - ferrites like @xmath1 ( bszgco)@xcite that are highly crystalline and their glassiness seems to be insensitive to disorder@xcite .

a common table - top demonstration of the effects of surface tension is to float a metal needle horizontally on water : even though the density of the needle is much greater than that of water , the needle is able to remain afloat because of the relatively large vertical component of surface tension . this effect is a matter of life or death for water - walking insects @xcite , and is also important in practical settings such as the self - assembly of small metallic components into macroscopic structures via capillary flotation forces @xcite . in this engineering setting an object should not only float when isolated at the interface , but must also remain afloat after it has come into contact with other interfacial objects , and portions of the meniscus that supported it have been eliminated . although the interactions that cause interfacial objects to come into contact and form clusters have been studied extensively ( see , for example , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the implications of such interactions on the objects ability to remain afloat have not been considered previously . here we consider the effects of these interactions via a series of model calculations that shed light on the physical and mathematical concepts that are at work in such situations . for simplicity , the calculations presented here are purely two - dimensional , though the same physical ideas apply to three - dimensional problems . perhaps the most natural way to characterise the effects of interaction is to ask how the maximum vertical load that can be supported by two floating cylinders varies as the distance between them is altered . we thus consider two cylinders of infinite length lying horizontally at the interface between two fluids of densities @xmath0 , as shown in figure [ 2cylchan ] . we assume that these cylinders are non - wetting so that the contact angle @xmath1 , a property of the three phases that meet at the contact line , satisfies @xmath2 . we non - dimensionalise forces per unit length by the surface tension coefficient , @xmath3 , and lengths by the _ capillary length _ , @xmath4 , and use non - dimensional variables henceforth . we wish to determine the maximum weight per unit length , @xmath5 , that can be supported by each of two identical cylinders with radius @xmath6 and centre centre separation @xmath7 . to remain afloat each individual cylinder must satisfy a condition of vertical force balance : their weight ( or other load ) must be balanced by the vertical contributions of surface tension and the hydrostatic pressure acting on the wetted surface of the cylinder . we assume that an external horizontal force is applied to maintain the separation of the cylinders and so do not consider the balance of horizontal forces explicitly . .,height=132 ] using the notation of figure [ 2cylchan ] , the vertical force balance condition may be written @xmath8 where @xmath9 are the contributions to the vertical upthrust provided by the deformation on each half of the cylinder separately , and @xmath10 is the height of the cylinders centres _ above _ the undeformed free surface . physically , the first term on the right hand side of is the vertical component of surface tension , and the second and third terms quantify the resultant of hydrostatic pressure acting on the wetted perimeter of the cylinder . the latter is given by the weight of water that would fill the dashed area in figure [ 2cylchan ] ( see * ? ? ? the angles @xmath11 and @xmath12 are determined by the interfacial shape , which is governed by the balance between hydrostatic pressure and the pressure jump across the interface associated with interfacial tension . this balance is expressed mathematically by the laplace young equation . in two dimensions this is @xmath13 where @xmath14 is the deflection of the interface ( again measured positive upwards ) from the horizontal , and subscripts denote differentiation . since the exterior meniscus extends to infinity , the first integral of is particularly simple in this instance and allows the height of the contact line , @xmath15 , to be related to the interfacial inclination , @xmath16 , via @xmath17 this , together with the geometrical condition @xmath18 , allows @xmath11 to be eliminated from in favour of @xmath19 and @xmath1 . for the interior meniscus , we simultaneously obtain @xmath12 and the shape @xmath14 , by using the matlab routine ` bvp4c ` to solve the nonlinear eigenproblem @xmath20 on @xmath21 $ ] . with the angles @xmath11 and @xmath12 calculated , @xmath22 can be determined from , and the maximum load that can be supported , @xmath23 , can be found numerically by varying @xmath10 . of particular interest is the dependence of @xmath23 on the cylinder separation , which is shown for several values of the _ bond number _ @xmath24 in figure [ modgraph ] . this plot includes the limiting case @xmath25 , corresponding to the application of two point forces to the interface . away when @xmath26 for several values of the bond number , @xmath24 . the dashed line shows the linear approximation for the limiting case @xmath25 when @xmath27.,height=151 ] the results presented in figure [ modgraph ] show that as the distance between two cylinders decreases , the maximum vertical load that can be supported by each cylinder decreases . physically , this result is intuitive since even though the interior meniscus is not completely eliminated in this instance , the vertical force that this meniscus can exert on the cylinder is diminished by the symmetry requirement that @xmath28 . in particular , for small @xmath29 and @xmath30 the total force that can be supported by each cylinder is around half of that which can be supported by an isolated cylinder . this corresponds to the simple physical picture that for small bond number , the restoring force is supplied primarily by the deformation of the meniscus @xcite ; when the interior meniscus is eliminated , the contact line length per cylinder , and hence the force that surface tension can provide , are halved . from this we expect that very dense objects that float when isolated at an interface might sink as they approach one another . since floating objects move towards one another due to capillary flotation forces ( see * for example ) , it seems likely that this effect may be ubiquitous for dense objects floating at an interface and may also have practical implications . for @xmath25 we can compute the asymptotic form of @xmath23 for @xmath27 by noting that for small separations the interior meniscus has small gradients and the laplace young equation may be approximated by @xmath31 , which has the solution @xmath32 . thus , the vertical force provided by the deformation is @xmath33 , which is extremised when @xmath34 . choosing the real root of this quartic corresponding to a maximum in @xmath5 and making consistent use of @xmath27 , @xmath23 can be expanded as a series in @xmath30 . we obtain @xmath35which compares favourably with the numerically computed results presented in figure [ modgraph ] . whilst the scenario considered in the previous section may be relevant in practical situations , it does not lend itself to particularly simple experimental validation . to allow for such a comparison , we now consider the equilibrium of two infinitely long , shallow strips of dimensional thickness @xmath36 , width @xmath37 , and density @xmath38 , floating with their long edges in contact so that the interior meniscus is completely eliminated . the configuration is shown schematically in figure [ 2psetup ] . here , we are no longer bound by a contact angle condition but instead assume that the meniscus is pinned to the uppermost corners of the strips . the additional complication of the strip s angle of inclination to the horizontal , @xmath39 , is determined by the balance of torques . ( this condition is satisfied automatically for shapes with circular cross - section and constant contact angle , as shown by @xcite . ) equating moments about the point of contact ( thereby eliminating the need to calculate the tension force that the strips exert on one another ) and balancing vertical forces , we obtain the conditions for equilibrium @xmath40 where @xmath41 is the appropriate ratio of the density of the strips to those of the surrounding fluids . after eliminating @xmath42 between and and using with the relation @xmath43 to eliminate @xmath16 , we have a single equation for @xmath39 given particular values of @xmath44 and @xmath10 . thus , for fixed @xmath44 and a given value of @xmath10 , we may solve for @xmath39 and deduce the corresponding value of @xmath42 from . by varying @xmath10 we are then able to calculate the maximum value of @xmath42 for which equilibrium is possible , much as before . the numerical results of this calculation are presented in figure [ 2ptouch ] . as the half - width of the strips , @xmath44 , is varied ( solid line ) . experimental results ( as described in text ) are shown by points @xmath45 ( strips that sink ) and @xmath46 ( strips that float).,height=188 ] also shown in figure [ 2ptouch ] are experimental results showing points in @xmath47 parameter space for which two identical strips remained afloat or sank upon touching . these experiments were performed with strips of stainless - steel shim of length @xmath48 with @xmath49 and thickness @xmath50 or @xmath51 . these were floated on aqueous solutions of @xmath52 , @xmath53 or @xmath54 methanol in air ( so that @xmath55 ) , allowing a wide range of values of @xmath44 and @xmath42 to be probed . the strips were then allowed to come into contact naturally via the mutually attractive flotation force @xcite . the data are plotted with horizontal and vertical error bars . the former indicate the uncertainty in the measurement of the strip widths . the latter indicate the uncertainty in the additional vertical force contribution of the ends ( since the strips are of finite length ) , which may be shown to be equivalent to an uncertainty in the effective value of @xmath42 . the agreement between our experiments and theory in this instance is very good . by adding additional strips to a floating pair of strips , a flexible raft is formed . given the analysis of the preceding sections it is natural to expect that as the raft is lengthened in this manner , there will come a point where its weight ( which scales with its total length ) exceeds the force that can be supplied by surface tension ( which is constant ) and so the raft should sink . the situation is complicated by the fact that the raft may bow in its middle , displacing a considerable amount of liquid in this region , as pointed out by @xcite . we now address the question of whether , for a raft of given weight per unit length , there is a maximum raft length before sinking occurs . we tackle this problem by treating the raft as a continuum , shown schematically in fig . [ raftsetup ] , and formulating an equation for the deformation of such a raft . this generalises the linear analysis of @xcite and allows us to consider situations in which interfacial deformations are no longer small , including the existence of a threshold length for sinking . we use a variational approach to determine the shape @xmath56 of the raft and the surrounding meniscus , though the same result may also be obtained by considering the force balance on an infinitesimal raft element . the non - dimensional arc - length , @xmath57 , is measured from the raft s axis of symmetry at @xmath58 , with the two ends of the raft being at @xmath59 . for simplicity , we neglect the intrinsic bending stiffness of the raft , although @xcite have shown that interfacial rafts do , in general , have some resistance to bending . the variational principle states that raft shapes must minimise the energy of the system over variations in @xmath60 and @xmath61 , subject to the constraint that @xmath62 . introducing a lagrange multiplier @xmath63 associated with this constraint , we find that equilibrium raft shapes extremise @xmath64\right)\mathrm{d}s,\ ] ] where @xmath42 was defined in and @xmath65 is the indicator function of the raft . the first two terms in the integral correspond to the gravitational energy of the displaced fluid and the raft , the third term is the surface energy of the uncovered liquid area , and the final term ensures that the constraint @xmath62 is satisfied . note that a small increase in arc - length such that @xmath66 increases the energy of the system so that the lagrange multiplier @xmath63 may be interpreted physically as the tension in the raft / meniscus . that the raft can support a tension at all may seem counterintuitive . it is a consequence of the attractive capillary interaction that would exist between two infinitesimally separated raft elements . requiring @xmath67 to be stationary with respect to variations in @xmath60 and @xmath61 yields differential equations for @xmath68 and @xmath69 . using the differential form of the constraint , @xmath70 , we may eliminate @xmath71 to obtain @xmath72 . this may be integrated using the boundary term from integration by parts at @xmath73 , the boundary conditions @xmath74 and @xmath75 as well as the continuity of @xmath71 at the raft edge , @xmath59 , to give @xmath76 , where @xmath77 . we now find the raft shape numerically by solving the nonlinear eigenproblem @xmath78 for @xmath61 , @xmath60 , @xmath79 on @xmath80 $ ] , and @xmath15 , using the matlab routine ` bvp4c ` . the results of this computation may be verified by calculation of the quantity @xmath81\cos\theta-1 , \label{cons_q}\ ] ] which is conserved and , from the boundary conditions , equal to @xmath82 . in the limit of small deformations reduces to the simpler linear form studied by @xcite in the context of determining typical raft profiles . here , however , we wish to determine whether a maximum raft length , @xmath83 , exists and if so find its value for a raft of given density @xmath42 . to investigate this , small deformation theory is inadequate since sinking is an essentially non - linear phenomenon . the symmetry condition @xmath84 ensures that @xmath85 and that @xmath86 , so that the centre of the raft may sink at most to its neutral buoyancy level . in what follows , it will be convenient to treat @xmath10 and @xmath42 as parameters giving rise to a particular raft semi - length @xmath87 ; we find @xmath88 ^ 2-\bigl[2-(h_0+y h_0 ^ 2/2{{\cal d}})^2\bigr]^2\bigr\}^{1/2}}\mathrm{d}y , \label{intlenexp}\ ] ] which follows by changing integration variables from @xmath57 to @xmath69 in @xmath89 . this allows us to consider the behaviour of @xmath90 for a given value of @xmath42 as @xmath10 is varied . the tension at the midpoint of the raft is given by @xmath91 , showing that the raft goes into compression if @xmath92 . physically this is unrealistic , corresponding to a divergence in @xmath93 . if @xmath94 , this situation is avoided automatically since @xmath95 but for @xmath96 we must consider this possibility ; we therefore consider these two cases separately . when @xmath94 , the centre of the raft may reach its neutral buoyancy depth @xmath97 without going into compression . numerical computation of the integral suggests that rafts grow arbitrarily long as @xmath98 ( see figure [ dres]@xmath99 ) . to show that this is the case , we consider the asymptotic behaviour of the integral in the limit @xmath100 , since @xmath101 . ] . this is done by splitting the range of integration into two sub - regions @xmath102 $ ] and @xmath103 $ ] , where @xmath104 is unspecified save for the condition that @xmath105 ( see * ? ? ? * ) . within these two regions , the two integrands may be simplified using approximations compatible with this gearing of @xmath104 , and the resulting integrals evaluated analytically . upon expanding these results for @xmath106 the leading order terms in @xmath104 cancel , yielding @xmath107 where @xmath108 . this result compares favourably with the numerical results in figure [ dres](a ) . in particular , notice that @xmath90 diverges logarithmically as @xmath109 ( i.e. as @xmath110 ) so that rafts of arbitrary length are possible . it also interesting to note that may be inverted to give an estimate of @xmath111 for given values of @xmath42 and @xmath90 a useful result when calculating raft shapes for large @xmath90 . that a raft of sufficiently low density can grow arbitrarily large in horizontal extent without sinking seems surprising at first glance . however , as new material is added to the raft , it may be accommodated at its neutral buoyancy level without the raft going into compression . therefore , the raft s ability to remain afloat is not jeopardised and it is almost obvious that these low density rafts may grow arbitrarily long without sinking . ) numerical results of the calculation of @xmath90 as a function of @xmath112 ( solid line ) compared to the asymptotic result for @xmath113 ( dashed line ) for the case @xmath114 . ( @xmath115 ) main figure : numerical results of the calculation of @xmath116 as a function of the density ratio @xmath117 ( solid line ) , together with the large @xmath118 asymptotic result @xmath119 ( dashed line ) . inset : rescaled graph comparing the numerically computed values ( points ) of @xmath116 with the asymptotic expansion ( solid line).,height=151 ] in this case , the raft can not reach its neutral buoyancy level , invalidating the argument just given to explain why , with @xmath94 , rafts may be arbitrarily large . we thus expect that a maximum raft length does exist and , further , that the limiting raft has @xmath120 . numerical computation of @xmath90 as a function of @xmath10 indicates that a critical half - length @xmath116 does exist , but that it is not attained with exactly this value of @xmath10 . instead , there is a competition between the raft sinking deep into the liquid ( to support its weight by increased hydrostatic pressure ) and having its ends a large distance apart ( i.e. lower pressure but over larger horizontal distances ) , and some compromise is reached . given the abrupt change in behaviour observed as @xmath42 increases past @xmath121 , we are particularly interested in the nature of this transition . numerical computations suggest that for @xmath122 , @xmath116 occurs when @xmath123 for some constant @xmath124 . motivated by this observation , we let @xmath123 and again split the domain of integration in into two regions @xmath125 $ ] and @xmath126 $ ] where @xmath127 . this allows us to calculate @xmath90 to leading order in @xmath128 , yielding @xmath129+o(\eta^2 ) , \label{lexp}\ ] ] where @xmath130 and @xmath131 are the complete elliptic integrals of the first and second kinds , respectively . the coefficient of @xmath128 in has a maximum for fixed @xmath128 at @xmath132 , where @xmath133 satisfies @xmath134 hence @xmath135 , and we obtain the asymptotic expression @xmath136 which compares very favourably with the numerically computed values of @xmath116 presented in the inset of figure [ dres](@xmath115 ) . for the limiting case @xmath137 , the above analysis breaks down since then @xmath138 and we lose the freedom to vary @xmath10 . however , by letting @xmath139 ( so that @xmath140 ) we take the limit @xmath141 of with @xmath113 fixed to find @xmath142+o(\epsilon ) . \label{etazero}\ ] ] this has a maximum value of @xmath143 at @xmath144 , which is the same value as that found from in the limit @xmath145 . it is also reassuring to note that , as @xmath146 with @xmath147 fixed , the expression in also gives @xmath148 . for completeness , we consider finally the limit @xmath149 . to leading order in @xmath150 , the integral for @xmath87 is given by @xmath151 this has a maximum value of @xmath150 at @xmath120 so that in the limit @xmath149 , @xmath152 . this is precisely as we should expect physically since large density objects can only float when the contribution of surface tension dominates that of the buoyancy due to excluded volume and , in particular , it must balance the weight of the raft . this asymptotic result compares favourably with the the numerical results presented in figure [ dres](@xmath115 ) . a direct comparison between the theoretical results outlined so far and experimental results is difficult since we have modelled the raft as a perfectly flexible continuum body of infinite extent along its axis of symmetry . despite these limitations , the theoretical raft shapes calculated via this model are in good agreement with simple experiments in which thin strips of stainless steel shim are laid side - by - side at an air water interface , as shown in figure [ raft ] even when the raft consists of only a small number of strips and we might not expect the continuum approximation to be valid . and @xmath90 : ( @xmath99 ) a complete raft with @xmath153 , @xmath154 ( @xmath115 ) one half of a raft with @xmath155 , @xmath156 and ( @xmath124 ) one half of a raft with @xmath155 , @xmath157 . the typical width of each individual strip is @xmath158 mm . the black region apparently above the raft is in fact a reflection of the black base of the confining tank from the meniscus at the edge of the tank , height=151 ] although this agreement is encouraging , our main interest lies more in whether there is a maximum length for such a raft to remain afloat , as predicted by the model . practical considerations mean it is difficult to produce strips of stainless steel shim narrower than about @xmath158 mm in the workshop , so the comparisons we are able to draw between our model and experiments can only be semi - quantitative . in spite of these limitations , we find that for stainless steel strips of length @xmath159 mm and thickness @xmath160 mm the maximum raft - length is @xmath161 mm for an aqueous solution of 25% methanol ( so that @xmath162 ) and @xmath163 mm for 15% methanol ( so that @xmath164 ) , which are certainly consistent with the corresponding theoretical results of @xmath165 mm @xmath166 mm and @xmath167 mm @xmath168 mm , respectively . here the length was increased by floating additional strips near the raft and allowing them to come into contact via the mutually attractive capillary flotation forces until the raft was no longer stable and sank . with @xmath153 and @xmath155 , we were able to add many strips without any sign of the raft sinking indicating that this process might be continued indefinitely . in this article , we have quantified the conditions under which objects can remain trapped at a fluid - fluid interface , and shown that when the deformation of the meniscus is suppressed by the presence of other objects the supporting force that can be generated decreases dramatically . for two small , parallel cylinders or strips , the maximum force that can be supported close to contact is only that provided by the contribution from the exterior meniscus and so sufficiently dense objects sink upon contact . a two - dimensional raft of touching , floating strips may compensate partially for this loss of meniscus by sinking lower into the fluid . for @xmath94 , this effect allows rafts of arbitrary length to remain afloat . for @xmath96 , there is a maximum length ( dependent on @xmath42 ) above which equilibrium is not possible . although the agreement between the experiments and theory presented here is good , our analysis was confined to two dimensions , whereas experiments must be carried out in the three - dimensional world . similarly , we have limited ourselves to considering the _ equilibrium _ of objects at an interface . we are currently studying the dynamics of sinking for the case of two touching strips considered in section [ 2strip ] , and find that a simple hydrodynamic model produces good agreement with experiments . we are grateful to david page - croft for his help in the laboratory and herbert huppert for comments on an earlier draft . dv and rjw are supported by the epsrc . pdm gratefully acknowledges the financial support of emmanuel college , cambridge .

we study the effect of interactions between objects floating at fluid interfaces , for the case in which the objects are primarily supported by surface tension . we give conditions on the density and size of these objects for equilibrium to be possible and show that two objects that float when well - separated may sink as the separation between the objects is decreased . finally , we examine the equilbrium of a raft of strips floating at an interface , and find that rafts of sufficiently low density may have infinite spatial extent , but that above a critical raft density , all rafts sink if they are sufficiently large . we compare our numerical and asymptotic results with some simple table - top experiments , and find good quantitative agreement .

a two - valley ( 2v ) two - dimensional ( 2d ) electron gas ( eg ) is the simplest model to describe electrons confined in solid state devices such as si - mosfets @xcite and certain quantum wells @xcite . in this 2d model , electrons interact via a @xmath0 potential in a uniform neutralizing background and possess an additional discrete degree of freedom ( the valley index ) . one may identify electrons with given spin projection and valley index as belonging to a different species or _ we focus on the case of two symmetric valleys , where the number of up ( down ) spin electrons is the same for both valleys . in this case at zero temperature the 2deg is completely characterized in terms of the coupling parameter @xmath1 and the spin polarization @xmath2 ( where @xmath3 is the total electron density , @xmath4 the bohr radius , @xmath5 the density of up ( down ) spin electrons ) . the interest in the properties of the 2deg has been strongly revived in the last years due to the experimental discovery of a previously unexpected metal - insulator transition @xcite in which the valley degree of freedom appears to play an important role @xcite . the transition takes place at low density , where an accurate treatment of electron correlation is crucial . in this respect , quantum monte carlo ( qmc ) simulations have provided over the years the method of choice for microscopic studies @xcite of the 2deg . we have recently provided an analytic expression of the correlation energy of the 2v2deg @xcite . here , we focus on the pair distribution functions ( pdfs ) , which are strictly related to the description of exchange and correlation properties of the system ( see _ e.g. _ ref . @xcite ) . as pointed out in ref . @xcite , pdfs may serve a variety of purposes , among which the estimate of finite thickness effects on the 2deg spin susceptibility @xcite , applications in dft calculations , or a test of the accuracy of hypernetted - chain calculations @xcite . the pdf is related to the probability of finding two electrons at positions @xmath6 and @xmath7 respectively . the _ component - resolved _ pdf @xmath8 of a multicomponent 2deg is defined as @xcite @xmath9 with @xmath10 , @xmath11 denoting creation and annihilation field operators , @xmath12 the expectation value on the ground state and @xmath13 the electron density of the component @xmath14 . the normalization is such that @xmath15 , in case there is neither exchange nor correlation . in a homogeneous and isotropic system , @xmath16 depends only on the relative distance @xmath17 and there is symmetry for index permutations ( @xmath18 ) . if @xmath19 is the concentration of the component @xmath14 , the total ( _ component - summed _ ) pdf @xmath20 reads @xmath21 in the 2v2deg , @xmath22 is a composite index which denotes the spin and valley ( respectively @xmath23 and @xmath24 ) degrees of freedom and spans the four cases @xmath25 . in general , for 2v there are ten different @xmath16 , but in the case of two symmetric valleys the number of different @xmath16 is two for @xmath26 and five for @xmath27 ( see also ref . @xcite ) . in the following we shall denote the different @xmath16 with one of the possible different labels ( _ e.g. _ for @xmath27 @xmath28 ) . most of the simulation details are the same as in ref . we performed fixed - phase dmc simulations ( for a review of qmc techniques see _ e.g. _ ref @xcite ) for @xmath29 and @xmath30 . to reduce the finite size effects , we used twist - averaged boundary conditions ( tabc ) @xcite , which also allow to change @xmath31 by flipping any number of spins at fixed number of electrons @xmath32 . to sample the pdfs we performed simulations for a system of @xmath33 electrons . time steps were chosen at the different @xmath34 to give an acceptance rate corresponding to @xmath35 . we did simulations with 320 walkers . the twist grid is the same as in ref . @xcite . as in ref . @xcite , we used a slater - jastrow trial wave function @xmath36 , but , here , we considered only plane - wave nodes , since the more accurate backflow ( bf ) nodes yield only slight modifications to the pdfs @xcite . besides , bf effects on the energy were shown to be bigger in the two - component case than in the four - component system @xcite . dmc provides the mixed estimate of an operator @xmath37 , _ i.e. _ @xmath38 ( with @xmath39 denoting the ground state of the system ) . if @xmath37 commutes with the hamiltonian @xmath40 , @xmath41 coincides with the expectation value on the true ground state @xmath39 . if @xmath37 does not commute with @xmath40 ( as in the case here considered ) , it is better to compute the extrapolated estimate @xmath42 ( where @xmath43 is the variational mc expectation value on @xmath36 ) which is accurate to second order in the difference @xmath44 between @xmath39 and @xmath36 . and @xmath45 . solid lines represent the total @xmath20 . ] . see labels . ] in fig . [ 2vspinresolved - z01]-[2vspinresolved - zno01 ] we show representative examples of 2v component - resolved pdfs for @xmath26 and finite @xmath31 respectively . all lenghts are given in units of @xmath46 . the component - resolved pdfs shown in fig . [ 2vspinresolved - z01 ] illustrate the tendency to a local order which favors electrons belonging to different species ( as e.g. @xmath47 and @xmath48 ) to get closer than electrons belonging to the same species . for intermediate spin polarizations ( see fig . [ 2vspinresolved - zno01 ] ) the component - resolved pdfs exhibit a richer structure than the @xmath26 cases , with qualitative features clearly related to the interplay of exchange and correlations ( for example , in the density range considered the diagonal pdf of the minority component in a strongly polarized system is found to be determined by exchange alone , to a very good approximation ) . the significant spin - polarization dependence seen in the component - resolved pdfs almost disappears in the total pdfs , particularly at large @xmath34 . this can be appreciated in fig . [ 2vspinsummed ] ( left panel ) , which shows the total pdfs for a high density ( @xmath49 ) and a low density ( @xmath50 ) case with zero and full spin polarization . the dependence of the total pdf on @xmath34 is depicted in fig . [ 2vspinsummed ] for @xmath51 . we see that even for @xmath34 as small as 1 the effect of exchange on the total pdf , which is expected to decrease with the number @xmath52 of ( equally populated ) components , is already very small for @xmath53 . and right panel : @xmath54 and @xmath51 ( increasing peaks for increasing @xmath34 ) . all lengths are in units of @xmath46 . ] full tabulations of the calculated pdfs are available upon request from the first author . 20 ando t , fowler a b and stern f 1982 _ rev . phys . _ * 54 * 437 shkolnikov y p , vakili k , de poortere e p and shayegan m 2004 * 92 * 246804 kravchenko s v and sarachik m p 2004 _ rep . phys . _ * 67 * 1 ( and references therein ) anissimova s , kravchenko s v , punnoose a , finkelstein a m and klapwijk t m 2007 _ nature phys . _ * 3 * 707 punnoose a and finkelstein a m 2005 _ science _ * 310 * 289 gunawan o , gokmen t , vakili k , padmanabhan m , de poortere e p and shayegan m 2007 _ nat . phys . _ * 3 * 388 tanatar b and ceperley d m 1989 _ phys . _ b * 39 * 5005 kwon y , ceperley d m and martin r m 1993 _ phys . b _ * 48 * 12037 rapisarda f and senatore g 1996 _ aust . j. phys . _ * 49 * 16 conti s and senatore g 1996 _ europhys . * 36 * ( 9 ) 695 varsano d , moroni s and senatore g 2001 _ europhys . * 53 * 348 attaccalite c , gori - giorgi p , moroni s and bachelet g b 2002 _ phys . rev . lett . _ * 88 * 256601 gori - giorgi p , moroni s and bachelet g b 2004 b * 70 * 115102 marchi m , de palo s , moroni s and senatore g 2008 the correlation energy and the spin susceptibility of the two - valley two - dimensional electron gas _ preprint _ cond - mat/0808.2569 giuliani g and vignale g 2005 _ quantum theory of the electron liquid _ ( cambridge : cambridge university press ) 35 and appendix a.4 de palo s , botti m , moroni s and senatore g 2005 _ phys . rev . lett . _ * 94 * 226405 ( 2005 ) dharma - wardana m w c and perrot f 2004 _ phys . _ b * 70 * 35308 foulkes m , mitas l , needs r j and rajagopal g 2001 _ rev . * 73 * 33 lin c , zong f h and ceperley d m 2001 _ phys . _ e * 64 * 016702

we present component - resolved and total pair distribution functions for a 2deg with two symmetric valleys . our results are based on quantum monte carlo simulations performed at several densities and spin polarizations .

low - mass pre - main - sequence ( pms ) stars are generally separated in two different classes , accreting classical ttauri stars ( cttss ) with broad h@xmath6 emission lines , blue continuum and near infrared excess and non - accreting weak - line ttauri stars ( wttss ) with narrow symmetric h@xmath6 emission lines ( e.g. * ? ? ? while cttss typically show large excess emission from the near - infrared to the millimeter , wttss often have no infrared ( ir ) excess at all . only a relatively small fraction of ttauri stars are observed in an intermediate transition state with little or no near - ir excess and significant far - ir excess . this clearly indicates that once the inner disk starts to dissipate , the entire disk disappears very rapidly @xcite . the missing near - ir excess combined with the clear presence of an outer disk is the defining characteristic of transition disks . however , a precise and generally accepted definition of what constitutes a transition disk object does not yet exist . the most conservative definition of transition disks , often labeled _ classical transition disks _ , consists of objects with no detectable near - ir excess , steeply rising slopes in the mid - ir , and large far - ir excesses ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? being less restrictive , objects with small , but still detectable , near - ir excesses ( e.g. * ? ? ? * ; * ? ? ? * ) can be included , until considering objects with decrement relative to the taurus median spectral energy distribution ( sed ) at any or all wavelengths ( e.g. * ? ? ? * ; * ? ? ? * ) . throughout this paper we follow the latter and broader definition . however , one has to be aware that this broad definition still is mostly sensitive to inner opacity holes but may overlook pre - transitional disks with a gap separating an optically thick inner disk from an optically thick outer disk . such systems have been identified from spitzer irs spectra @xcite , but can be missed by photometric selection alone . the _ spitzer _ space telescope generated a huge database containing ir observations of pms stars in star - forming regions . most importantly , _ spitzer _ products such as the catalogs of the _ cores to disks ( c2d)_del@xmath7document.pdf ] and _ gould belt spitzer ( gb ) _ legacy projects @xcite provide seds from 3.6 to 24 @xmath8 m for large numbers of pms stars . one of the most interesting results concerning transition disk studies with _ spitzer _ has been the great diversity of sed morphologies ( see * ? ? ? * for a review ) . the widespread of ir sed morphologies found in transition disk objects can not be adapted to the classical taxonomy to describe young stellar objects ( ysos ) such as the class i , ii , iii definitions from @xcite . @xcite quantified the richness of sed morphologies in terms of two parameters based on the sed shapes considering the longest wavelength at which the observed flux is dominated by the stellar photosphere , @xmath9 , and the slope of the infrared excess , @xmath10 , computed from @xmath9 to 24 @xmath8 m . studying the diverse population of transition disks is key for understanding circumstellar disk evolution as much of the diversity of their sed morphologies is likely to arise from different physical processes dominating the disk s evolution . evolutionary processes that may play an important role include viscous accretion @xcite , photoevaporation @xcite , the magneto - rotational instability ( _ mri _ ) @xcite , grain growth and dust settling @xcite , planet formation @xcite and dynamical interactions between the disk and stellar or substellar companions @xcite . as discussed by @xcite , one can distinguish between some of these processes if certain observational constraints , in addition to the seds , are available . to this end , we are performing an extensive ground - based observing program to obtain estimates for the disk masses ( from submillimeter photometry ) , accretion rates ( from the velocity profiles of the h@xmath6 line ) , and multiplicity information ( from ao observations ) of _ spitzer_-selected disks in several nearby star - forming regions . our recently completed study of ophiuchus objects ( * ? ? ? * hereafter paper i ) confirms that transition disks are indeed a very heterogeneous group of objects with a wide range of sed morphologies , disk masses ( @xmath11 0.5 to 40 m@xmath1 ) and accretion rates . since the properties of the transition disks in our sample point towards different processes driving the evolution of each disk , we have been able to identify strong candidates for the following disk categories : ( giant ) planet - forming disks , circumbinary disks , grain - growth dominated disks , photoevaporating disks , and debris disks . we here follow the same approach as in paper i in performing multiwavelength observations to derive estimates on disk masses , accretion rates , and multiplicity . we present submillimeter wavelength photometry ( from apex ) , high - resolution optical spectroscopy ( from the clay , and du pont telescopes ) , and adaptive optics near - ir imaging ( from the vlt ) for _ spitzer_-selected transition circumstellar disks located in the following star forming regions : a ) lupus : i , iii , iv , v , vi , b ) corona australis ( cra ) , and c ) scorpius ( scp ) . the lupus clouds constitute one of the main southern nearby low - mass star - forming regions containing the following sub - clouds at slightly different distances : lupus i , iv , v , vi at 150 @xmath12 20 pc and lupus iii at 200 @xmath12 20 pc @xcite . the clouds are situated in the lupus - scorpius - centaurus ob association spanning over 20 deg in the sky . their population is dominated by mid m - type pms stars , but some very late m stars or substellar objects have been found as well thanks to _ spitzer _ capabilities ( see * ? ? ? * for a review ) . in general , the ages of the lupus clouds are estimated to be @xmath13 myr , @xcite . however , a comprehensive analysis using _ spitzer _ irac and mips observations in combination with near - ir ( 2mass ) data has been performed for lupus i , iii , and iv by the _ c2d _ legacy project @xcite and lupus v , vi by the _ gould belt _ legacy project @xcite and revealed a significant difference between the sub - clouds . while lupus i , iii , and iv are dominated by class ii ysos , lupus v , vi mostly contain class iii objects . this has been interpreted as a consequence of lupus v , vi being a few myrs older than lupus i , iii , and iv by @xcite . in any case , the lupus star - forming regions represent an excellent test - bed for theories of circumstellar disk evolution as their stellar members should span all evolutionary stages . the scorpius clouds @xcite lie on the edge of the lupus - scorpius - centaurus ob association , just north of the well studied ophiuchus molecular cloud , but it is highly fragmentary and presents much lower levels of star - formation . in fact , the _ gould belt _ project only finds 10 ysos candidates in the 2.1 sq deg . mapped by irac and mips ( hatchell et al . , in preparation ) . the age of scp is estimated to be @xmath14 myr @xcite . the cra star - forming region , also mapped by the _ gould belt _ project @xcite , contains an embedded association known as the coronet , a relatively isolated cluster containing haebe stars and ttauri stars @xcite . it is situated at a distance of 150 @xmath12 20 pc out of the galactic plane , at the edge of the _ gould belt _ ( see * ? ? ? * and references therein ) . with an age of @xmath15 1 myr , the coronet is younger than the lupus clouds and has been claimed to host an intriguingly high fraction of classical transition disks of @xmath16 @xcite . however , @xcite convincingly shows that the dust emission in ttauri stars of spectral type m is very small short ward of @xmath17 m which might mimic an inner hole , the defining feature of typical transition disk systems . based on this finding , @xcite estimate a much smaller fraction of transition disks in coronet , of @xmath18 . we have systematically searched the catalogs of the _ c2d _ and _ gould belt _ legacy projects applying the broad transition disk definition described in detail in paper i to the lupus i , iii , iv , v , vi , scp , and cra clouds . in brief , we select systems that fulfill the following criteria . 1 . have spitzer colors [ 3.6]-[4.5 ] @xmath11 0.25 , which excludes fulls disks " , i.e. , optically thick disks extending inward to the dust sublimation radius except in cases with significant dust settling in inner disks around m stars @xcite . 2 . have spitzer colors [ 3.6]-[24 ] @xmath19 1.5 , to ensure that all targets have very significant excesses ( @xmath19 5 - 10 @xmath20 ) , unambiguously indicating the presence of circumstellar material . 3 . have s / n @xmath21 7 in 2mass , irac , mips ( 24 @xmath8 m ) bands to only include targets with reliable photometry . 4 . have k@xmath22 @xmath11 11 mag , driven by the sensitivity of our near - ir adaptive optics observations and to avoid extragalactic contamination . 5 . are brighter than r = 18 mag according to the usno - b1 @xcite , driven by the sensitivity of our optical spectroscopy observations . compared to the c2d sample discussed in @xcite our sample might be slightly biased against very low mass stars and deeply embedded objects because of this brightness limit . these selection criteria result in a primary target list of 60 objects that we did follow - up using different observational facilities to characterize our transition disk candidate sample . we performed multiwavelength ( optical , infrared , and submillimeter ) observations of our targets with the aim to identify which physical process is primarily responsible for their transition disk nature . high resolution optical spectra can be used to estimate spectral types and accretion rates from the velocity dispersion of the h@xmath6 . near - ir images allow to identify multiple star systems down to projected separations of 0.06 - 007@xmath23 , corresponding to @xmath24 au at distances of @xmath25 pc . from single dish submillimeter observations we inferred disk masses . in the following section , we describe in detail the observations performed and the data reduction . we obtained high resolution ( r @xmath19 20,000 ) spectra for our entire sample using 2 different telescopes : magellan / clay and du pont located at las campanas observatory in chile . we observed 49 of our 60 targets with the magellan inamori kyocera echelle ( mike ) spectrograph on the 6.5-m clay telescope . the observations were performed on 2009 april 2728 and 2010 june 1113 . since the ccd of mike s red arm has a pixel scale of 0.13@xmath23/pixel , we binned the detector by a factor of 3 in the dispersion direction and a factor of 2 in the spatial direction , thus reducing the readout time and readout noise . we used an 1@xmath23 slit width . the resulting spectra covered 49005000 at a resolving power of 22,000 . this corresponds to a resolution of @xmath260.3 at the location of the h@xmath6 line , and to a velocity dispersion of @xmath2614 km s@xmath5 . for each object , we obtained a set of 3 or 4 spectra , with exposure times ranging from 3 to 10 minutes each , depending on the brightness of the targets . the data analysis was carried out with iraf software . after bias subtraction and flat - field corrections with milky flats , the spectra were reduced using the standard iraf package imred : echelle . the remaining 11 targets were observed with the echelle spectrograph on the 2.5-m irne du pont telescope . the observations were performed in 2009 may 1416 , and we used an 1@xmath23 slit width . the ccd s scale is 0.26@xmath23/pixel , and we consequently applied a 2@xmath272 binning . the wavelength coverage of the obtained spectra ranged between 4000 and 9000 at a resolving power of 32,000 in the red arm . this corresponds to a resolution of @xmath260.2 and a velocity dispersion of @xmath269.4 km s@xmath5 in the vicinity of h@xmath6 . for each object we obtained a set of 3 to 4 spectra with exposure times ranging from 10 to 15 minutes each , depending on the brightness of the target . the data analysis was carried out with iraf . after bias subtraction and flat - field corrections with milky flats , the spectra were reduced using the standard package imred : echelle . high spatial resolution near - ir observations of our 60 targets were obtained with naco ( the nasmyth adaptive optics systems ( naos ) and the near - ir imager and spectrograph ( conica ) camera at the 8.2-m telescope yepun ) , which is part of the european southern observatory s ( eso ) very large telescope ( vlt ) in cerro paranal , chile . the data were acquired in service mode during the eso s observing period 083 ( 2009 april 1 september 30 ) . to take advantage of the near - ir brightness of our targets , we used the infrared wavefront sensor and the n90c10 dichroic to direct 90@xmath28 of the near - ir light to the adaptive optics systems and 10@xmath28 of the light to the science camera . we used the s13 camera ( 13.3 mas / pixel and 14@xmath2714@xmath23 field of view ) and the double rdrstrd readout mode . the observations were performed through the k@xmath29 and j - band filters at 5 dithered positions per filter . the total exposure times ranged from 1 to 50 s for the k@xmath29-band observations and from 2 to 200s for the j - band observations , depending on the brightness of the target . the data were reduced using the jitter software , which is part of eso s data reduction package eclipse . as discussed in the following section , our spectroscopic observations showed that our initial sample of 60 transition disk candidates was highly contaminated by asymptotic giant branch ( agb ) stars . the 17 bona fide pms stars were observed with the atacama pathfinder experiment ( apex ) , the 12-m radio telescope located in llano de chajnantor in chile . the observations were performed during period 083 ( 083.f-0162a-2009 , 9.2 hrs ) and period 085 ( e-085.c-0571d-2010 , 30.9 hrs ) . we used the apex - laboca camera @xcite at 870 @xmath8 m ( 345 ghz ) in service mode aiming for detections of the dust continuum emission . the nominal laboca beam is full width at half - maximum @xmath30 and the pointing uncertainty is @xmath31 . to obtain the lowest possible flux limit , the most sensitive part of the array was centered on each source . the observations were reduced using the bolometer array data analysis package boa@xmath32 . for both observing runs , skydips were performed hourly and combined with radiometer readings to obtain accurate opacity estimates . the absolute flux calibration follows the method outlined by @xcite and is expected to be accurate to within 10@xmath28 . the absolute flux scale pointing calibrators were determined through observations of either iras16342 - 38 or g34.3 while planets were used to focus the telescope . the telescope pointing was checked regularly with scans on nearby bright sources and was found to be stable within 3 ( rms ) . the period 083 observations were performed using compact mapping mode with raster spiral patterns . the weather conditions were excellent with precipitable water vapor levels below @xmath33 mm . eight sources ( objects # 1 , 2 , 5 , 9 , 12 , 15 , 16 , 17 ) were observed . on - source integrations of 64 minutes were performed to achieve an rms @xmath26 7 mjy / beam . the brightest object of the whole sample ( # 2 ) was the only source detected at submillimeter wavelengths in period 083 . during the longer period 085 observing run , the beam switching mode using the wobbling secondary and mapping mode were employed . during this period , the remaining nine sources were observed and object # 12 was re - observed with higher sensitivity . the weather conditions were favorable with precipitable water vapour levels below 1.2 mm . the wobbler observations of each target consist of a set of two loops of 10 scans per target , reaching a total on - source observing time of 48 minutes . an average rms @xmath26 4 mjy / beam was obtained . in the case of a signal detection on - source position , we took a few maps in order to check for emission contamination from the off - position . in all cases , the contamination was discarded and we confirmed the detection of six sources ( # 3 , 7 , 8 , 10 , 11 , and 12 ) . agb stars are surrounded by shells of dust and thus have small , but detectable , ir excesses . spitzer_-selected yso samples from _ c2d _ and _ gould belt _ catalogs are therefore contaminated by agb stars . using high resolution optical spectra , we discovered that 43 objects of our candidates are agb stars , while the remaining 17 targets are spectroscopically confirmed ttauri stars . we separated contaminating agb stars from genuine transition disk ttauri stars in the same way as in paper i , i.e. based on the presence / absence of emission lines associated with chromospheric activity and/or accretion and the presence of the li 6707 absorption line indicating stellar youth . the coordinates , _ spitzer _ names , the usno - b1 r - band magnitude , and the near to mid - ir fluxes of the agb stars contaminating our sample of transition disks are compiled in table [ t : giants ] . as shown in table [ t : cont ] the fractional contamination due to agb stars of our color selected transition disk candidates differs significantly between the different clouds . the number of transition disk candidates is far too small in the case of lupus i and scp to draw any conclusions . our lupus iii , iv , and cra samples are contaminated by a fraction of agb stars that is more or less consistent with the contamination in ophiuchus ( see paperi , section 4.1.2 ) . the small number of transition disks in cra seems to be in contradiction with the larger sample identified by @xcite . however , our selection criteria contain relatively strong brightness constraints ( in particular @xmath34 ) due to the design of our follow - up program which excludes most of the systems listed by them . in addition , as mentioned in the introduction , a large fraction of the transition disk candidates of @xcite might be classical m - dwarf ttauri stars with intrinsically little near ir excess due to the small color contrast between the disk and the stellar photosphere @xcite . apparently , lupusv and vi are dramatically more contaminated than lupus iii , iv , i.e. all the color - selected transition disk candidates are in fact agb stars . this high percentage of contamination is perhaps related to the position in the galaxy ( see table [ t : cont ] ) . the lupus complex occupies 334 @xmath11 l @xmath11 352 , + 5 @xmath11 b @xmath11 + 25 , i.e. observing lupusv and vi we are looking towards the galactic center closer to the plane . in contrast , cra and ophiuchus ( paper i ) are located at higher galactic latitudes . in any case , the absence of any spectroscopically confirmed transition disk in lupusv , vi puts doubts on the finding of ( * ? ? ? * see their section 5.1 ) that the high fraction of classiii lada systems can _ not _ be explained by contamination . so far all classiii yso candidates from these clouds that have been followed up spectroscopically are clearly contaminating background giants . our sample of transition disks in lupus v , vi shares 30 classiii objects and one classii object with the sample investigated by @xcite . all these 31 objects turned out to be agb stars which means that at least @xmath35 and potentially much more of the classiii objects from @xcite are not ysos but giant stars . this result also questions the conclusion of @xcite that lupus v , vi are significantly older than lupus i , iii . the generally large fraction of giant stars in our sample of transition disk candidates allows to investigate possible refinements of our color selection algorithm . figure [ f : ccont ] shows the color - color diagram of the 60 selected southern transition disk targets of this paper . all transition disk candidates in our sample with @xmath36-[24 ] < 1.8 $ ] turned out to be giant stars . this agrees quite well with the results obtained for the ophiuchus sample where 4/6 transition disk candidates with [ 3.6]-[24 ] @xmath11 1.8 had to be classified as giant stars ( see paperi , fig.1 ) . consequently , one may derive an estimate of the contamination of yso catalogs due to background giant stars by applying this simple color cut . figure[f : c2d ] and [ f : gb ] show the transition disk candidates and agb candidates for both the _ c2d _ and _ gb _ catalogs including all star forming regions . note , that we here apply color selection criteria only , i.e. the requirements of @xmath37 mag and @xmath38 mag that have been used to define the transition disk candidate sample for our multiwavelength follow - up program are not incorporated . instead , here we are interested in estimating the fraction of yso candidates in a given cloud that are likely to be agb stars based on their very low @xmath39 m excess . the resulting rough estimates of giant star contamination are given in table [ giants - all ] separated by catalog and cloud . according to these estimates , the agb contamination is expected to vary significantly ranging from @xmath40 per cent . this shows that agb contamination can have an important impact on studies that are based on the pure numbers of ysos as provided by the _ c2d _ and _ gb _ catalogs . for example , star formation rates as determined e.g. in @xcite might become significantly smaller if agb contamination is taken into account . apparently , applying the new more restrictive color selection could also significantly increase the success - rate of identifying ysos directly from color selection criteria and future follow - up studies may take this into account . the _ spitzer _ and alternative names , 2mass and _ spitzer _ fluxes , and the usno - b1 r - band magnitudes and the relevant information derived from our follow - up observations for the remaining 17 bona fide transition disk candidates are listed in tables [ t : mags ] and [ t : obs ] . in what follows , we use the data discussed in section [ obs ] to characterize our sample of transition disks . in oreder to determine the spectral types of the transition disks in our sample we use the equations by @xcite that empirically relate the spectral type with the strength of the tio5 molecular band . the uncertainty of this method is estimated to be @xmath260.5 subclasses . for most of our transition disk objects , estimates of the spectral types have been provided previously @xcite . the spectral types obtained by us and those given in the literature are listed in table [ t : obs ] and we find good agreement . all but one system ( # 2 ) have been classified as m - dwarfs . for target # 2 , we adopt the spectral type k0 given by @xcite . binarity can play an important role in the context of transition disks as the presence of a close stellar companion may cause the inner hole , i.e. some of the transition disks in our sample might actually be nothing else but circumbinary disks . some systems in our sample have been previously identified as wide binaries . @xcite carried out an optical survey of the lupus i , iii , iv regions using bands rc , ic , zwi of the wide - field imager ( wfi ) attached to the eso 2.2-m telescope at la silla observatory . visual inspection of the images revealed that objects # 1 , # 3 , # 4 , and # 9 are wide binary systems with companions at projected separations of 420 , 1000 , 600 , and 560 au considering the distance of the lupus clouds . we have newly identified six multiple systems by visual inspection of the naco images , objects # 6 , # 7 , # 11 , # 13 , # 16 , and # 17 ( see figure [ f : mult ] ) . the projected separations are 0.7@xmath23 , 0.4@xmath23 , 1.15@xmath23 , 1.8@xmath23 , 0.5@xmath23 , 0.5@xmath23 corresponding to 140 , 80 , 230 , 234 , 75 , 75 au at the corresponding distances . object # 13 is a triple system , i.e. a binary with an additional faint companion at 3@xmath23 ( 390 au ) . for each binary system , we searched for additional tight companions by comparing each other s point - spread functions ( psfs ) . the psf pairs were virtually identical in all cases , except for target # 11 . the south - west component of this target has a perfectly round psf , while the south - east component , @xmath261.5@xmath23 away , is clearly elongated ( see figure [ f : mult ] ) . since variations in the psf shape are not expected within such small angular distances and this behavior is seen in both the j- and k@xmath22-band images , we conclude that target # 11 is likely to be a triple system . we have also searched in the literature for additional companions in our sample that our vlt observations could have missed . in addition to multiple systems discussed above , we find that # 14 has been reported by @xcite as a binary system with a projected separation of 0.13@xmath23 ( corresponding to 20 au ) and flux ratio of 0.7 using speckle interferometry at the new technology telescope in 2001 . we see no evidence for a bright companion in our naco images ( see figure [ f : mult ] ) . however , since our ao images were taken 8 years later than the speckle data , it is possible that the projected separation had changed enough in the intervening years for the binary to become unresolved . hence , our sample consists of eleven multiple systems , i.e. nine binaries ( objects # 1 , 3 , 4 , 6 , 7 , 9 , 14 , 16 , 17 ) , and two triples ( objects # 11 and 13 ) . only in the cases of the close b / c pair in object # 11 and # 14 the binary separation is small enough to suspect that the companions might have tidally disrupted the circumbinary disk thereby causing the inner hole inferred from the sed . however , in neither case the circumbinary nature can be confirmed because it is unknown whether the ir excess in object # 11 originates in the wide a component or the close b / c pair and the multiplicity of object # 14 is not confirmed by our observations . we therefore only consider these two objects to be circumbinary disk _ candidates_. table [ t : obs ] lists the projected angular separations of the systems . @xcite modeled the ir and submillimeter seds of circumstellar disks and found a linear relation between the submillimeter flux and the disk mass that has been calibrated by @xcite who obtained @xmath41~m_{\rm{jup}},\end{aligned}\ ] ] where @xmath42 is the distance to the target . as described in paper i , disk masses obtained with the above relation are within a factor of 2 of model derived values , which is certainly good enough for the purposes of our survey . however , one should keep in mind that larger systematic errors can not be ruled out @xcite as long as strong observational constraints on the grain size distributions and the gas - to - dust ratios are lacking . adopting distances of 150 pc to lupus iv , 200 pc to lupus iii @xcite , 130 pc to scp ( hatchell et al . in preparation ) , and 150 pc to cra @xcite we use equation ( [ eq : mass ] ) to estimate disk masses for the 17 systems in our sample ( see table [ t : derived ] ) . 50@xmath28 ( i.e. 7/17 ) of the targets have been detected at 8510 @xmath8 m ( table [ t : obs ] ) . the corresponding disks masses range from @xmath43 m@xmath1 . adopting a flux value of 3@xmath20 for targets with non - detected emission , we derive upper limits for the remaining targets of @xmath44 m@xmath1 . most of our targets have disk masses @xmath45 m@xmath1 , but 5 targets have disk masses typical for cttss ( @xmath263 10 m@xmath1 ) . the most massive disks are detected around , with 9 and 6 m@xmath1 , respectively . the accretion rate is the second crucial parameter necessary to distinguish between the different mechanisms that may form inner opacity holes in circumstellar transitions disks . most pms stars show h@xmath6 emission , either generated from chromospheric activity or magnetospheric accretion @xcite . while non - accreting objects show rather narrow ( @xmath46 km s@xmath5 ) and symmetric line profiles of chromospheric origin , the large - velocity magnetospheric accretion columns produce broad ( @xmath47 km s@xmath5 ) and asymmetric line profiles . as in paperi we estimate the accretion rates of our transition disk systems according to the empirical relation obtained by @xcite , i.e. @xmath48 which is supposed to be relatively well calibrated for velocity widths covering , which corresponds to accretion rates of @xmath49 . however , the empirical dividing line between accreting and non - accreting objects has been placed slightly shifted by different authors at @xmath50 between 200 km s@xmath5 @xcite and 270 km s@xmath5 @xcite . for systems with @xmath51 km s@xmath5 we therefore separate accreting and non - accreting objects based on the ( a)symmetry of the h@xmath6 emission line profile and take into account the spectral type because accreting lower mass stars tend to have narrower h@xmath6 emission lines . to measure the h@xmath6 velocity width @xmath50 we considered for each system the spectral range that corresponds to h@xmath52 km / s . the continuum plus emission profile were fitted with a gaussian plus parabolic profile . the parabolic fit was then used to normalized the spectrum . a single gaussian profile was sufficient here , being the emission single or double - peaked , as at this stage we were only interested in obtaining a good parabolic fit for the normalization . once the continuum had been normalized we measured @xmath50 at 10 per cent of the peak intensity . the obtained velocity dispersion of the h@xmath6 emission lines and the corresponding accretion rate estimates are given in table [ t : obs ] and table [ t : derived ] , respectively . the obtained accretion rates should be considered order - of - magnitude estimates due to the large uncertainties associated with equation ( [ eq : acc ] ) and the intrinsic variability of accretion in ttauri stars . our sample shows a large diversity of h@xmath6 signatures . five targets are classified as non - accreting objects that clearly show symmetric and narrow h@xmath6 emission line profiles ( @xmath11 200 km s@xmath5 , see figure [ f : acc1 ] ) as expected from chromospheric activity . for all these non - accreting objects , we estimate an upper limit of the accretion rate , i.e. . we classify the remaining 12 transition disk objects as accreting . the majority of them ( 10 ) show clearly broad and asymmetric emission - line profiles ( see figure [ f : acc2 ] ) . however , targets # 8 and # 12 represent borderline cases with a rather small velocity dispersion for accreting systems ( @xmath53 km s@xmath5 ) and not clearly asymmetric line profiles . such borderline systems require a more detailed discussion . both stars are of late spectral type ( m4-m5 ) and very low - mass stars tend to have narrower h@xmath6 lines than higher mass objects because of their lower accretion rates @xcite and their lower gravitational potentials @xcite . object # 12 additionally shows @xmath54 m excess emission indicating the presence of an inner disk . given all the available data , we classify targets # 8 and # 12 as accreting objects , but warn the reader that their accreting nature is less certain than that of the rest of the objects classified as cttss . as accretion in ttauri stars can well be episodic , mutli - epoch spectroscopy would be useful to unambiguously identify the accreting nature of these two systems . the mass accretion rates estimated for the 12 disks classified as accreting systems range from with the collected information presented in the previous sections we have at hand the following information of the _ spitzer_-selected transition disks in our sample : * detailed seds that we quantify with the two - parameter scheme introduced by @xcite , which is based on the longest wavelength at which the observed flux is dominated by the stellar photosphere , @xmath9 , we compare the extinction - corrected sed with nextgen models @xcite normalized to the j - band and choose @xmath9as the longest wavelength at which the stellar photosphere contributes over 50% of the total flux . the uncertainty of @xmath9is roughly one sed point . ] , and the slope of the ir excess , @xmath10 , computed as @xmath55 between @xmath9 and @xmath56@xmath8 m . * multiplicity information from the literature and ao ir imaging . * disk mass estimates based on measured submillimeter flux . * accretion rate estimates derived from h@xmath6 line profiles . this information allows to separate the sample according to the physical processes that are the most likely cause of the inner opacity hole : grain growth , planet formation , photoevaporation , or close binary interactions . in what follows we briefly review each process that might be responsible for the formation of the inner opacity holes , describe our criteria for classifying transition disks , and discuss the corresponding sub - samples of transition disks obtained . the presence of accretion in classical transition disk objects raises the question how the inner disk can be cleared of small grains while gas remains in the dust hole . the two mechanisms that can explain the coexistence of accretion and inner opacity holes are grain growth and dynamical interactions with ( sub)stellar companions . due to both , the higher densities and the faster relative velocities of particles in the inner parts of the disk , this disk region offers much better conditions for dust agglomeration than the outer parts of the disk . therefore , significant grain growth should start in the inner disk regions . as soon as the grains grow to sizes exceeding the considered wavelength ( @xmath57 ) , the opacity decreases until an inner opacity hole forms . early models by @xcite taking into account only dust coagulation predict much too short timescales of the order of @xmath58yrs to clear the entire disk of small grains , which is inconsistent with observed seds of most classical ttauri stars . a more reliable picture combines coagulation and collisional fragmentation or erosion of large dust aggregates @xcite . as a gradual transition between the inner and outer disk is predicted by grain growth and dust settling models @xcite , grain growth dominated disks should have @xmath59 ( i.e. , falling mid - ir seds ) while @xmath60 , associated with the size of the hole , can differ over a rather wide range of values . although grain growth does not directly affect the gas , it may increase accretion because the inner opacity hole can lead to efficient ionization and trigger the _ mri _ instability @xcite . among the 17 transition disks in our sample , nine objects are accreting and are associated with @xmath10 @xmath61 . the corresponding seds are shown in figure [ f : grain - seds ] . the grain growth candidate systems in our sample could be confounded with accreting classical ttauri m - stars as predicted by @xcite . however , most of our grain growth dominated disks have seds close to the stellar photosphere up to @xmath62 m and we therefore do not expect significant contamination by non transition disks . the only exceptions being objects # 5 and 10 with a small value of @xmath9@xmath63 and to some extend # 4 and # 11 ( @xmath9@xmath64 ) . we recommend the reader to keep in mind the uncertain classification of these four systems . compared with the oph sample ( paper i ) the accretion rates obtained for grain growth dominated disks are slightly smaller ( i. e. @xmath65m@xmath4yr@xmath5 ) . this might be related to the slightly lower stellar masses or to advanced viscous evolution as discussed in @xcite . the grain growth dominated disks are located in the lupus iii , iv ( 8) and the cra ( 1 ) star - forming regions . the truncation of the disk as the result of dynamical interactions with companions was first proposed by @xcite . more recently , it has been shown that most pms stars are in multiple systems with a lognormal semi - major axis distribution centered at @xmath66au ( e.g. , * ? ? ? . a significant fraction of the binaries in the star - forming regions considered here should therefore be tight binaries with separations of @xmath67au . disks in such close binary system will be tidally truncated at @xmath68 the binary separation and a circumbinary disk with an inner hole is formed @xcite . the corresponding sed is that of a transition disk . however , the circumbinary nature does not exclude additional evolutionary processes to be at work and we therefore provide an additional classification based on the disk properties only ( see table [ t : derived ] ) . identification of companions that may cause the formation of a circumbinary disk is possible either due to high - resolution imaging or by measuring radial velocity variations . as described in sect . [ s : naco ] , we identified 2 circumbinary disk candidates among our 17 transition disk systems . one of them , object # 11 was discovered by inspecting the naco images obtained with the vlt , while the close binary nature of object # 14 has been discovered by @xcite using speckle interferometry at the ntt . object # 11 shows signs of strong accretion and has a sed with @xmath69 in agreement with grain growth . it is currently not clear under which conditions gap - crossing streams can exist and allow accretion onto the central star to proceed , but signs of accretion in circumbinary systems @xcite indicate that accretion is likely to continue . on the other hand , object # 14 is a non - accreting system such as the known binary coku tau@xmath704 @xcite . according to its l@xmath71/l@xmath72 ratio we classify this system as a circumbinary / photoevaporation disk candidate . as a final note of caution , we would like to stress that both objects discussed above ( # 11 and # 14 ) are circumbinary disk _ candidates_. as all but one of our targets are m - type stars , most companions potentially responsible for their transition disk seds are expected to lie at closer separations than those probed by the ao images . therefore , our sample of circumbinary disk candidates is incomplete and heavily biased towards large separations . methods more sensitive to closer companions such as aperture masking and/or radial velocity observations are required to draw firm conclusions on circumbinary disk fractions . the most exciting way to produce a transition disk sed is by giant planet formation . according to early models as well as recent numerical simulations , the formation of giant planets involves the formation of gaps and holes in the circumstellar disk @xcite . as in the case of ( sub)stellar companions it is uncertain if and to what extent accretion proceeds in the presence of a forming giant planet . therefore , the most important sign of ongoing planet formation remains a sharp inner hole , usually corresponding to @xmath10@xmath73 ( i.e. , a rising mid - ir sed ) . however , although very useful , the definition of @xmath10 is incomplete , as the sed may also steeply rise at wavelengths longer than @xmath74 m . a spectacular example illustrating this is given by object # 3 . while @xmath10@xmath75 , the sed steeply rises between @xmath74 m and @xmath76 m . furthermore , object # 3 shows clear signs of accretion ( @xmath77 m@xmath4/yr ) and of harboring a relatively massive disk ( m@xmath71@xmath78 m@xmath1 ) . since this is a very atypical object , we verified that the large 70 @xmath8 m flux is not contaminated by extended emission from the molecular cloud . we examined the 24 and 70@xmath8 m mosaics and verified that the detections are consistent with point sources at the source location ( see figure [ f : tran6 ] ) . a more typical transition disk system that might represent a currently planet forming disk is target # 15 with a clearly positive value of @xmath10 and a high accretion rate . a borderline case between grain growth and planet forming disks is object # 2 . a high accretion rate combined with @xmath10@xmath79 could be consistent with both scenarios . keeping in mind the ambiguity , we classify object # 2 as a planet forming disk candidate because it could potentially be an extremely interesting object . the sed of object # 2 might be explained by a discontinuity in the grain size distribution rather than an inner opacity hole . while the inner part of the disk still contains small grains , outer regions of the disk might be dominated by slightly larger dust particles . such a scenario is in excellent agreement with the predictions of numerical simulations performed by @xcite . they show that the planet disk interaction at the outer edge of the gap cleared by a planet can act as a filter passed by small particles only which produces a discontinuity in the dust particle size . to firmly establish its nature object # 2 deserves further follow - up observations ( e.g. , high resolution imaging with alma ) . the seds of the three candidates for ongoing giant planet formation in our sample are shown in figure [ f : planet - seds ] . the hosting forming giant planets candidates are located in the lupus iii , iv ( 2 ) and the cra ( 1 ) star - forming regions . the second main class of transition disks are those that do not show signs of accretion . in such disks the inner opacity hole , i.e. the lack of small dust particles in the inner disk regions , is likely to coincide with a gas hole , i.e. the inner disk is completely drained . the most important process for clearing the inner disk in transitions disks that do not accrete is photoevaporation ( e.g. * ? ? ? * ) . according to this model , extreme - ultraviolet ( euv ) photons , originating in the stellar chromosphere , ionize and heat the circumstellar hydrogen which is then partly lost in a wind . this process is supposed to work in all circumstellar disks but becomes important only when the accretion rate drops to values similar to the photoevaporation rate . then , the inner disk drains on the viscous timescale supported by the generation of the _ mri _ @xcite . once an inner hole has formed , the inner disk rim is efficiently radiated and the entire disk should therefore quickly disappear . photoevaporating disks should have negligible accretion @xcite . to separate photoevaporating disks from debris disks , we require the disk luminosity to be higher than that seen in the brightest bona - fide debris disks , i.e. l@xmath71/l@xmath72 @xmath80 @xcite . we thus obtained @xmath81 m upper limits from the noise of the @xmath81 m _ spitzer _ mosaics at the source location and calculated l@xmath71/l@xmath72 for our sample by integrating the stellar fluxes and disk fluxes over frequency ( see section 5.1.3 in paper i for details of the @xmath81 m data analysis and the l@xmath71/l@xmath72 calculation ) . we classify three transition disks as photoevaporating disk candidates with negligible accretion ( m@xmath82 @xmath11 10@xmath83 yr@xmath5 ) but l@xmath71/l@xmath72 @xmath80 ( table [ t : derived ] ) . according to our submillimeter measurements , all these three systems have small disk masses ( @xmath84 m@xmath1 , table [ t : derived ] ) . in fact , for all photoevaporation candidates we could only derive upper limits on the disk mass . the seds of the three systems classified as photoevaporating disk objects are shown in figure [ f : photo - seds ] . the photoevaporated disks are located in the lupus iii , iv ( 2 ) and cra ( 1 ) star - forming regions . photoevaporation can be considered as a transition stage between primordial and debris disks . debris disks contain a small amount of dust and are gas - poor . we find two debris disk candidates , i.e. non - accreting systems with l@xmath71/l@xmath72@xmath85 , among our 17 transition disks ( see figure [ f : debris - seds ] ) . the debris disks are located in the cra ( 1 ) and scp ( 1 ) star - forming regions . in the previous sections , we presented detailed follow - up observations of 60 _ spitzer_-selected transition disk candidates located in the southern star - forming regions lupus i , iii , iv , v , vi , cra , and scp . optical spectroscopy revealed that only 17 systems of these candidates are genuine transition disk ttauri stars . deriving estimates for the accretion rates , disk masses , and multiplicity of these 17 systems we classified them as dominated by grain growth ( 9 ) , giant planet formation ( 3 ) , photoevaporation ( 3 ) , or being in the final debris disk stage ( 2 ) . two of these transition objects , one grain growth ( # 11 ) and one photoevaporating ( # 14 ) , are circumbinary disk candidates , which offers the possibility of tidal truncation as mechanism responsible for an inner hole in the common / shared disk . combining these results with those presented in paperi , we now have at hand well - defined and well characterized samples of transition disks from several different star - forming regions . figure [ f : alp - lam ] summarizes the properties of these samples based on @xmath10 and @xmath9 . the main aim of these series of papers is to progress with our understanding of circumstellar disk evolution and to compare transition disk samples of different clouds is key in this respect . table [ t : fractions ] shows the fractions of different types of transition disks for ophiuchus ( age @xmath15 @xmath86 myr , ( * ? ? ? * and references therein ) ) , cra ( age @xmath15 1 myr , @xcite ) , and lupus i , iii , iv ( age @xmath87myr , @xcite ) . all yso candidates followed up spectroscopically located in lupus v , vi turned out to be agb stars . these clouds have been recently estimated to be a few myrs older @xcite based on the dominance of class iii systems . as we have shown in section [ contamination ] , at least @xmath88% of the claimed class iii systems located in lupus v , vi are very likely to be agb stars . this reduces the fraction of classiii objects to values similar to those obtained for lupus iii . based on this , lupus v , vi , and iii could well be of a very similar age . the main result that can be obtained from table [ t : fractions ] clearly is that young clouds ( @xmath89 myr ) contain a mixture of grain growth , photoevaporating , debris , and tidally disrupted transition disks . it is clear that all states of disk evolution are already present at this age range , which implies that different disks evolve at different rates and/or through different evolutionary paths . an important difficulty in constraining disk evolution is that stellar ages obtained from isochrones are very uncertain for individual systems . an analysis of the stellar age distributions of each disk category is therefore postponed to paper iii ( cieza et al . 2012 , apj submitted ) , where we discuss a larger sample of well characterized transition disk objects including the systems presented here . the general picture of photoevaporation is the following . in very young disks , the accretion rate largely exceeds the evaporation rate and the disk evolves virtually unaffected by photoevaporation . as the accretion rate is decreasing with time , the disk necessarily reaches the time when the accretion rate equals the photoevaporation rate and the outer disk is no longer able to resupply the inner disk with material . at this point , the inner disk drains on the viscous timescale ( @xmath0 @xmath58 yr ) and an inner hole of a few au in radius is formed in the disk . the inner disk edge is now directly exposed to the euv radiation and the disk rapidly photoevaporates from the inside out . early models of euv photoevaporation predict evaporation rates of @xmath90m@xmath4/yr @xcite . more recent simulations taking into account x - ray @xcite and/or far - ultraviolet ( fuv ) irradiation @xcite in addition to the euv photons , largely exceed these early predictions , reaching photoevaporation rates of the order of @xmath91m@xmath4/yr ( see also * ? ? ? as in steady state accretion disks the mass transfer through the disk is roughly proportional to the mass accretion rate onto the star , a crucial prediction of the photoevaporation model is that high photoevaporation rates imply high disk masses at the time the inner disk is drained . in particular , models with efficient x - ray photoevaporation predict a significant population of relatively massive ( @xmath267 m@xmath1 ) non - accreting transition disks @xcite . figure [ f : photo - mass ] shows the upper limits ( derived from submillimeter non detections ) on the disk masses of the photoevaporating transition disks in all the clouds we considered so far . even taking into account uncertainties in our classification of photoevaporation candidates , it is evident that large numbers of non - accreting but massive disks do not exist . this indicates that photoevaporation is less efficient than predicted by the models described above . however , one has to take into account that the sample of transition disks considered here contains low - mass stars only while model calculations have been performed exclusively for more massive stars @xmath92m@xmath4 . therefore , either a more homogeneous sample of photoevaporating disk systems covering a larger range of host star masses ( earlier spectral types ) or simulations of photoevaporation for disks around low - mass stars are required to provide a final answer on this issue . of course , our classification of transition disk objects is based on rather rough empirical relations and requires to carefully consider possible caveats . an obvious uncertainty concerns our multiplicity survey . the method of direct detection of companions is obviously more sensitive to binaries with large separations and low inclinations . our naco observations are sensitive to projected separations of @xmath93au given the distance to our targets and depending on the intrinsic distribution of orbital separations we may therefore miss a significant fraction of close binaries . to overcome this observational bias we are currently performing radial velocity measurements of our targets using vlt / uves . the method of detecting radial velocity variations is more sensitive to small separations and high inclinations and therefore complements the imaging results presented here . we will present the results in a forthcoming paper . however , the fact that only 6 of the 43 transition disks studied herein and in paper i are circumbinary disk candidates strongly suggests that binaries at the peak of their separation distribution ( @xmath26 30 au ) do not result in transition disk objects as such stellar binaries would be easily detectable by our ao observations . instead , they are likely to destroy the disk rather quickly @xcite . another uncertainty in our classification procedure is the rather ad - hoc separation between photoevaporating and debris disk systems by using a limit in l@xmath71/l@xmath72 . however , there is a physical and not only phenomenological difference between these two types of transition disks . photoevaporating disks are dissipating primordial disks and should have gas rich outer disks while the debris disks should be gas poor . molecular line observations with the atacama large millimeter / submillimeter array ( alma ) of non - accreting disks will be able to distinguish between the two types of objects . a huge problem related to the process of photoevaporation is that the mass loss rates predicted by different models differ by up - to two orders of magnitude ( see e.g. * ? ? ? the disk mass at the time photoevaporation opens a hole in the disk is directly connected to the photoevaporation rates . measuring the disk masses of photoevaporating disks could therefore significantly constrain theoretical models of photoevaporation . however , all of the photoevaporting disk candidates remain undetected and we can only put upper limits to their masses . fortunately , alma will be much more sensitive than all presently available telescopes and will soon be able to measure the masses of many bona fide photoevaporating disks . alma should also be able to measure , through high resolution continuum observations at multiple wavelengths , the radial dependence in the grain size distribution expected in the grain - growth dominated disks . finally , the recent identification using the aperture masking technique of what seems to be forming planets within the inner cavities of the transition disks around t cha @xcite and lkca 15 @xcite strongly encourages to obtain similar observations for the three planet - forming disk candidates identified herein , objects # 2 , # 3 , and # 15 . any system with a planet still embedded in a primordial disk would represent an invaluable laboratory to study planet formation with current and future instrumentation . we have carried out a multifrequency study of _ spitzer_-selected yso transition disk candidates located in the lupus complex ( 53 ) , cra ( 5 ) , and scp ( 2 ) . we obtained submillimeter observations ( apex ) , optical high resolution echelle spectroscopy ( clay / mike , du pont / echelle ) , and nir images ( from ao imaging vlt / naco ) . after deriving spectral types of each target , 43 agb stars were removed ( lupus complex ( 41 ) , cra ( 1 ) , and scp ( 1 ) ) , leaving a sample of 17 genuine transition disk systems . we find that the vast majority of agb stars have [ 3.6]-[24 ] @xmath11 1.8 , underscoring the need for a spectroscopic confirmation of yso candidates with small 24@xmath8 m excesses . the data obtained for the 17 transition systems allows to estimate multiplicity , stellar accretion rates , and disk masses thereby allowing to identify the physical mechanism that is most likely to be responsible for the formation of the inner opacity hole . the observational results of this study can be summarized as follows : 1 . the derived spectral classification indicates that all but one ( object # 2 , k0 ) central star are m - type stars , in agreement with previous results @xcite . 12/17 targets are accreting objects ( i.e. asymmetric h@xmath6 profile having a velocity width 200 km s@xmath5 at 10@xmath28 of peak intensity ) . 3 . @xmath26 50@xmath28 of the sample are multiple systems and among them , two triple systems . two binary systems have small projected separations and are therefore candidates to host a circumbinary disk . 7/17 targets have flux detection in the submillimeter . for the remaining systems , we derive and upper limit of the disk mass ( corresponding to a flux of 3 @xmath27 rms ) . the estimated disk masses for the detected objects cover the range 2 m@xmath110 m@xmath1 . combining the derived estimates of disk masses , accretion rates and multiplicity with the sed morphology and fractional disk luminosity ( l@xmath71/l@xmath72 ) allows to classify the disks as strong candidates for the following categories : * 9/17 grain growth - dominated disks ( accreting objects with negative sed slopes in the mid - ir , @xmath10@xmath11 0 ) . * 3/17 photoevaporating disks ( non - accreting objects with disk mass @xmath11 3 m@xmath1 , but ) . * 2/17 debris disks ( non - accreting objects with disk mass @xmath11 2.1 m@xmath1 and ) . * 2/17 circumbinary disks ( a binary tight enough to accommodate both components within the inner hole ) . * 3/17 giant planet forming disk ( accreting systems with seds indicating sharp inner holes ) . inspecting in more detail the different sub - clouds analyzed in this study we find the same heterogeneity of the transition disk population in lupus iii , iv , cra as in our previous analysis of transition disks in ophiuchus ( * ? ? ? * paperi ) . we therefore conclude that photoevaporation , giant planet formation , and grain growth produce inner holes on similar timescales . not a single transition disks has been found in lupus i , v , vi . all 33 candidates that have been spectroscopically followed up turned out to be agb stars which questions the recent interpretation of @xcite that lupus i , v , vi might be relatively old star forming regions dominated by classiii objects . in addition , our detailed observational analysis of transition disks provides clear constraints on theoretical models of disk photoevaporation by the central star . according to the large evaporation rates predicted by recent models ( i.e. see * ? ? ? * ) , large numbers of massive photoevaporating transition disks systems should exist . in contrast to this prediction , all photoevaporating disk candidates identified in this work and paperi contain very little mass , indicating much smaller evaporation rates at least for the low - mass stars considered here . similarly , the low incidence of circumbinary transition disk candidates ( @xmath26 10@xmath28 ) supports the idea that most disks are destroyed rather quickly by companions at @xmath94 au separations . finally , we emphasize that the 43 transition disk systems discussed in this work and in paperi represent the currently largest and most homogeneous sample of well - characterized transition disks . further investigating these systems with new observing capabilities such as alma therefore holds the potential to significantly improvement our understanding of the physical processes driving circumstellar disk evolution . gar was supported by alma / conicyt ( grant 31070021 ) and eso / comit mixto . mrs acknowledges support from millennium science initiative , chilean ministry of economy : nucleus p10 - 022-f and fondecyt ( grant 1100782 ) . lac acknowledges support provided by nasa through the _ sagan _ fellowship program . arm thanks for financial support from fondecyt in the form of grant number 3110049 , eso / comite mixto and gemini / conicyt ( 32080023 ) . asc was supported by grants from consejo nacional de investigaciones cientficas y tcnicas de la repblica argentina , agencia nacional de promocin cientfica y tecnolgica and universidad nacional de la plata ( argentina ) . we finally thank dr . giorgio siringo and dr . c. de breuck for assistance with performing the apex observations and the corresponding data reduction . we are also grateful for the support of the staff at las campanas observatory . a special thank is given to dr . m. orellana and evelyn puebla for their help during the first las campanas observing run . apex is a collaboration between the max - planck - institut fur radioastronomie , the european southern observatory , and the onsala space observatory . this work makes use of data obtained with the _ spitzer _ space telescope , which is operated by jpl / caltech , under a contract with nasa . _ facilities _ : _ spitzer _ ( irac , mips ) , vlt : yepun , magellan : clay , du pont ( echelle ) spitzer _ c2d _ systems classified into yso candidates , agb candidates , and transition disk candidates according to simple color cuts based on the results of our spectroscopic follow - up program ( see text for more details ) . , width=480 ] -band images of the six multiple systems that have been detected with our vlt - ao observations ( objects # 6 , 7 , 11 , 13 , 16 , and 17 ) and of object # 14 , which has been identified as a close binary with a 0.13@xmath23 separation from speckle observations in 2001 @xcite . the putative companion in object # 14 remains unresolved by our 2009 observations . targets # 11 and 13 are triple systems . in the former case , the tighter components are not fully resolved , but their presence can be inferred from the highly elongated image ( lower right panel ) . , width=480 ] normalized average h@xmath6 profiles for non - accreting objects with clear h@xmath6 emission . the horizontal dashed line indicates the 10% peak intensity , where @xmath95v is measured . the velocity widths are @xmath11 200 km s@xmath5 and the line profiles are symmetric.,width=480 ] normalized h@xmath6 profiles of the 12 accreting objects . these systems are considered accreting because either the velocity width is @xmath19 200 km s@xmath5 or the line profile is asymmetric . note that objects # 8 and 12 represent borderline systems as the @xmath95v@xmath96km / s and the line is not clearly asymmetric ( see text for more details).,width=480 ] 1.1 to 5.1 m@xmath1 , and accretion rates from @xmath2 to @xmath97 . the filled circles are detections , while the arrows represent 3@xmath20 limits . the open squares correspond to the observed optical and near - ir fluxes before being corrected for extinction as in paper i. for each object , the average of the two r - band magnitudes ( from the usno - b1 catalog ) listed in table [ t : mags ] has been used . the classification of object # 5 and 10 as grain growth dominated is slightly uncertain as classical ttauris stars of late m - types can have similar seds @xcite . the solid line represents the stellar photosphere normalized to the extinction - corrected j band . the dotted lines correspond to the median mid - ir sed of cttss calculated by @xcite . the dashed lines are the quartiles . , width=480 ] m images of object # 3 . we find no evidence for extended emission from the molecular cloud . in both mosaics , the detections are consistent with a point source at the location of the target ( marked by the crosshairs ) . see the electronic edition of the journal for a color version of this figure . , width=480 ] @xmath73 and , the other two systems are somewhat peculiar : target # 3 shows a very steep rise in flux observed at , which indicates a very large inner hole and being relatively close to a full disk but with signs for a small and sharp inner hole . disk masses are 9.1 , 5.6 and for objects # 2 , # 3 , and # 15 ; respectively . the solid line as well as the dashes and dotted lines are the same as in fig.[f : grain - seds ] . , width=288 ] . none of these systems have been detected at submillimeter wavelength and consequently only upper limits for the disk masses could be derived . we conclude that photoevaporation seems to be less efficient than has recently been suggested . , width=302 ] /l@xmath72 @xmath98 10@xmath99 . the detection of the gas component in photoevaporating disks ( e.g. with alma ) may lead to a more physically motivated separation between the two sub - samples . the solid line as well as the dashes and dotted lines are the same as in figure [ f : grain - seds ] . [ f : debris - seds ] , width=302 ] vs @xmath9 for transition disks identified in paperi and the present work . the locations of the ophiuchus transition disk sample from this work ( top panel ) and paperi ( bottom panel ) cover very similar ranges in the @xmath10 versus @xmath9 plane . different symbols indicate different formation processes of the inner opacity hole . in general , the lupus , cra , and scp data confirm our previous findings : planet forming disks have large values of @xmath10 and @xmath9 , grain growth dominated disks should have small @xmath10 , but cover the entire range of @xmath9 ; and debris disks have extremely low values of @xmath10 and the ir excess starts at long @xmath9 . however , two systems clearly show that @xmath10 and @xmath9 alone can not fully characterize transition disks . we identified one planet forming disk candidate with @xmath10@xmath100 but indications of a sharp hole at longer wavelength ( object # 3 ) and one planet forming candidate with @xmath9 = 4.5 @xmath8 m ( object # 2 ) have been found . , width=302 ] rrcrrrrrrrrrrrcc 1 & 234.51292 & -33.23269 & sstc2d_j153803.1 - 331358 & 13.35 & 3.91e+02 & 6.91e+02 & 6.83e+02 & 3.47e+02 & 2.20e+02 & 1.82e+02 & 1.26e+02 & 3.56e+01 & lup i & 1 + 2 & 235.64750 & -34.37292 & sstc2d_j154235.4 - 342223 & 14.17 & 5.16e+02 & 8.86e+02 & 8.43e+02 & 4.00e+02 & 2.66e+02 & 2.03e+02 & 1.40e+02 & 4.96e+01 & lup i & + 3 & 239.93868 & -41.91590 & sstc2d_j155945.3 - 415457 & 13.18 & 3.64e+02 & 7.92e+02 & 8.86e+02 & 5.78e+02 & 3.35e+02 & 3.23e+02 & 3.80e+02 & 2.68e+02 & lup iv & 1 + 4 & 240.37369 & -42.13432 & sstc2d_j160129.7 - 420804 & 15.74 & 7.74e+01 & 1.57e+02 & 1.60e+02 & 9.11e+01 & 5.86e+01 & 4.89e+01 & 3.80e+01 & 1.79e+01 & lup iv & 1 + 5 & 240.62463 & -41.85307 & sstc2d_j160229.9 - 415111 & 17.91 & 3.95e+01 & 8.07e+01 & 8.73e+01 & 4.99e+01 & 3.01e+01 & 2.55e+01 & 1.82e+01 & 5.15e+00 & lup iv & 1 + 6 & 242.19953 & -38.83361 & sstc2d_j160847.9 - 385001 & 13.71 & 1.12e+03 & 1.93e+03 & 1.89e+03 & 9.86e+02 & 4.50e+02 & 4.37e+02 & 3.14e+02 & 1.41e+02 & lup iii & + 7 & 242.39212 & -39.22835 & sstc2d_j160934.1 - 391342 & 15.11 & 5.36e+02 & 1.20e+03 & 1.43e+03 & 5.33e+02 & 3.99e+02 & 4.66e+02 & 2.83e+02 & 5.35e+01 & lup iii & 1 + 8 & 242.50045 & -38.90031 & sstc2d_j161000.1 - 385401 & 17.47 & 2.25e+02 & 4.13e+02 & 5.55e+02 & 3.66e+02 & 1.76e+02 & 2.68e+02 & 2.06e+02 & 9.72e+01 & lup iii & 1 + 9 & 242.85827 & -39.18979 & sstc2d_j161126.0 - 391123 & 16.29 & 6.96e+02 & 1.43e+03 & 1.71e+03 & 1.01e+03 & 5.99e+02 & 5.56e+02 & 3.62e+02 & 1.05e+02 & lup iii & 1 + 10 & 243.21550 & -38.70443 & sstc2d_j161251.7 - 384216 & 14.06 & 5.97e+02 & 1.04e+03 & 1.07e+03 & 6.37e+02 & 3.49e+02 & 3.06e+02 & 2.02e+02 & 7.43e+01 & lup iii & 1 + 11 & 244.92350 & -37.78246 & sstgbs_j16194163 - 3746568 & 16.36 & 6.48e+01 & 1.27e+02 & 1.33e+02 & 8.56e+01 & 5.64e+01 & 4.77e+01 & 3.26e+01 & 1.02e+01 & lup v & 2 + 12 & 245.00856 & -41.62400 & sstgbs_j16200205 - 4137264 & 15.37 & 6.17e+02 & 1.24e+03 & 1.35e+03 & 7.27e+02 & 4.28e+02 & 3.60e+02 & 2.31e+02 & 9.00e+01 & lup vi & 2 + 13 & 245.03960 & -41.43343 & sstgbs_j16200950 - 4126003 & 15.78 & 7.69e+02 & 1.77e+03 & 2.05e+03 & 1.40e+03 & 8.77e+02 & 7.88e+02 & 5.20e+02 & 1.61e+02 & lup vi & 2 + 14 & 245.13167 & -37.51133 & sstgbs_j16203160 - 3730407 & 16.96 & 1.40e+03 & 2.78e+03 & 3.38e+03 & 2.07e+03 & 1.24e+03 & 1.27e+03 & 9.10e+02 & 2.11e+02 & lup v & 2 + 15 & 245.22704 & -36.91195 & sstgbs_j16205449 - 3654430 & 16.03 & 1.89e+02 & 3.86e+02 & 4.12e+02 & 2.24e+02 & 1.16e+02 & 9.97e+01 & 6.43e+01 & 2.51e+01 & lup v & 2 + 16 & 245.35836 & -37.51710 & sstgbs_j16212600 - 3731015 & 16.69 & 1.62e+02 & 4.21e+02 & 6.41e+02 & 7.57e+02 & 5.29e+02 & 5.18e+02 & 3.87e+02 & 1.61e+02 & lup v & 2 + 17 & 245.37467 & -37.03926 & sstgbs_j16212991 - 3702213 & 14.84 & 1.68e+02 & 3.15e+02 & 3.31e+02 & 1.89e+02 & 9.94e+01 & 8.37e+01 & 5.79e+01 & 5.69e+01 & lup v & 2 + 18 & 245.41990 & -41.37274 & sstgbs_j16214077 - 4122218 & 15.98 & 1.26e+02 & 2.51e+02 & 2.64e+02 & 1.58e+02 & 8.63e+01 & 7.36e+01 & 4.86e+01 & 1.92e+01 & lup vi & 2 + 19 & 245.45722 & -41.11716 & sstgbs_j16214973 - 4107017 & 15.35 & 1.20e+0 & 2.30e+02 & 2.65e+02 & 1.92e+02 & 1.34e+02 & 1.08e+02 & 7.48e+01 & 2.40e+01 & lup vi & 2 + 20 & 245.48435 & -37.22131 & sstgbs_j16215624 - 3713167 & 17.18 & 1.95e+0 & 3.89e+02 & 4.34e+02 & 2.43e+02 & 1.31e+02 & 1.15e+02 & 7.38e+01 & 2.84e+01 & lup v & 2 + 21 & 245.50680 & -37.27693 & sstgbs_j16220163 - 3716369&16.20 & 5.48e+0 & 1.06e+02 & 1.09e+02 & 6.50e+01 & 4.23e+01 & 3.53e+01 & 2.48e+01 & 8.24e+00 & lup v & 2 + 22 & 245.57457 & -36.97127 & sstgbs_j16221789 - 3658165&16.80 & 2.68e+02 & 5.24e+02 & 5.83e+02 & 3.42e+02 & 1.95e+02 & 1.68e+02 & 1.08e+02 & 3.44e+01 & lup v & 2 + 23 & 245.63924 & -41.05499 & sstgbs_j16223341 - 4103179 & 16.34 & 5.72e+02 & 1.32e+03 & 1.50e+03 & 8.22e+02 & 4.95e+02 & 4.26e+02 & 2.87e+02 & 1.21e+02 & lup vi&2 + 24 & 245.67774 & -37.35645 & sstgbs_j16224265 - 3721232&16.08 & 1.97e+02 & 4.34e+02 & 4.97e+02 & 3.61e+02 & 2.74e+02 & 2.47e+02 & 2.61e+02 & 1.98e+02 & lup v & 2 + 25 & 245.76534 & -37.49477 & sstgbs_j16230368 - 3729411 & 16.02 & 6.05e+01 & 1.18e+02 & 1.20e+02 & 6.64e+01 & 4.17e+01 & 3.55e+01 & 3.12e+01 & 1.86e+01 & lup v & 2 + 26 & 245.79592 & -41.28620 & sstgbs_j16231101 - 4117103&16.48 & 1.14e+02 & 2.64e+02 & 3.00e+02 & 1.67e+02 & 1.15e+02 & 9.21e+01 & 6.16e+01 & 1.79e+01 & lup vi & 2 + 27 & 245.81152 & -40.28753 & sstgbs_j16231476 - 4017151 & 16.96 & 1.69e+02 & 3.56e+02 & 4.07e+02 & 2.20e+02 & 1.36e+02 & 1.15e+02 & 8.15e+01 & 2.86e+01 & lup vi & 2 + 28 & 245.81661 & -41.06693 & sstgbs_j16231598 - 4104009 & 13.11 & 1.97e+03 & 4.16e+03 & 4.86e+03 & 1.80e+03 & 1.31e+03 & 1.12e+03 & 7.36e+02 & 2.41e+02 & lup vi & 2 + 29 & 245.85994 & -37.86896 & sstgbs_j16232638 - 3752082 & 17.31 & 1.20e+02 & 2.35e+02 & 2.99e+02 & 3.24e+02 & 2.47e+02 & 2.33e+02 & 1.74e+02 & 4.96e+01 & lup v & 2 + 30 & 245.88526 & -37.87672 & sstgbs_j16233246 - 3752361 & 15.74 & 4.72e+02 & 9.78e+02 & 1.06e+03 & 5.68e+02 & 3.23e+02 & 2.85e+02 & 1.92e+02 & 7.09e+01 & lup v & 2 + 31 & 245.91108 & -37.52745 & sstgbs_j16233865 - 3731388 & 15.80 & 1.27e+02 & 2.39e+02 & 2.53e+02 & 1.43e+02 & 7.59e+01 & 7.05e+01 & 5.16e+01 & 3.14e+01 & lup v & 2 + 32 & 245.95431 & -40.43823 & sstgbs_j16234903 - 4026176 & 17.12 & 9.57e+01 & 2.08e+02 & 2.33e+02 & 1.64e+02 & 1.13e+02 & 9.72e+01 & 7.79e+01 & 2.52e+01 & lup vi & 2 + 33 & 245.99615 & -37.89800 & sstgbs_j16235907 - 3753528 & 14.28 & 2.23e+03 & 4.60e+03 & 5.14e+03 & 2.51e+03 & 1.61e+03 & 1.41e+03 & 9.50e+02 & 3.19e+02 & lup v & 2 + 34 & 246.11110 & -37.96131 & sstgbs_j16242666 - 3757407 & 17.52 & 3.13e+02 & 6.67e+02 & 7.63e+02 & 4.10e+02 & 2.14e+02 & 2.02e+02 & 1.29e+02 & 5.62e+01 & lup v & 2 + 35 & 246.23294 & -40.19118 & sstgbs_j16245590 - 4011282 & 15.15 & 1.06e+03 & 2.09e+03 & 2.65e+03 & 1.43e+03 & 9.49e+02 & 9.54e+02 & 6.74e+02 & 1.78e+02 & lup vi & 2 + 36 & 246.27875 & -38.05595 & sstgbs_j16250690 - 3803214 & 14.99 & 1.34e+03 & 2.71e+03 & 2.98e+03 & 1.64e+03 & 9.41e+02 & 8.20e+02 & 5.39e+02 & 1.78e+02 & lup v & 2 + 37 & 246.46860 & -40.31346 & sstgbs_j16255246 - 4018484 & 17.39 & 1.23e+02 & 2.63e+02 & 2.93e+02 & 1.23e+02 & 7.96e+01 & 7.38e+01 & 4.68e+01 & 1.28e+01 & lup vi & 2 + 38 & 246.49321 & -40.16614 & sstc2d_j16255837 - 4009581 & 16.88 & 5.17e+02 & 1.20e+03 & 1.44e+03 & 7.77e+02 & 4.23e+02 & 4.06e+02 & 2.63e+02 & 1.05e+02 & lup vi & 2 + 39 & 246.55582 & -39.83173 & sstc2d_j16261339 - 3949542 & 14.83 & 1.68e+02 & 3.01e+02 & 3.55e+02 & 2.15e+02 & 1.60e+02 & 1.34e+02 & 1.01e+02 & 3.70e+01 & lup vi & 2 + 40 & 246.60631 & -39.74646 & sstgbs_j16262551 - 3944472 & 15.44 & 4.76e+01 & 9.31e+01 & 9.09e+01 & 4.70e+01 & 3.20e+01 & 2.52e+01 & 2.94e+01 & 1.53e+01 & lup vi & 2 + 41 & 246.96060 & -39.80278 & sstgbs_j16275054 - 3948100 & 16.20 & 3.15e+02 & 6.61e+02 & 7.35e+02 & 4.71e+02 & 2.81e+02 & 2.39e+02 & 1.59e+02 & 5.74e+01 & lup vi & 2 + 42 & 252.27333 & -15.62027 & sstc2d_j16490560 - 1537129 & 17.92 & 1.40e+02 & 6.63e+02 & 1.12e+03 & 1.00e+03 & 7.78e+02 & 7.42e+02 & 5.17e+02 & 1.46e+02 & scp & 4 + 43 & 285.78812 & -36.95611 & sstgbs_j19030915 - 3657220 & 17.45 & 6.88e+01 & 1.97e+02 & 2.43e+02 & 1.63e+02 & 1.21e+02 & 1.01e+02 & 6.75e+01 & 2.04e+01 & cra & 3 [ t : giants ] cccccccc lupus i & 339 , 16 & 2 & 0 & 2 & 100 & 1.54 & 150 @xmath12 20 + lupus iii & 340 , 9 & 15 & 10 & 5 & 33 & 1.54 & 200 @xmath12 20 + lupus iv & 336 , 8 & 5 & 2 & 3 & 60 & 1.54&150 @xmath12 20 + lupus v & 342 , 9 & 16 & 0 & 16 & 100 & 10 & 150 @xmath12 20 + lupus vi & 342 , 6 & 15 & 0 & 15 & 100 & 10 & 150 @xmath12 20 + cra & 0,-19 & 5 & 4 & 1 & 20 & 1 & 150 @xmath12 20 + scp & 250,18 & 2 & 1 & 1 & 50 & 5 & 130 @xmath12 20 + oph ( paper i ) & 353,18 & 34 & 26 & 8 & 24 & 2 & 150 @xmath12 20 + c|c|c|c|c & ysoc & agb & td & agb + & & candidates & candidates & candidates in td region + & & whole sample & & in td region + & # & % & # & % + + & 29 & 10.4 & 7 & 28.6 + * lup i * & 20 & 15 & 8 & 25 + * lup iii * & 79 & 18.9 & 18 & 11.1 + * lup iv * & 12 & 25 & 5 & 20 + * oph * & 297 & 7.7 & 52 & 15.38 + * per * & 387 & 2.6 & 56 & 10.7 + * ser * & 262 & 6.5 & 60 & 10 + + & 174 & 1.7 & 28 & 7.1 + * cra * & 45 & 4.4 & 7 & 14.2 + * ic5146 * & 163 & 2.4 & 24 & 4.1 + * lup v * & 44 & 47.7 & 22 & 36.3 + * lup vi * & 46 & 67.3 & 21 & 57.1 + * serp - aquila * & 1442 & 28.6 & 641 & 32.6 + * cham i * & 93 & 1 & 17 & 5.9 + * cham iii * & 4 & 75 & 1 & 100 + * musca * & 13 & 84.6 & 5 & 80.0 + * ceph * & 119 & 2.5 & 19 & 10.5 + * sco * & 9 & 11 & 4 & 25 + rrcrrrrrrrrrrrc 1 & sstc2d_j160026.1 - 415356 & ..... & 15.62 & 15.54 & 2.97e+01 & 3.70e+01 & 3.25e+01 & 2.15e+01 & 1.68e+01 & 1.42e+01 & 1.63e+01 & 2.40e+01 & @xmath11 50 & lup iv + 2 & sstc2d_j160044.5 - 415531 & v*mylup & 11.22 & 11.06 & 2.63e+02 & 3.44e+02 & 3.05e+02 & 1.77e+02 & 1.41e+02 & 1.40e+02 & 2.13e+02 & 5.90e+02 & 1.05e+03 & lup iv + 3 & sstc2d_j160711.6 - 390348 & sz91 & 13.61 & 13.89 & 6.03e+01 & 9.13e+01 & 7.67e+01 & 3.86e+01 & 2.47e+01 & 1.72e+01 & 1.09e+01 & 9.72e+00 & 5.02e+02 & lup iii + 4 & sstc2d_j160752.3 - 385806 & sz95 & 13.66 & 14.02 & 6.28e+01 & 7.89e+01 & 6.61e+01 & 4.21e+01 & 3.18e+01 & 2.73e+01 & 2.96e+01 & 3.00e+01 & @xmath11 50 & lup iii + 5 & sstc2d_j160812.6 - 390834 & sz96 & 12.98 & 13.66 & 1.42e+02 & 1.87e+02 & 1.74e+02 & 1.68e+02 & 1.13e+02 & 1.38e+02 & 1.73e+02 & 2.41e+02 & 1.54e+02 & lup iii + 6 & sstc2d_j160828.4 - 390532 & sz101 & 13.52 & 13.53 & 1.10e+02 & 1.32e+02 & 1.17e+02 & 7.98e+01 & 5.55e+01 & 4.16e+01 & 3.29e+01 & 2.41e+01 & @xmath11 50 & lup iii + 7 & sstc2d_j160831.5 - 384729 & lup338 & 12.70 & 13.03 & 2.15e+02 & 2.75e+02 & 2.37e+02 & 1.49e+02 & 9.25e+01 & 6.98e+01 & 5.16e+01 & 2.85e+01 & @xmath11 50 & lup iii + 8 & sstc2d_j160841.8 - 390137 & sz107 & 15.29 & 15.47 & 5.05e+01 & 5.76e+01 & 5.01e+01 & 2.65e+01 & 2.02e+01 & 1.41e+01 & 9.24e+00 & 1.07e+01 & @xmath11 50 & lup iii + 9 & sstc2d_j160855.5 - 390234 & sz112 & 14.57 & 14.68 & 6.32e+01 & 7.87e+01 & 6.90e+01 & 4.87e+01 & 3.80e+01 & 3.04e+01 & 2.48e+01 & 1.24e+02 & 1.20e+02&lup iii + 10 & sstc2d_j160901.4 - 392512 & .... & 14.83 & 14.89 & 3.62e+01 & 5.45e+01 & 5.09e+01 & 4.47e+01 & 3.40e+01 & 3.06e+01 & 2.58e+01 & 4.22e+01 & 1.14e+02 & lup iii + 11 & sstc2d_j160954.0 - 392328 & lup359 & 12.96 & 13.30 & 1.59e+02 & 2.13e+02 & 1.94e+02 & 1.35e+02 & 1.01e+02 & 8.44e+01 & 7.93e+01 & 9.65e+01 & @xmath11 50 & lup iii + 12 & sstc2d_j161029.6 - 392215 & ... & 15.69 & 15.79 & 2.66e+01 & 3.18e+01 & 2.88e+01 & 1.91e+01 & 1.42e+01 & 1.15e+01 & 1.09e+01 & 3.37e+01 & 1.10e+02 & lup iii + 13 & sstc2d_j162209.6 - 195301 & ... & 14.48 & 14.27 & 1.44e+02 & 2.08e+02 & 1.84e+02 & 1.09e+02 & 7.15e+01 & 5.05e+01 & 3.27e+01 & 1.59e+01 & @xmath11 50 & scp + 14 & sstgbs_j190029.1 - 365604 & crapms8 & 13.80 & 13.78 & 7.84e+01 & 1.06e+02 & 9.89e+01 & 5.08e+01 & 3.60e+01 & 2.62e+01 & 1.83e+01 & 3.59e+01 & @xmath11 100 & cra + 15 & sstgbs_j190058.1 - 364505 & cra-9 & 13.49 & 13.57 & 1.12e+02 & 1.61e+02 & 1.40e+02 & 5.81e+01 & 4.38e+01 & 3.18e+01 & 2.48e+01 & 1.78e+02 & @xmath11 100 & cra + 16 & sstgbs_j190129.0 - 370148 & g-94 & 15.53 & 14.95 & 3.53e+01 & 4.25e+01 & 3.62e+01 & 1.95e+01 & 1.38e+01 & 9.67e+00 & 6.52e+00 & 2.92e+00 & @xmath11 50 & cra + 17 & sstgbs_j190311.8 - 370902 & cra-35 & 17.20 & 16.82 & 2.46e+01 & 3.24e+01 & 3.09e+01 & 2.12e+01 & 1.58e+01 & 1.21e+01 & 1.06e+01 & 1.18e+01 & @xmath11 50 & cra + [ t : mags ] rcccrcrrrcc 1 & 240.10887 & -41.89877 & clay & m5.25,m1@xmath101 & 1,2 & 0.47 & 162 & @xmath11 21 & 7 & 2.8 + 2 & 240.18554 & -41.92534 & dupont & k0 & 7 & 0.44 & 532 & 100 & 5 & + 3 & 241.79833 & -39.06326 & dupont & m1.5,m0.5 & 1,6 & 0.41 & 283 & 34.5 & 2.9 & 5 + 4 & 241.96800 & -38.96840 & dupont & m3.25,m1.5 & 1,6 & 0.46 & 321 & @xmath11 9.9 & 3.3 & 3 + 5 & 242.05258 & -39.14264 & dupont & m2,m1.5 & 1,6 & 0.5 & 233 & @xmath11 21 & 7 & + 6 & 242.11837 & -39.09229 & dupont & m5,m4 & 1,6 & 0.28 & 343 & @xmath11 10.8 & 3.6 & 0.7 + 7 & 242.13146 & -38.79148 & dupont & m2.25,m2 & 1,4 & 0.25 & 382 & 6.7 & 2.2 & 0.4 + 8 & 242.17413 & -39.02695 & dupont & m5.75,m5.5 & 1,6 & & 200 & 9.7 & 2.5 & + 9 & 242.23133 & -39.04276 & dupont & m6,m6 & 1,6 & & 189 & @xmath11 21 & 7 & 2.8 + 10 & 242.25583 & -39.41997 & clay & m4,m4 & 1,2 & 0.45 & 369 & 31.4 & 3.4 & + 11 & 242.47496 & -39.39109 & dupont & m2.75,m1.5 & 1,4 & 0.40 & 336 & 16.7 & 3.3 & 1.15 + 12 & 242.62321 & -39.37076 & clay & m4.5,m4 & 1,2 & 0.52 & 180 & 23.2 & 4.7 & + 13 & 245.54000 & -19.88357 & clay & m3.7 & 1 & 0.55 & 132 & @xmath11 10.8 & 3.6 & 1.8,3 + 14 & 285.12113 & -36.93437 & dupont & m4,m3 & 1,5 & 0.30 & 93 & @xmath11 10.5 & 3.5 & 0.132 + 15 & 285.24187 & -36.75139 & clay & m0.75 & 1 & 0.48 & 440 & @xmath11 21 & 7 & + 16 & 285.37088 & -37.03011 & dupont & m3.75,m3.5 & 1,3 & & 83 & @xmath11 21 & 7 & 0.5 + 17 & 285.79929 & -37.15055 & clay & m5.0 & 1 & 0.51 & 205 & @xmath11 21 & 7 & 0.5 . [ t : obs ] lrrrrrcrc 1 & @xmath11 -11 & @xmath11 1.9 & 420 & 4.50 & -0.80 & -2.1 & 0.06 & photo . disk , lup iv + 2 & -7.7 & 9.1 & & 4.50 & -0.17 & -2.4 & 2.37 & giant planet , lup iv + 3 & -10.1 & 5.6 & 1000 & 8.00 & -2.18 & -2.6 & 0.39 & giant planet , lup iii + 4 & -9.7 & @xmath11 1.6 & 600 & 4.50 & -1.05 & -2.2 & 0.30 & grain growth@xmath102 , lup iii + 5 & -10.6 & @xmath11 3.4 & & 2.20 & -0.93 & -1.4 & 0.86 & grain growth@xmath102 , lup iii + 6 & -9.5 & @xmath11 1.8 & 140 & 5.80 & -1.42 & -2.6 & 0.43 & grain growth , lup iii + 7 & -9.1 & 1.1 & 80 & 8.00 & -1.55 & -2.9 & 1.27 & grain growth , lup iii + 8 & -11 & 1.6 & 76 & 5.8 & -0.86 & -2.8 & 0.17 & grain growth@xmath102 , lup iii + 9 & @xmath11 -11 & @xmath11 3.4 & 560 & 4.50 & -0.31 & -1.9 & 0.22 & photo . disk , lup iii + 10 & -9.3 & 5.1 & & 2.20 & -1.19 & -1.5 & 0.27 & grain growth@xmath102 , lup iii + 11 & -9.6 & 2.7 & 230@xmath103 & 4.50 & -1.05 & -2.3 & 0.96 & circumbinary / gr - grow@xmath102 , lup iii + 12 & -11 & 3.8 & & 5.80 & -0.28 & -2.1 & 0.11 & grain growth@xmath102 , lup iii + 13 & @xmath11 -11 & @xmath11 0.8 & 234,390@xmath103 & 8.00 & -1.67 & -3.7 & 0.48 & circumbinary / debris , scp + 14 & @xmath11 -11 & @xmath11 1 & 20 & 5.80 & -0.42 & -2.7 & 0.14 & circumbinary/ photo . disk , cra + 15 & -8.6 & @xmath11 2 & & 8.00 & 0.76 & -2.4 & 0.46 & giant planet , cra + 16 & @xmath11 -11 & @xmath11 2 & 75 & 8.00 & -1.74 & -3.2 & 0.07 & debris disk , cra + 17 & -11 & @xmath11 2 & 75 & 5.80 & -1.06 & -2.3 & 0.06 & grain growth , cra + [ t : derived ]

transition disk objects are pre - main - sequence stars with little or no near - ir excess and significant far - ir excess , implying inner opacity holes in their disks . here we present a multifrequency study of transition disk candidates located in lupus i , iii , iv , v , vi , corona australis , and scorpius . complementing the information provided by _ spitzer _ with adaptive optics ( ao ) imaging ( naco , vlt ) , submillimeter photometry ( apex ) , and echelle spectroscopy ( magellan , du pont telescopes ) , we estimate the multiplicity , disk mass , and accretion rate for each object in our sample in order to identify the mechanism potentially responsible for its inner hole . we find that our transition disks show a rich diversity in their sed morphology , have disk masses ranging from @xmath0 1 to 10 m@xmath1 and accretion rates ranging from @xmath0 @xmath2 to @xmath3 m@xmath4 yr@xmath5 . of the 17 bona fide transition disks in our sample , 3 , 9 , 3 , and 2 objects are consistent with giant planet formation , grain growth , photoevaporation , and debris disks , respectively . two disks could be circumbinary , which offers tidal truncation as an alternative origin of the inner hole . we find the same heterogeneity of the transition disk population in lupus iii , iv , and corona australis as in our previous analysis of transition disks in ophiuchus while all transition disk candidates selected in lupus v , vi turned out to be contaminating background agb stars . all transition disks classified as photoevaporating disks have small disk masses , which indicates that photoevaporation must be less efficient than predicted by most recent models . the three systems that are excellent candidates for harboring giant planets potentially represent invaluable laboratories to study planet formation with the atacama large millimeter / submillimeter array ..

the stress - energy tensor @xmath6 is said to obey the weak energy condition ( wec ) if @xmath7 for all timelike vectors @xmath8 . this condition is obeyed by all known forms of classical matter , but is violated in quantum field theory @xcite in which the renormalised energy density may become arbitrarily negative at points of spacetime . if extended regions of large negative energy density occur in nature , a variety of exotic phenomena might be possible , ranging from violations of the second law of thermodynamics and cosmic censorship to the creation of time machines and ` warp drive ' . accordingly , it is important to understand the extent to which the weak energy condition may be violated . in a series of papers @xcite , ford , roman and pfenning have studied quantum field theory in various flat and curved spacetimes and established lower bounds ( known as _ quantum inequalities _ ) on the time averaged energy densities measured by observers . the results have been employed to argue against the possibility of traversable wormholes @xcite , warp drive @xcite and also to discuss the rle of negative energy densities in the process of black hole evaporation @xcite . to be specific , consider a free real scalar field of mass @xmath9 in @xmath0-dimensional minkowski space , and let @xmath10 denote the renormalised stress - energy tensor . quantum inequalities provide lower bounds on the averaged expected energy density @xmath11 measured by a stationary observer at the spatial origin @xmath12 , where @xmath13 is a non - negative sampling function and the angle brackets denote the expectation value in the quantum state @xmath14 . the sampling function employed by ford _ is the lorentzian function peaked at @xmath15 @xmath16 in which @xmath17 sets the timescale over which sampling occurs . with this sampling function , ford and roman have shown @xcite that in four dimensional minkowski space the averaged energy density @xmath18 obeys the bound @xmath19 for all ( sufficiently well - behaved ) states @xmath14 and any @xmath20 , where the real - valued function @xmath21 is independent of @xmath14 and is positive and strictly decreasing on @xmath22 with @xmath23 . in two dimensions , they obtained the corresponding result @xmath24 where @xmath25 and the @xmath26 are modified bessel functions of the second kind . in the present paper , we will improve and generalise these bounds to cover more general sampling functions in minkowksi space of arbitrary dimension . some progress in this direction has already been made by flanagan @xcite in the case of 2-dimensional massless field theory . flanagan derived an optimal lower bound on @xmath18 given by @xmath27 for any smooth non - negative sampling function @xmath13 , where @xmath28 denotes the support of @xmath13 .. ] flanagan s argument depends critically on special features of 2-dimensional massless field theory . in this paper , we will consider arbitrary smooth , non - negative , even sampling functions @xmath13 of rapid decrease at infinity ( including the possibility of compact support ) , and will obtain the slightly weaker bound @xmath29 in the 2-dimensional massless case . however , our argument has the virtue of generalising directly to both massive field theory in two dimensions and to massive and massless field theory in four dimensional minkowski space . our general quantum inequality in @xmath30-dimensional minkowski space , is the bound @xmath31 for all ( sufficiently well - behaved ) quantum states @xmath14 , where @xmath32 , the hat denotes the fourier transform and the constant @xmath33 is equal to the area of the unit @xmath34-sphere divided by @xmath35 . the bounded non - negative functions @xmath36 are defined ( for @xmath37 ) on @xmath38 by @xmath39 and obey @xmath40 and @xmath41 as @xmath42 . we plot @xmath43 and @xmath44 in figures 1 and 2 . the derivation of eq . ( [ eq : qi ] ) is quite simple and depends mainly on the canonical commutation relations and the convolution theorem . although we will argue formally here , we expect that the elementary nature of the argument will facilitate a fully rigorous treatment . we begin by stating our conventions . in @xmath30-dimensional minkowski space , @xmath45 denotes the metric with signature @xmath46 . the klein gordon equation is @xmath47 and the quantum field @xmath48 is defined by @xmath49 where @xmath50 , the @xmath30-vector @xmath51 has components @xmath52 , and the annihilation and creation operators @xmath53 and @xmath54 obey the canonical commutation relations @xmath55=[a^\dagger({{\bf k}}),a^\dagger({{\bf k}}')]=0 \qquad [ a({{\bf k}}),a^\dagger({{\bf k}}')]=(2\pi)^n\delta({{\bf k}}-{{\bf k}}').\ ] ] the classical energy density of a field @xmath56 is @xmath57 from which the renormalised ( normal ordered ) quantum energy density at position @xmath58 is easily shown to be @xmath59 . \label{eq : t00}\end{aligned}\ ] ] finally , the fourier transform @xmath60 of a function @xmath13 on @xmath61 is defined by @xmath62 let @xmath13 be a smooth , even and non - negative function on @xmath61 which decays rapidly at infinity [ including the possibility that @xmath13 is compactly supported ] . using @xmath63 to denote the pointwise square root of @xmath13 [ i.e. , @xmath32 ] , the function @xmath65 defined by @xmath66 on @xmath61 is smooth , real - valued and even , decays rapidly at infinity and obeys @xmath67 where the convolution @xmath68 is defined by @xmath69 now let @xmath70 be a real valued function on @xmath71 , growing no faster than polynomially . we use the annihilation and creation operators of the scalar field in @xmath30-dimensional minkowski space to define two families @xmath72 of operators on the fock space of the minkowski vacuum by @xmath73 using the commutation relations and symmetrising the integrand in @xmath74 and @xmath75 , we calculate @xmath76 where @xmath77 and the functions @xmath78 and @xmath21 are given by @xmath79 and @xmath80 the expressions for @xmath78 and @xmath21 may be simplified , using the fact that @xmath65 is even , to obtain @xmath81 and , similarly , @xmath82 since the right hand side of eq . ( [ eq : oo ] ) is ( formally ) a manifestly positive operator , we conclude that the expectation value @xmath83 obeys the following bound @xmath84 in all sufficiently well behaved states @xmath14 . as we will see , this bound provides the key to our derivation of the quantum inequality ( [ eq : qi ] ) . we conclude this section with two remarks . firstly , we have argued rather formally and have interchanged orders of integration at will . nonetheless , we expect that if @xmath13 has rapid decay at infinity ( e.g. , if @xmath13 is a schwartz test function ) our result can be established rigorously for a dense set of states in the folium of the usual minkowski vacuum , and conceivably for all hadamard states on the usual field algebra . secondly , we do not claim that eq . ( [ eq : bd ] ) is the sharpest bound that can be placed on the expectation value of @xmath85 , and in fact do not expect that the bound is actually attained by any reasonable state @xmath14 , as @xmath14 would necessarily belong to the kernel of all @xmath86 s ( except , perhaps , for a set of measure zero ) . the result derived above allows a simple proof of the energy inequalities . with @xmath13 as in section [ sect : pos ] , define @xmath87 so that @xmath88 . by eq . ( [ eq : t00 ] ) and the fact that @xmath60 is even , we have @xmath89 with @xmath78 and @xmath21 given by eqs . ( [ eq : fkk ] ) and ( [ eq : gkk ] ) . clearly , @xmath90 is a finite sum of operators of the form @xmath85 . we therefore apply the bound ( [ eq : bd ] ) for the various cases @xmath91 and add the results , obtaining @xmath92 for all ( sufficiently well - behaved ) states @xmath14 . here , @xmath33 is equal to the area of the unit @xmath34-sphere divided by @xmath35 [ with the convention that @xmath93 , that is , @xmath94 if we now make the change of variables @xmath95 we find @xmath96 where the functions @xmath97 are defined by eq . ( [ eq : qn ] ) , thus completing the derivation of our general quantum inequality eq . ( [ eq : qi ] ) . we note that each @xmath97 is a positive function with @xmath40 and @xmath41 as @xmath42 . for @xmath98 , @xmath97 is strictly increasing on @xmath22 , while @xmath99 exhibits a maximum near @xmath100 and decreases thereafter see figures [ fig1 ] and [ fig2 ] . to conclude this section , we consider the scaling behaviour of ( [ eq : qi ] ) . let @xmath101 be the scaled function @xmath102 this function has the same integral over @xmath61 as @xmath13 but with @xmath103 times the characteristic width . it is easy to see that @xmath104 from which it follows that the bound for sampling function @xmath105 at mass @xmath9 is equal to @xmath106 times the bound for sampling function @xmath13 at mass @xmath107 . this is the expected scaling behaviour and is also exhibited by the quantum inequalities ( [ eq : qifr4 ] ) and ( [ eq : qifr2 ] ) of ford and roman . furthermore , it is clear that the value of the bound ( [ eq : qi ] ) tends to zero for each fixed @xmath13 as @xmath108 . thus we have @xmath109 and we recover the averaged weak energy condition in the limit ( cf . in this section , we briefly discuss the most interesting cases of the general quantum inequality derived above , and compare our results with those of flanagan @xcite and ford and roman @xcite . as mentioned in the introduction , flanagan has derived an apparently optimal quantum inequality for massless 2-dimensional field theory @xcite . substituting @xmath110 and @xmath111 into our bound eq . ( [ eq : qi ] ) and using the fact that @xmath112 as @xmath42 , we find @xmath113 the integrand is an even function in @xmath114 , so we may extend the range of integration to the whole of @xmath61 and then employ parseval s theorem to yield a @xmath115-space version of the quantum inequality @xmath116 which should be compared with flanagan s bound ( [ eq : qiflan ] ) . our bound is seen to be weaker by a factor of @xmath5 ; there is no contradiction because we do not expect our bound to be optimal . since flanagan s bound is six times stronger than that of ford and roman when applied to the lorentzian sampling function , our bound is accordingly four times stronger in this case . in the 2-dimensional massive case , the quantum inequality ( [ eq : qi ] ) becomes @xmath117 with @xmath43 given by @xmath118 figure [ fig1 ] shows that @xmath99 exhibits a maximum value of approximately @xmath119 near @xmath100 . we can not exclude the possibility that massive fields in @xmath3-dimensions can exhibit slightly stronger negative energy densities than massless fields can ( by a factor of at most @xmath119 ) . interestingly , a similar phenomenon occurs in ford and roman s treatment @xcite so it would be worthwhile to determine whether this is indeed so , or whether the peak is an artifact of the argument ( as ford and roman suggest ) . just as in the 2-dimensional case discussed above , the 4-dimensional quantum inequality takes a particularly simple form for massless fields : @xmath120 in particular , for the lorentzian function we have @xmath121 which is @xmath122 of ford and roman s result ( [ eq : qifr4 ] ) in this case [ recall that @xmath23 ] . this entails a slight tightening of the constraints on traversable wormholes @xcite . finally , we state the form of eq . ( [ eq : qi ] ) in the 4-dimensional massive case . in terms of the fourier transform of @xmath13 , we have @xmath124 where @xmath125 as figure [ fig2 ] makes clear , the function @xmath44 is bounded between zero and unity . accordingly the bound ( [ eq : qi4massless ] ) is also a lower bound for massive fields : in 4 dimensions [ indeed , in any spacetime dimension greater than or equal to three ] the effect of introducing a mass can not decrease the averaged energy density below the massless bound . we have given a simple derivation of a quantum inequality for the free real scalar field in minkowski space of any dimension , which allows more general sampling functions than previously possible . in particular , our derivation allows for compactly supported sampling functions and therefore removes any remaining doubt that the quantum inequalities of ford and roman might rely on subtle large scale effects to cancel local negative energy densities . in conclusion , we make various remarks . first , the derivation given here has been somewhat formal and lacking in mathematical rigour . however , we hope that our argument is simple enough that a rigorous formulation might be established without too much difficulty , and intend to return to this issue elsewhere . two elements of our discussion need to be more precisely specified : namely the class of sampling functions and the class of quantum states for which eq . ( [ eq : qi ] ) is valid . we have required that the sampling function @xmath13 be even primarily for convenience , and expect that this condition may be removed . in addition , we may speculate as to whether it is necessary that @xmath13 be smooth . of course , a necessary condition on @xmath13 is is that the integral on the right hand side of eq . ( [ eq : qi ] ) should converge ; this amounts to a smoothness condition on @xmath13 which becomes more stringent as the spacetime dimension increases ( and which is always satisfied if @xmath13 is actually smooth ) . thus , it may be that the smoothness of @xmath13 could be relaxed to @xmath126 where @xmath127 depends on @xmath128 . however , it is clearly important that @xmath13 has some degree of continuity . as an example , suppose @xmath13 is the characteristic function for the interval @xmath129\subset{{\bbb r}}$ ] . we have @xmath130 from which it follows that the integral in eq . ( [ eq : qi ] ) diverges for any @xmath37 . thus the quantum inequality provides no information in this case , which is consistent with results of garfinkle ( quoted by yurtsever in @xcite see particularly footnote [ 1 ] therein ) that the integral of @xmath131 over sharply defined boxes in spacetime can be unboundedly negative . as mentioned above , the class of quantum states for which eq . ( [ eq : qi ] ) holds must also be clarified . it is likely that quantum inequalities will hold for a dense class of states in the fock space built on the minkowski vacuum ; more generally , we hope that such inequalities might be established for the class of ( globally ) hadamard states @xcite . second , we have seen that our bound is weaker by a factor of @xmath5 than the optimal bound proposed by flanagan @xcite , which was derived using special features of 2-dimensional massless field theory . it would be interesting to investigate whether our argument could be improved to replicate flanagan s result and perhaps to obtain optimal bounds for the massive case and also higher dimensional spacetimes .

we generalise results of ford and roman which place lower bounds known as quantum inequalities on the renormalised energy density of a quantum field averaged against a choice of sampling function . ford and roman derived their results for a specific non - compactly supported sampling function ; here we use a different argument to obtain quantum inequalities for a class of smooth , even and non - negative sampling functions which are either compactly supported or decay rapidly at infinity . our results hold in @xmath0-dimensional minkowski space ( @xmath1 ) for the free real scalar field of mass @xmath2 . we discuss various features of our bounds in @xmath3 and @xmath4 dimensions . in particular , for massless field theory in @xmath3-dimensional minkowski space , we show that our quantum inequality is weaker than flanagan s optimal bound by a factor of @xmath5 . [ section ]

groups of galaxies represent a more common , more typical environment than their well - studied cousins massive clusters . most galaxies in the local universe lie in groups ( e.g. turner & gott 1976 ) and groups lie at a transition between the densest environments in the universe , cluster cores , and the field , making them important environments in the study of galaxy evolution ( e.g. zabludoff & mulchaey 1998 ) . groups are also important environments for diagnosing the origin of the non - gravitational heating of the intracluster medium ( icm ) . from low - redshift studies , non - gravitational heating of the icm appears to be proportionally more important in groups than in clusters , leading to a steepening of the @xmath2 relation for group mass systems ( e.g. helsdon & ponman 2000 ) and excess group entropy compared to the self - similar expectations based on massive clusters ( e.g. ponman et al . 2003 ; sun et al . 2009 ) . in fact , the evolution of the group x - ray scaling relations with redshift can help to constrain models of non - gravitational heating ( balogh et al . 2006 ) , distinguishing , for example , between excess entropy that is constant with redshift ( i.e. preheating ) versus entropy that is tied to halo cooling time ( i.e. increasing with time ) . groups of galaxies , and in particular their evolution with redshift , have been relatively poorly studied compared to massive clusters . their faint x - ray luminosities and low galaxy densities make groups difficult to detect outside of the local universe , and we have only recently begun to study the evolution of this important environment with redshift ( e.g. mulchaey et al . 2006 ; jeltema et al . 2006 , 2007 , 2008a ; willis et al . 2005 ; pacaud et al . 2007 ; wilman et al . 2005a , b ; gerke et al . 2007 ) . for example , only a handful of groups / poor clusters at moderate redshifts ( @xmath8 ) have both deep enough x - ray data and optical spectroscopy to be able to look at the relationships between group x - ray temperature , luminosity and group velocity dispersion ( jeltema et al . 2006 , 2007 , 2008a ; mulchaey et al . 2006 ; willis et al . 2005 ; gastaldello et al . 2008 ) . in this paper , we seek to extend studies of the x - ray emission from groups to higher redshifts using the deep _ chandra _ data in the extended groth strip ( egs ) . these data cover a @xmath9 deg@xmath10 field with a nominal exposure time of 200 ksec . the area covered represents an increase by a factor of eight over a previous search for groups in a single egs chandra pointing @xcite which yielded no detections . the depth of the _ chandra _ data allows us to both detect and to determine average temperatures for groups at high redshifts ( @xmath11 ) . specifically , we conduct an x - ray search for groups through their extended x - ray emission . in addition to the _ chandra _ coverage , the egs field was one of four fields targeted by the deep2 spectroscopic survey ( davis et al . 2003 ) , giving the added benefit of deep optical spectroscopic coverage . in particular , the deep2 spectroscopy in the egs is magnitude limited , but not colour - selected , giving a large galaxy sample at all redshifts and allowing us to optically identify and determine velocity dispersions for x - ray - detected group candidates ( davis et al . these data allow us to take a first look at the @xmath2 , @xmath3 , and @xmath4 scaling relations for high - redshift groups and poor clusters , and in future work we will investigate the properties of galaxies in these systems . while previous x - ray surveys have found a few high - redshift groups , these currently lack this depth of optical spectroscopy @xcite . our data reduction , group selection and optical identification procedures are presented in 2 and 3 . in 4 we present our main results including the discovery of a `` supergroup '' at @xmath5 ( 4.1 ) , three high - redshift groups ( 4.2 ) at @xmath0 , and several group agn ( 4.3 ) . the high - redshift systems represent the first x - ray detections of deep2 spectroscopically selected groups @xcite , and we present a first look at the scaling relations of systems of this mass at high redshifts . our results are summarised in 5 , and in the appendix we discuss the one x - ray - selected group for which no optical identification was possible . throughout the paper , we assume a cosmology of @xmath12 km s@xmath13 mpc@xmath13 , @xmath14 , @xmath15 and @xmath16 . the extended groth strip was observed with a series of eight _ chandra _ acis - i pointings , one in cycle 3 ( nandra et al . 2005 ) and seven in cycle 6 ( larid et al . 2009 ) , as a part of the all - wavelength extended groth strip international survey ( aegis ) . the number of observations per acis - i pointing varies between three and ten , but all fields have approximately the same total depth of 200 ksec . details about the survey and the aegis x - ray point source catalogue can be found in laird et al . ( 2009 ) . here we present the first extended x - ray source list . to search for extended sources ( see 3.1 ) we utilise the combined , soft band ( 0.5 - 2 kev ) images presented in laird et al . ( 2009 ) , and we refer the reader to this work for a detailed discussion of the reduction procedure . in brief , these images were created starting from the level 1 event files ; known aspect offsets were removed , bad pixels and afterglows were detected and removed , and the cti and time dependent gain corrections were applied . the observations taken in cycle 6 ( seven of the eight pointings ) were taken in vfaint mode , and for these observations the additional vfaint mode background cleaning was applied . background flare time periods were removed following the procedure described in nandra et al . ( 2007 ) and excluding time periods where the background exceeded 1.4 times the quiescent rate . from these cleaned event files , images and exposure maps were created in several energy bands ; here we focus on the soft band ( 0.5 - 2 kev ) for group detection as this band maximises the signal - to - noise for soft , thermal sources like groups . images and exposure maps of overlapping observations were aligned and merged using the centroids of bright sources in the field , creating one merged image per acis - i pointing . for group detection , we utilise these merged images and exposure maps . for each detected candidate group , we also extract spectra to determine the average source luminosity and temperature ( 3.3 ) . in the spectroscopic analysis , we wish to consider each observation separately , allowing us to create observation specific spectral response files ( rmfs and arfs ) . here we followed a very similar data reduction path to create individual observation level 2 , flare filtered event files to the one outlined above . however , we employ a slightly more restrictive flare filtering technique matching the prescriptions of markevitch et al . the filtering excluded time periods when the count rate , excluding point sources and group emission , was not within 20% of the quiescent rate in the 0.3 - 12 kev band . in practice , most of the observations have little contamination from background flares and the difference in flare filtering is negligible . in total , the aegis _ chandra _ data were divided in to 42 observations ; for nearly all of these , the net flare filtered exposure times differ by less than a couple percent and only two observations have net exposure times differing by more than 20% . for each group candidate identified below , we extracted spectra , after the removal of point sources , within the radius where the total group flux was above the background . point sources are detected using wavdetect with spatial scales of 1 , 2 , 4 , 8 , and 16 pixels . they were typically excluded using the output wavdetect 3@xmath17 ellipses , though for some sources embedded within the group emission or where two point sources had a very small separation the source regions were adjusted by hand . background spectra were extracted from a large annular region surrounding the source region , adjusting as necessary to avoid chip edges . we used the ciao script specextract to create source and background spectra , rmf and arf files for each observation of a given group candidate . as the total net group counts are typically quite low , we employ the fitting procedure for low count data described in willis et al . ( 2005 ) and applied to the _ xmm _ large - scale structure survey ( xmm - lss ) . each spectrum is binned such that the associated background spectrum has at least five counts per bin . spectra are fit within xspec ( v12.4.0ao ) using the c - statistic , which is considerably more accurate than the @xmath18 statistic for data with low counts per bin ( cash 1979 ) . the c - statistic within xspec ( cstat ) is slightly modified to account for statistical fluctuations in the background , and the slight binning of the data and backgrounds employed here alleviates biases in the model parameters that can occur when many of the spectral bins contain zero background counts @xcite . to detect extended x - ray sources , we employ the voronoi tessellation and percolation algorithm ( vtp ; ebeling & wiedenmann 1993 ) run on the soft band images and exposure maps . specifically , we utilise the ciao tool vtpdetect with a false positive threshold of @xmath19 and a minimum of 20 source counts . to consider a source as extended , we required that the average of the vtpdetect ellipse minor and major axes exceed three times the 95% encircled energy radius at the source off - axis distance and that the source have a signal - to - noise @xmath20 once nearby point sources were removed . candidate extended sources were then examined by eye to confirm their extended nature ; in particular , closely spaced point sources can lead to false detections . we also examined by eye the full ( 0.5 - 7 kev ) and hard ( 2 - 7 kev ) band images to search for possible contamination from hard point sources that may be faint in the soft band images . a similar methodology was used to find groups and clusters in the cdf - n @xcite and previously in the single _ chandra _ pointing in the egs taken in cycle 3 @xcite . in total , our search for extended x - ray sources yielded seven candidate groups , listed in table 1 . we detect extended sources to an observed flux limit of about @xmath21 ergs @xmath22 s@xmath13 . all of the detected sources are found in regions of fairly high effective exposure ( @xmath23 ksec ) , giving a survey area of @xmath24 deg ( see figure 2 in laird et al . ( 2009 ) ) and an extended source density of 13.2 deg@xmath25 . given the small number statistics and differences in selection technique this density is consistent within the errors with previous surveys . for example , the chandra deep fields north and south each detect one extended source near our flux limit in a single acis - i pointing @xcite , while the the xmm cosmos survey @xcite and projection of the rosat deep cluster survey @xcite to lower fluxes give somewhat higher densities of 20 - 35 deg@xmath25 at our flux limit . to identify potential optical group counterparts for the extended x - ray sources , we constructed a catalogue of galaxy groups in the egs using the deep2 spectroscopic data in the aegis region . unlike the other three deep2 fields where high - redshift ( @xmath0 ) galaxies were pre - selected based on colour for spectroscopic follow - up , in aegis , galaxies of all redshifts were targeted for spectroscopy @xcite , allowing us to search for groups at @xmath26 . we used the voronoi - delaunay method ( vdm ) group - finder described in gerke et al . ( 2005 ) , with the same parametrization described in that paper . this group - finder identifies groups by searching for galaxy overdensities in redshift space ; our initial group catalogue used only galaxies with good ( quality 3 or 4 ) deep2 redshifts . we then searched this catalogue for groups that fall near each x - ray source on the sky . by inspecting these groups member galaxies in the imaging data , and comparing to the x - ray contours , we identified a best - match optical counterpart for each x - ray source . the group - finder parametrization is optimised for the galaxy sampling rate of the overall deep2 survey , which is less dense than in egs . the optical group catalogue may , therefore , contain false - positives or interloper contamination . however , here we are simply using this catalogue as a tool to match already identified x - ray groups , and as described in 3.3 , we refine the selection of group member galaxies when determining the velocity dispersions . the best optical group matches to each extended x - ray source are shown in figures 1 and 2 compared to the _ chandra _ soft band x - ray contours . here we show both the contours of the diffuse x - ray emission ( in black ) and x - ray point sources ( in green ) . for the purposes of illustration , the diffuse contours were created by first filling point source regions ( from wavdetect ) using the ciao tool dmfilth through poisson sampling with the distribution mean determined from a local background region . the filled image is then adaptively smoothed with the tool csmooth and exposure corrected . one of our x - ray candidates , group 3 , unfortunately falls outside of the deep2 spectroscopic coverage and no optical identification was possible . we discuss this group in more detail in the appendix . three of the extended x - ray sources are identified as low - redshift , low x - ray luminosity groups ( @xmath27 ; figure 1 ) . in particular , the three low - redshift systems form part of a superstructure or `` supergroup '' at @xmath28 in the southern half of the aegis field . the remaining three group candidates are likely high - redshift , x - ray bright groups ( @xmath0 ; figure 2 ) . these three systems represent the first x - ray confirmation of spectroscopically selected , high - redshift deep2 groups . in one case , group 5 , two deep2 groups are good matches to the x - ray source position , one at @xmath29 ( figure 2 ) and one at @xmath30 ( figure 3 ) . here we consider the high - redshift system to be a better match both in terms of the location of the group members , including the likely central galaxy , and the average velocity dispersion and inferred x - ray luminosity . as shown in figure 3 , the galaxies in the @xmath30 group are offset with respect to the x - ray contours , and this group has a fairly low velocity dispersion of 232 km s@xmath13 . however , the x - ray emission does show an extension in the direction of three members of the @xmath30 system , one bright spiral galaxy with two smaller companions , indicating that the x - ray emission may be partially due to this foreground group . for each identified group , we determine the average x - ray luminosity and temperature from the _ chandra _ spectra . we fit the spectra from each observation of a given group candidate jointly to an absorbed thermal plasma model ( mekal ) with the absorption fixed at the galactic value @xcite , which ranges from @xmath31 @xmath22 , and the metallicity fixed at 0.3 solar @xcite . in most cases , the spectra are fit in the @xmath32 kev band . in all but one case , we are able to constrain the x - ray temperature . group 2 is unfortunately very faint . for this source , we restrict the spectrum to the 0.5 - 2 kev range to increase the signal - to - noise and quote the best - fitting luminosity for a fixed temperature of 1.5 kev ; we include the uncertainty in the temperature in the luminosity errors by varying the temperature between 0.5 and 10 kev . the average temperatures and x - ray luminosities of each group within the spectral extraction radius ( @xmath33 ) are listed in table 2 . for group 3 , we do not know the group redshift , but the spectrum still allows us to estimate the temperature . for this group , we left the redshift as a free parameter in the spectral fit when determining the temperature and temperature errors . the best - fitting temperature is not highly redshift dependent . we confirmed that leaving the redshift free gave reasonable results by refitting with a fixed redshift which we varied between 0.05 and 1.0 ; the best - fitting temperature was always within the errors quoted in table 2 . in order to compare to other group samples , we wish to compare the luminosities within a similar radius . therefore , in the scaling relations presented in 4 , we project the group luminosities to @xmath34 , the radius at which the group density is 500 times the critical density . we estimate @xmath34 from the average temperature using the relation @xmath35 mpc ( table 2 ; willis et al . 2005 ; finoguenov , reiprich , & bhringer 2001 ) . we extrapolate the luminosities from the spectral extraction radius based on a @xmath36-model surface brightness distribution with @xmath37 and @xmath38 @xmath39 kpc , typical values found for clusters and x - ray luminous intermediate - redshift groups ( e.g. jones & forman 1999 ; jeltema et al . the high - redshift aegis groups are detected to at least half of @xmath34 , and therefore , the extrapolation in luminosity is not large ( see table 2 ) . however , the luminosities of the low - redshift groups change by factors of @xmath40 due to the small detection radii . given the size of this correction , we choose to be conservative in our estimates of the luminosity errors . for all groups ( low and high redshift ) , we include the effect of the uncertainty in @xmath34 due to the uncertainty in the group temperature in the projected luminosity errors . we also bound the error from the unknown @xmath36-model parameters by using @xmath41 and @xmath42 @xmath39 kpc , which may be more appropriate values for low temperature groups @xcite ; this effectively sets the lower limits on our projected luminosities . for each group , we also determine the velocity dispersion of the member galaxies . in addition to the deep2 redshifts , in this step , we included galaxies with good - quality redshifts from the sloan digital sky survey ( adelman - mccarthy et al . 2008 ) , using data from the nyu value - added galaxy catalogue ( blanton et al . 2005 ) , and galaxies with redshifts that were obtained in follow - up observations of the deep2 sample with the hectospec spectrograph on the mmt ( coil et al . 2009 ; willmer et al . , in preparation ) . with the addition of these galaxies and because the group - finder we used was calibrated for the regions of deep2 outside of the egs , we refined the membership of each optical counterpart `` by hand , '' by searching for galaxies within a redshift - space cylinder , oriented with its axis along the line of sight , with radius and length chosen to produce a well - formed group on the sky , with a reasonably gaussian velocity distribution . there is inevitably some arbitrariness in the choice of this redshift - space cylinder , but we have checked explicitly that our results are not very sensitive to the details of the cylinder we choose in each case . in particular , systematic errors on the group velocity dispersions arising from the choice of cylinder are smaller than the statistical uncertainties due to discrete sampling of the velocity distribution . we computed these statistical uncertainties using the same procedure as in fang et al . ( 2007 ) . in brief , we use monte - carlo simulations to determine the probability distribution @xmath43 of measured velocity dispersions @xmath44 , given a true dispersion @xmath45 and a number of samples @xmath46 . one can then use bayes s theorem to derive the likelihood distribution @xmath47 of @xmath45 given an observed @xmath44 and @xmath46 , and our error bars on the group velocity dispersion reflect the 68% confidence region of this distribution . the on - sky distributions of galaxies in the optical - counterpart groups are shown in figures 1 and 2 , and the velocity dispersions and memberships of the groups are compiled in table 1 . group in the field . the x - ray emission shows an extension to the northeast in the direction of three members of the @xmath30 system , but the group as a whole is offset to the east of the x - ray emission . the image is 1 @xmath39 mpc comoving on a side for @xmath30 . , width=264 ] [ cols="<,^,^,^,^,^,^",options="header " , ] columns 3 and 4 list the x - ray spectral extraction radius and the net spectral counts for each group . columns 5 - 7 list the average group temperature and luminosity ( soft band and bolometric ) within @xmath33 . column 8 lists @xmath34 calculated as described in 3.3 , and the final column lists the multiplicative factor applied to extrapolate the measured luminosities to @xmath34 . values in italics indicate that the temperature was fixed at 1.5 kev . @xmath48 for group 2 the spectral extraction radius was reduced with respect the the total group extent to avoid a chip edge present in some of the observations . the three extended x - ray sources associated with low - redshift groups are shown in figure 1 . these three systems lie in the southern portion of the aegis _ chandra _ coverage , and they all appear to lie in a superstructure at @xmath28 . the galaxy distribution in this superstructure is shown in the right panel of figure 1 . two of the groups lie close together in both position and velocity space , while the northern most group lies in the foreground at @xmath7 . a couple of other associations of @xmath49 group - scale systems , or supergroups , have been found previously ( gonzalez et al . 2005 ; brough et al . 2006 ) , and these systems may be the precursors of massive clusters formed through multiple mergers . however , despite the obvious large - scale structure in the egs seen in figure 1 , we find that even the closer two groups ( 1 and 2 ) are unlikely to be bound given their projected distance ( @xmath50 mpc ) and relative velocity ( @xmath51 km s@xmath13 ) ( beers , geller , & huchra 1982 ) . notable from figure 1 ( left ) is the fact that all three groups have luminous elliptical galaxies at their x - ray centres , similar to other x - ray - bright , low - redshift groups @xcite . these groups are fairly x - ray faint having measured bolometric x - ray luminosities of @xmath52 @xmath53 ergs s@xmath13 and luminosities within @xmath34 of @xmath54 @xmath53 ergs s@xmath13 . the detected x - ray emission in all cases appears to extend beyond the optical extent of the central galaxy , consistent with group - scale emission . for all three groups the detected emission extends beyond 60 kpc , the criterion used to by osmond & ponman ( 2004 ) to separate galactic halos from group - scale halos , as shown by the surface brightness profiles in figure 4 . and @xmath42 @xmath39 kpc convolved with the psf at the group location . at a redshift of @xmath55 , 10 pixels is 6.6 @xmath39 kpc . , title="fig:",width=264 ] and @xmath42 @xmath39 kpc convolved with the psf at the group location . at a redshift of @xmath55 , 10 pixels is 6.6 @xmath39 kpc . , title="fig:",width=264 ] and @xmath42 @xmath39 kpc convolved with the psf at the group location . at a redshift of @xmath55 , 10 pixels is 6.6 @xmath39 kpc . , title="fig:",width=264 ] for the two low - redshift groups for which we can measure temperatures , group 1 and group 4 , we find @xmath56 and @xmath57 , respectively . the third group , group 2 , is the lowest velocity dispersion system in our sample and was previously identified in the sdss c4 dr3 cluster catalogue ( miller et al . 2005 ; von der linden et al . 2007 ) . , @xmath3 , and @xmath4 scaling relations for low - redshift groups in aegis ( red squares ) compared to x - ray luminous , low - redshift groups from the gems sample ( open stars ; osmond & ponman 2004 ) . fits show the best - fitting relations for the gems groups ( dashed line ; helsdon & ponman , in preparation ) and low - redshift clusters ( dotted line ; markevitch 1998 ) . luminosities for all systems are bolometric ( 0.01 - 100 kev ) and calculated within @xmath34 . , title="fig:",width=302 ] , @xmath3 , and @xmath4 scaling relations for low - redshift groups in aegis ( red squares ) compared to x - ray luminous , low - redshift groups from the gems sample ( open stars ; osmond & ponman 2004 ) . fits show the best - fitting relations for the gems groups ( dashed line ; helsdon & ponman , in preparation ) and low - redshift clusters ( dotted line ; markevitch 1998 ) . luminosities for all systems are bolometric ( 0.01 - 100 kev ) and calculated within @xmath34 . , title="fig:",width=302 ] , @xmath3 , and @xmath4 scaling relations for low - redshift groups in aegis ( red squares ) compared to x - ray luminous , low - redshift groups from the gems sample ( open stars ; osmond & ponman 2004 ) . fits show the best - fitting relations for the gems groups ( dashed line ; helsdon & ponman , in preparation ) and low - redshift clusters ( dotted line ; markevitch 1998 ) . luminosities for all systems are bolometric ( 0.01 - 100 kev ) and calculated within @xmath34 . , title="fig:",width=302 ] in figure 5 , we plot the @xmath2 , @xmath3 , and @xmath4 scaling relations for the low - redshift aegis groups ( red squares ) compared to the relations observed for other low - redshift groups and clusters with extended x - ray emission @xcite . within the uncertainties on the individual group properties and the scatter observed in these scaling relations , the aegis groups are consistent with other low - redshift groups . however , group 4 appears to have a temperature which is high compared to its x - ray luminosity . as discussed in 4.3 , this group has a central agn which could contribute to heating the gas . as noted above , the x - ray emission for all three low - redshift groups is centred on a luminous elliptical galaxy , and despite the extended nature of the emission the observed x - ray luminosities are not significantly different from those found for isolated , massive elliptical galaxies . as an indication of how much of the detected x - ray emission might be due to the central group galaxy , in figure 6 , we compare the detected x - ray luminosities to those of nearby elliptical galaxies . the x - ray luminosity of galaxies is known to correlate with optical / near - ir luminosity , although with large scatter , indicating a correlation between x - ray emission and galaxy stellar mass . here we consider the correlation between @xmath58 and @xmath59 , because @xmath60 band luminosity is more tightly correlated with stellar mass than bluer bands . @xmath61 band luminosities for the central galaxies are taken from 2mass . to compare to galaxy samples from the literature , here we use the x - ray luminosity in the @xmath62 kev band , and we do not assume that the x - ray emission extends beyond the detection radius ( i.e. we use the luminosity measured within @xmath33 rather than projecting to @xmath34 ) . in figure 6 , we compare the @xmath63 relation for the aegis systems ( red squares ) to a sample of satellite early - type galaxies ( central galaxies are excluded ) in low - redshift clusters and groups @xcite observed with _ chandra _ and to a sample of field early - types observed with _ rosat _ @xcite . as noted in @xcite , there appears to be an offset between satellite galaxies in groups and clusters and more isolated ellipticals in this relation with satellite galaxies having lower average x - ray luminosities for their k - band luminosities . this offset may indicate that some of the hot gas has been stripped from galaxies in groups and clusters , but higher resolution x - ray observations of a field sample are needed to check this result . we might expect central galaxies to have similar or brighter x - ray emission to field galaxies . we find , however , that the aegis systems span the full range of @xmath63 shown by field , group , and cluster galaxies . two of the three low - redshift aegis groups have x - ray luminosities which are completely consistent with the k - band luminosities of their central galaxy without the need for group x - ray emission . one group , group 1 , with the faintest central galaxy may have excess x - ray emission but is also consistent within the observed scatter with the @xmath63 relation of field galaxies . for this group in particular , we see from figure 1 that the detected x - ray emission extends beyond the central galaxy and includes other group members . however , it is clear that the detected x - ray emission for all of the low - redshift aegis groups likely has a significant contribution from , or is dominated by , the central galaxy . relation for the aegis systems ( red squares ) is compared to non - central early - type galaxies in groups and clusters ( diamonds ; jeltema et al . 2008b , sun et al . 2007 ) and to early - type galaxies in the field ( stars ; ellis & osullivan 2006 ) . the best - fitting relations are shown for the group and cluster galaxies ( solid line ) and the field galaxies ( dashed line ) . here the x - ray luminosities are in the 0.5 - 2 kev band and are not extrapolated . , width=302 ] the three high - redshift groups with extended x - ray emission are shown in figure 2 . these groups are all at redshifts above @xmath64 . they represent the first x - ray confirmation of spectroscopically selected deep2 groups and have velocity dispersions at the high end of those found for groups in the deep2 survey @xcite . a previous search for x - ray emission from deep2 groups based the single _ chandra _ pointing in the aegis region taken in a03 yielded no detections and concluded that the deep2 groups appear to be underluminous in x - rays compared to low - redshift groups with similar velocity dispersions @xcite . however , this one pointing contained relatively few groups ( 7 ) , all of which had only 3 - 6 member galaxies making their velocity dispersions uncertain . in fact , we do not detect any groups in this pointing . the high - redshift systems detected here have bolometric x - ray luminosities of a few times @xmath65 @xmath53 ergs s@xmath13 and temperatures of @xmath66 kev , placing them in the massive group to poor cluster regime of galaxy associations . in particular , the highest redshift system in our sample has a velocity dispersion of @xmath67 km s@xmath13 and @xmath68 kev ; at a redshift of @xmath1 , it is one of only a few such low - mass systems at @xmath69 observed in x - rays @xcite . , @xmath3 , and @xmath4 scaling relations for high - redshift groups in aegis ( @xmath0 ; red squares ) compared to x - ray luminous intermediate - redshift groups ( @xmath8 ) from the rdcs and xmm - lss samples ( blue circles ; jeltema et al . 2008a , 2007 ; pacaud et al . 2007 ; willis et al . 2005 ) and low - redshift groups from the gems sample ( open stars ; osmond & ponman 2004 ) . fits show the best - fitting relations for the intermediate - redshift groups ( blue solid line ) , the low - redshift gems groups ( dashed line ; helsdon & ponman , in preparation ) and low - redshift clusters ( dotted line ; markevitch 1998 ) . luminosities for all systems are bolometric ( 0.01 - 100 kev ) and calculated within @xmath34 . , title="fig:",width=302 ] , @xmath3 , and @xmath4 scaling relations for high - redshift groups in aegis ( @xmath0 ; red squares ) compared to x - ray luminous intermediate - redshift groups ( @xmath8 ) from the rdcs and xmm - lss samples ( blue circles ; jeltema et al . 2008a , 2007 ; pacaud et al . 2007 ; willis et al . 2005 ) and low - redshift groups from the gems sample ( open stars ; osmond & ponman 2004 ) . fits show the best - fitting relations for the intermediate - redshift groups ( blue solid line ) , the low - redshift gems groups ( dashed line ; helsdon & ponman , in preparation ) and low - redshift clusters ( dotted line ; markevitch 1998 ) . luminosities for all systems are bolometric ( 0.01 - 100 kev ) and calculated within @xmath34 . , title="fig:",width=302 ] , @xmath3 , and @xmath4 scaling relations for high - redshift groups in aegis ( @xmath0 ; red squares ) compared to x - ray luminous intermediate - redshift groups ( @xmath8 ) from the rdcs and xmm - lss samples ( blue circles ; jeltema et al . 2008a , 2007 ; pacaud et al . 2007 ; willis et al . 2005 ) and low - redshift groups from the gems sample ( open stars ; osmond & ponman 2004 ) . fits show the best - fitting relations for the intermediate - redshift groups ( blue solid line ) , the low - redshift gems groups ( dashed line ; helsdon & ponman , in preparation ) and low - redshift clusters ( dotted line ; markevitch 1998 ) . luminosities for all systems are bolometric ( 0.01 - 100 kev ) and calculated within @xmath34 . , title="fig:",width=302 ] figure 7 shows the @xmath2 , @xmath3 , and @xmath4 scaling relations for the high - redshift aegis groups . as in figure 5 , we compare the aegis systems to the scaling relations observed for low - redshift groups and clusters . here we also show the relations for x - ray luminous , intermediate - redshift groups / poor clusters at @xmath8 derived primarily from two samples : dedicated optical and x - ray follow - up of groups discovered in the _ rosat _ deep cluster survey ( rdcs ; jeltema et al . 2008a , 2007,2006 ; mulchaey et al . 2006 ) and the xmm - lss ( pacaud et al . 2007 ; willis et al . the best - fitting scaling relations for the intermediate - redshift systems are taken from @xcite . the intermediate - redshift group sample shows a similar range in x - ray luminosity and temperature to the higher - redshift aegis groups . an @xmath70 correction is applied to the @xmath2 and @xmath3 relations to account for the expected self - similar evolution with redshift of the scaling relations as calculated within a spherical overdensity radius ( e.g. bryan & norman 1998 ) . the high - redshift aegis groups agree quite well with the scaling relations found for lower - redshift systems , with no indications of significant evolution . they are particularly well - matched to the intermediate - redshift groups , showing a similar flatter relationship in @xmath3 and steeper relationship in @xmath4 compared to local groups and clusters . from these three x - ray selected systems , there is no indication that deep2 groups are underluminous in x - rays for their velocity dispersions ; however , there are other deep2 groups with similar velocity dispersions and redshifts within the _ chandra _ coverage that are not found by our x - ray selection . the x - ray detected group at @xmath1 is a more massive system , but taking the two x - ray groups at @xmath71 with @xmath72 km s@xmath13 as a model , there are five groups not detected in x - rays with @xmath73 km s@xmath13 , @xmath74 , and at least five member galaxies in the deep2 catalogue within the aegis region @xcite . here we limit the sample to groups with at least five members to reduce the uncertainty in the velocity dispersions . for five members and @xmath75 km s@xmath13 , the 1@xmath17 error range is @xmath76 km s@xmath13 . the x - ray properties of optical spectroscopically selected groups , including upper limits and average luminosities through stacking , will be considered in detail in a future paper after the completion of a new aegis group catalogue taking advantage of newer spectroscopy and optimised for the aegis spectroscopic selection ( gerke et al . 2009 , in preparation ) . as mentioned in 3.2 , the deep2 group selection in the aegis region may have a higher false positive rate due to the denser galaxy sampling compared to the rest of the deep2 survey . in addition , deeper x - ray data with a total exposure time of 800 ksecs is currently being taken for @xmath77% of the field and will be used in future studies . as stated above , at low redshifts x - ray luminous groups are typically found to have bright elliptical galaxies near the x - ray peak . the same is true of the low - redshift aegis groups presented here . at intermediate - redshifts ( @xmath8 ) , follow - up of x - ray - selected groups from the rdcs , found a range of central galaxy properties for systems of similar x - ray luminosity to the high - redshift aegis groups @xcite . more specifically , only a couple of the rdcs groups have single , dominant central galaxies . in some groups the central galaxy has multiple luminous nuclei , while other groups have no clear brightest cluster galaxy ( bcg ) , containing instead groups of bright galaxies and/or a brightest galaxy that is significantly offset from the x - ray centre . kpc on a side and centred on the approximate x - ray centre . group member galaxies are marked with black circles . all of the groups have plausible central galaxies or galaxy pairs . , title="fig:",width=226 ] kpc on a side and centred on the approximate x - ray centre . group member galaxies are marked with black circles . all of the groups have plausible central galaxies or galaxy pairs . , title="fig:",width=226 ] kpc on a side and centred on the approximate x - ray centre . group member galaxies are marked with black circles . all of the groups have plausible central galaxies or galaxy pairs . , title="fig:",width=226 ] unfortunately , the peak of the x - ray emission is not well - determined for the faint , high - redshift aegis groups , but here we consider whether there are group members which might be central galaxies . in figure 8 , we show optical images of the group galaxies within 100 @xmath39 kpc of the x - ray centre as determined by vtpdetect . all three high - redshift groups contain potential central galaxies ( i.e. consistent with the x - ray peak ) . for group 5 , in particular , acs imaging is available and shows a bright elliptical group member near the x - ray centre whose colours place it on the red sequence . the lack of acs imaging for the other two groups makes the assessment of galaxy morphology more challenging . group 6 also appears to have a bright , red sequence , likely early - type galaxy near the x - ray centre ; this galaxy has a smaller companion nearby in both position and velocity space . the highest - redshift system , group 7 , contains a group of similar magnitude galaxies near the centre . closest to the centre of this group is a galaxy pair . both galaxies in the pair fall at the bright end of the green valley in colour - magnitude space , unlike the other central galaxies , and at least one appears asymmetric / disky at the resolution of the cfht image . we may , therefore , be observing an earlier stage of bcg evolution in the highest redshift group . figure 9 shows the position of the likely central group galaxies on the colour - magnitude diagram ( left ) as well as their spectra ( right ) . while the centrals of groups 5 and 6 show strong stellar absorption features , typical of passive galaxies , both galaxies in the central pair of group 7 show strong [ oii ] emission , indicative of ongoing star - formation or agn activity . in particular , the spectrum of galaxy 7b also shows weak [ neiii ] and [ nev ] emission indicating the presence of an agn in this green valley central galaxy . the right panel of figure 2 shows that for all three groups , the galaxies near the x - ray peak also lie near the centre of the group velocity distribution , as one would expect . in summary , all of the x - ray detected aegis groups have likely central galaxies or galaxy associations . in two groups , the central galaxy may have one or more companions . in the highest redshift group at @xmath1 , the central galaxy pair appears to have younger stellar populations than are typically seen for centrals in x - ray luminous groups at lower redshifts , and one member of this pair hosts an agn seen through its optical emission lines . . groups 5 and 6 lie on the bright end of the red sequence , suggesting that these are red - and - dead objects , as typical of local groups . the central pair of group 7 , by contrast , lies at the bright end of the `` green valley '' between the red and blue populations , suggesting that these galaxies may not have fully ceased star - formation . right : optical spectra for each of the central galaxies . centrals 6 and 7b were observed with the deimos spectrograph on the keck ii telescope , as part of the deep2 galaxy redshift survey ; centrals 5 and 7a were observed with the mmt . each spectrum has been shifted to its rest frame , and prominent spectral features are labelled . while the centrals of groups 5 and 6 show strong stellar absorption features , typical of passive galaxies , both galaxies in the central pair of group 7 show strong [ oii ] emission , indicative of ongoing star - formation or agn activity . the presence of both [ neiii ] and [ nev ] emission in the spectrum of 7b indicate that this galaxy does in fact host an agn.,width=680 ] as is evident from figures 1 and 2 , in addition to diffuse x - ray emission , we detect x - ray point sources associated with several group member galaxies . in particular , for all three low - redshift groups we detect point sources associated with the central elliptical galaxy . these central x - ray sources have relatively low luminosities of @xmath78 @xmath53 ergs s@xmath13 . the central location of these sources within their respective elliptical galaxies and their relatively high x - ray luminosities compared to low mass x - ray binaries ( lmxbs ) , however , make them likely low - luminosity agn . the central galaxy in group 4 shows a pair of x - ray point sources with broad band ( 0.5 - 10 kev ) luminosities of @xmath79 and @xmath80 @xmath53 ergs s@xmath13 , with the latter slightly offset from the centre of the galaxy . in addition , the 1.4 ghz vla data shows extended , bright radio emission associated to the central galaxy , so this group almost certainly hosts a central agn or dual agn , given the pair of x - ray point sources . figure 10 shows the central galaxy overlaid with contours of radio ( red ) , diffuse x - ray ( black ) , and x - ray point source ( green ) emission . the radio emission is extended but not aligned with the optical axis , and a large , faint , asymmetric radio structure appears to the southeast along the jet axis . from the aegis20 radio catalogue , the central radio emission has a luminosity of @xmath81 w hz@xmath13 @xcite . the mmt spectrum of this galaxy does not show significant [ oiii ] emission , but does show weak h@xmath82 emission offset by @xmath83 km s@xmath13 from the galaxy . interestingly , this group has a relatively high x - ray temperature compared to the @xmath2 relation . kpc on a side . as in figure 1 , contours of diffuse x - ray emission are shown in black , and x - ray point sources are shown in green . all contours are linearly spaced . bottom : same for the high - redshift group group 5 . here the hst - acs i - band image is shown and the scale is 400 @xmath39 kpc on a side.,title="fig:",width=302 ] kpc on a side . as in figure 1 , contours of diffuse x - ray emission are shown in black , and x - ray point sources are shown in green . all contours are linearly spaced . bottom : same for the high - redshift group group 5 . here the hst - acs i - band image is shown and the scale is 400 @xmath39 kpc on a side.,title="fig:",width=302 ] the central source in group 2 has a luminosity of @xmath84 @xmath53 ergs s@xmath13 , while the central source in group 1 has a somewhat lower luminosity of @xmath85 @xmath53 ergs s@xmath13 . unfortunately , radio data are not available for groups 1 and 2 . spitzer data for the bcg in group 1 show a potential , though not conclusive , excess at 8 @xmath86 m and 24 @xmath86 m . in addition to the central sources in the low - redshift groups , we detect three x - ray point sources with lower luminosities of @xmath87 @xmath53 ergs s@xmath13 associated with spiral galaxies showing spectral signatures of ongoing star formation . in the high - redshift groups , we detect two x - ray point sources associated with group member galaxies in the @xmath71 groups and one additional x - ray source within the redshift range of group 5 but outside the group radius . the two group sources have broad band x - ray luminosities greater than @xmath65 @xmath53 ergs s@xmath13 and the neighbouring source has @xmath88 @xmath53 ergs s@xmath13 , making them difficult to explain as anything other than agn . overall , we find an x - ray agn fraction in @xmath71 groups of @xmath89% for galaxies brighter than @xmath90 with @xmath91 @xmath53 ergs s@xmath13 ( 2 out of 27 ) , consistent within the sizeable errors with the cluster fractions of @xmath92% ( @xmath93 and @xmath94 ergs s@xmath13 ) found at @xmath95 by eastman et al . ( 2007 ) and @xmath96% ( @xmath93 and @xmath91 ergs s@xmath13 ) found by kocevski et al . ( 2008 ) at @xmath97 . our agn fraction is marginally higher than the cluster fraction of @xmath98% at low redshift ( sivakoff et al . 2008 ) . a previous study in the aegis field of x - ray agn associated with high - redshift , optically selected deep2 groups found an excess of agn in groups , but once the host galaxy properties were accounted for , the x - ray agn fraction in groups was consistent with the field ( georgakakis et al . 2008 ) . one of the high - redshift x - ray point sources is associated with the likely central elliptical galaxy in group 5 . this galaxy also hosts a bright , bent double radio source with @xmath99 w hz@xmath13 shown in figure 10 . the bent nature of this source may indicate that it is moving with respect to the icm ( e.g. owen & rudnick 1976 ) ; however its line - of - sight velocity relative to the group is small . alternatively , it has been suggested that bent radio sources may indicate a recent group / cluster merger ( e.g. burns et al . in fact , the galaxy distribution surrounding group 5 shows that it may be part of a larger - scale / filamentary structure , and there is a potential second peak in the velocity distribution at slightly higher redshift , supporting the possibility of a merger . fainter radio sources ( @xmath100 between @xmath101 w hz@xmath13 and @xmath102 w hz@xmath13 ) are also found in the other two high - redshift groups associated to the central galaxy in group 6 , the x - ray agn in group 6 , and both the central pair and the galaxy south of this pair in group 7 . as noted above , one the the central galaxies in group 7 , 7b , also shows optical emission lines indicating an agn . in summary , all of the central group galaxies , and several other groups members , show signs of activity in x - ray , radio , and/or optical emission . in particular , the central galaxies in groups 4 and 5 host both x - ray and radio jet sources . we have conducted an x - ray search for groups of galaxies in the extended groth strip utilising the chandra coverage of @xmath9 deg@xmath10 , part of the aegis survey . groups of galaxies are selected based on the presence of extended x - ray emission and then matched against optically selected groups from the deep2 spectroscopy in the field . in total , we find seven extended x - ray sources , and we identify optical counterparts for six . three of these groups lie at @xmath0 and represent the first x - ray detections of high - redshift deep2 groups . the other three groups are low - luminosity , low - redshift groups which form part of a larger structure or `` supergroup '' at @xmath28 in the southern portion of the aegis field . the seventh group candidate unfortunately lies outside of the current spectroscopy , but as discussed in the appendix , it is also a high - redshift group candidate . our main results are summarised below . _ low - redshift groups : _ the three low - redshift extended x - ray sources are all centred on luminous elliptical galaxies in groups at @xmath28 , one of which was previously identified in the sdss c4 cluster catalogue . together the deep2 , mmt , and sdss spectroscopy reveal a large - scale structure in the southern portion of the aegis field dominated by these three groups . the groups are relatively low - mass systems with @xmath103 kev and @xmath104 km s@xmath13 . when projected to @xmath34 , their x - ray luminosities are consistent with the total group velocity dispersion ; however , their detected x - ray luminosities are similar to those found for individual massive elliptical galaxies . _ high - redshift groups : _ for the high - redshift systems , we find x - ray temperatures in the range of @xmath66 kev and bolometric x - ray luminosities of a few times @xmath65 @xmath53 ergs s@xmath13 , placing them in the massive group to poor cluster regime of galaxy associations . the hottest system with @xmath105 kev lies at a redshift of 1.13 , making it one of only a few systems of its mass known at @xmath106 . the extensive deep2 spectroscopy supplemented by additional redshifts from mmt allow us to identify @xmath107 members for each group and to determine group velocity dispersions . these data allow us the first look at the @xmath2 , @xmath3 , and @xmath4 scaling relations for systems in this mass range at high redshifts . we find no evidence for evolution in these scaling relations when comparing the x - ray - selected deep2 groups to x - ray luminous groups at lower redshifts . however , there are x - ray undetected deep2 groups in the field with similar velocity dispersions , which will be studied in detail in a future paper . similar to x - ray luminous , low - redshift groups , all of the detected systems have potential central galaxies , though two of the central galaxies appear to have close companions . the likely central galaxies in the two groups at @xmath108 are bright , red galaxies with strong stellar absorption features typical of passive bcgs . the central galaxy pair in the highest redshift system at @xmath1 , however , both show [ oii ] emission and lie at the bright end of the green valley in colour - magnitude space . here we may be witnessing an early stage of bcg formation . _ group agn : _ we find x - ray point sources associated to all of the central galaxies in the low - redshift groups and one of the centrals in the high - redshift groups . we also detect radio sources near the centres of all of the groups with radio observations ( all three high - redshift and one low - redshift group ) . particularly convincing and intriguing examples of central agn activity include the bright bent - double radio source and x - ray point source at the center of group 5 at @xmath6 , the extended radio and double x - ray point sources associated to the central galaxy in the lowest - redshift group , group 4 , and the bright green valley pair of galaxies in group 7 at @xmath1 , one of which shows optical agn emission lines . considering all group galaxies , we find an x - ray - selected agn fraction in @xmath71 groups of @xmath89% for galaxies brighter than @xmath90 with @xmath91 @xmath53 ergs s@xmath13 , consistent with previous studies of high - redshift clusters @xcite . + we would like to sincerely thank the anonymous referee for their comments which led to several improvements in the paper . we would also like to thank members of the aegis team , particularly d. rosario and rob ivison , for their advice and support with the multiwavelength data . support for this work was provided by the national aeronautics and space administration through _ chandra _ award number ar9 - 0017x issued by the _ chandra _ x - ray observatory centre , which is operated by the smithsonian astrophysical observatory for and on behalf of the national aeronautics space administration under contract nas8 - 03060 . t.e.j . is grateful for support from the alexander f. morrison fellowship , administered through the university of california observatories and the regents of the university of california . b.f.g . was supported by the u.s . department of energy under contract number de - ac3 - 76sf00515 . support for m.c.c . was provided by nasa through the _ spitzer space telescope _ fellowship program . e.s.l . acknowledges financial support from the uk science and technology facilities council . this study makes use of data from aegis , a multiwavelength sky survey conducted with the chandra , galex , hubble , keck , cfht , mmt , subaru , palomar , spitzer , vla , and other telescopes and supported in part by the nsf , nasa , and the stfc . as discussed in section 2.2 , one of our extended x - ray group candidates , group 3 , could not be identified optically , because it falls outside the deep2 spectroscopic region . a ned search also reveals no known galaxy redshifts within a couple of arcminutes of the group , though interestingly there is a radio source about @xmath109 from the x - ray centre . here we consider this extended x - ray source in more detail . the top panel of figure a1 shows the _ chandra _ 0.5 - 2 kev contours overlaid on the cfht r - band image . there are two galaxies lying very close to the peak of the x - ray emission . using the cfht bri photometry , we estimate photometric redshifts for these galaxies and find @xmath110 ( @xmath111 ) for the brighter galaxy and @xmath112 for the fainter galaxy . as mentioned in 3.3 , we are able to estimate a temperature for this group of @xmath113 kev by leaving the redshift as a free parameter . this temperature implies a massive group / poor cluster size system . in fact , this system is one of the hottest and highest flux sources in our sample . in principle , requiring that the group lie on the @xmath2 relation can constrain its redshift , and we illustrate this comparison in the bottom panel of figure a1 . here we refit the spectrum and determine the temperature and luminosity for four assumed , fixed redshifts , @xmath114 and @xmath115 . the luminosities are projected to @xmath34 for the assumed redshift as described in 3.3 . from figure a1 , the @xmath2 relation for this group favours a moderate to high redshift , with @xmath116 and @xmath117 best matching the relation . however , within the temperature errors and the scatter in the @xmath2 relation a wide range of redshifts are possible . relation for group 3 for different assumed redshifts , @xmath118 ( red squares ) compared to the group and cluster samples shown in figure 7 . luminosities for all systems are bolometric ( 0.01 - 100 kev ) and calculated within @xmath34 . , title="fig:",width=226 ] relation for group 3 for different assumed redshifts , @xmath118 ( red squares ) compared to the group and cluster samples shown in figure 7 . luminosities for all systems are bolometric ( 0.01 - 100 kev ) and calculated within @xmath34 . , title="fig:",width=302 ] adelman - mccarthy , j. k. , et al . 2008 , apjs , 175 , 297 anders , e. , & grevesse , n. 1989 , geochimica et cosmochimica acta , 53 , 197 balogh , m. l. , babul , a. , voit , g. m. , mccarthy , i. g. , jones , l. r. , lewis , g. f. , & ebeling , h. 2006 , mnras , 366 , 624 bauer , f. e. , et al . 2002 , aj , 123 , 1163 beers , t. c. , geller , m. j. , & huchra , j. p. 1982 , apj , 257 , 23 blanton , m. r. , et al . 2005 , aj , 129 , 2562 brough , s. , forbes , d. a. , kilborn , v. a. , couch , w. , & colless , m. 2006 , mnras , 369 , 1351 bryan , g. l. , & norman , m. l. 1998 , apj , 495 , 80 burns , j. o. , loken , c. , roettiger , k. , rizza , e. , bryan , g. , norman , m. l. , gmez , p. , & owen , f. n. 2002 , new astronomy review , 46 , 135 cash , w. 1979 , apj , 228 , 939 coil , a. l. , et al . 2009 , apj , submitted davis , m. , et al . 2003 , proc . spie , 4834 , 161 davis , m. , et al . 2007 , apj , 660 , l1 eastman , j. , martini , p. , sivakoff , g. , kelson , d. d. , mulchaey , j. s. , & tran , k .- v . 2007 , apj , 664 , l9 ebeling , h. , & wiedenmann , g. 1993 , physical review e , 47 , 704 ellis , s. c. , & osullivan , e. 2006 , mnras , 367 , 627 fang , t. , et al . 2007 , apj , 660 , l27 finoguenov , a. , et al . 2007 , apjs , 172 , 182 finoguenov , a. , reiprich , t. h. , bhringer , h. 2001 , a&a , 368 , 749 gastaldello , f. , trevese , d. , vagnetti , f. , & fusco - 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we present the discovery of seven x - ray emitting groups of galaxies selected as extended x - ray sources in the 200 ksec _ chandra _ coverage of the all - wavelength extended groth strip international survey ( aegis ) . in addition , we report on agn activity associated to these systems . for the six extended sources which lie within the deep2 galaxy redshift survey coverage , we identify optical counterparts and determine velocity dispersions . in particular , we find three massive high - redshift groups at @xmath0 , one of which is at @xmath1 , the first x - ray detections of spectroscopically selected deep2 groups . we also present a first look at the the @xmath2 , @xmath3 , and @xmath4 scaling relations for high - redshift massive groups . we find that the properties of these x - ray selected systems agree well with the scaling relations of similar systems at low redshift , although there are x - ray undetected groups in the deep2 catalogue with similar velocity dispersions . the other three x - ray groups with identified redshifts are associated with lower mass groups at @xmath5 and together form part of a large structure or `` supergroup '' in the southern portion of the aegis field . similar to other x - ray - luminous groups , all of the low - redshift systems are centred on massive elliptical galaxies , and all of the high - redshift groups have likely central galaxies or galaxy pairs . interestingly , the central galaxies in the highest redshift system show indications of ongoing star formation . all of the central group galaxies host x - ray point sources , radio sources , and/or show optical agn emission . particularly interesting examples of central agn activity include a bent - double radio source plus x - ray point source at the center of a group at @xmath6 , extended radio and double x - ray point sources associated to the central galaxy in the lowest - redshift group at @xmath7 , and a bright green valley galaxy ( part of a pair ) in the @xmath1 group which shows optical agn emission lines . galaxies : clusters : general x - rays : galaxies : clusters galaxies : active

discovery of so called colossal magnetoresistance ( cmr ) in doped lanthanum manganites of the type la@xmath0a@xmath1mno@xmath2 ( where @xmath10 is a divalent alkaline - earth element like ca , sr , ba ) @xcite has fundamental importance for solid state physics and offers promissing application in advanced technology . the most pronounced effect of cmr ( _ i.e. _ , an extremely large negative magnetoresistance ( mr ) ) was found in la@xmath0ca@xmath1mno@xmath2 films with @xmath11 . in the concentration range @xmath12 the la@xmath0ca@xmath1mno@xmath2 is a ferromagnet with a rather high conductivity at temperatures far below the curie temperature @xmath5 . the resistance strongly increases with temperature and has a peak at a temperature @xmath13 which is close to @xmath14 in samples with fairly perfect crystalline structure . the mr is maximal near @xmath5 . above @xmath13 ( in the paramagnetic state ) the resistance has semiconducting behavior and the mr is much less . below @xmath14 the mr strongly decreases with decreasing temperature , and it is believed that the mr must go to zero on approaching @xmath15 k in fairly good crystals@xcite . the common explanation of the cmr is usually provided in the frame of the double - exchange ( de ) model @xcite which is based on the assumption of the appearance of mn@xmath16 ions with substitution of la@xmath17 by a divalent cation . it is believed that in this case a ferromagnetism results from the strong ferromagnetic exchange between mn@xmath17 and mn@xmath16 . this model , however , can not explain the many features of the resistivity behavior of manganites in both the ferromagnetic and the paramagnetic states . therefore , in succeeding theoretical works additional physical mechanisms ( mainly still in the frame of de model ) were considered . the possible influence of strong electron - phonon coupling ( jahn - teller distortion ) , polaronic effects ( magnetic or lattice polarons ) , nearly half - metallic nature of ferromagnetism in the manganites , electron localization , phase separation and other effects were considered ( see refs . and references therein ) . in spite of extensive experimental and theoretical efforts a clear understanding of cmr in the manganites is not yet available . the reason is that the knowledge of even the basic electronic properties of doped manganites is still far from complete . for example , one can find in the literature conflicting experimental claims regarding the nature of holes in doped manganites at @xmath18 . the de model is based on the assumption that the holes in doped manganites correspond to mn@xmath16 ions arising among the regular mn@xmath17 ions due to the doping . but some investigations give strong evidence that the holes are located mainly on the oxygen ions @xcite ( _ i.e. _ , the holes are of o @xmath19 character ) . on the other hand there is experimental evidence ( see ref . and references therein ) that holes doped into lamno@xmath2 are mainly of mn @xmath20 character . one of the important questions in physics of the cmr manganites is the nature of the rather high - conducting state below @xmath14 . the only sure assumption at present is that the charge carriers at low temperatures can be considered to be quasifree . whether the doped cmr manganites in the ferromagnetic state should be regarded as conventional bad ( disordered ) metals or as just heavily doped degenerate semiconductors has been argued @xcite . it is known , however , that manganites do not behave like conventional non - ferromagnetic metals . for example , the decrease in resistivity with decreasing temperature of fairly good crystalline manganites is too large to be attributed , as in conventional metals , to the electron - phonon interaction @xcite . it follows also from the known experimental data@xcite that a clear correlation exists between transport properties and magnetism in doped manganites . namely , the resistance @xmath21 of manganites in the ferromagnetic state is a function of the magnetization @xmath22 which in turn depends on the temperature and magnetic field : @xmath23 $ ] . in manganites the conductivity increases with the enhancement of ferromagnetic order . this is actually the source of the huge resistivity decrease at the paramagnetic - ferromagnetic transition and the cmr . this correlation is most pronounced in good quality crystals , but persists to some degree even in rather disordered systems , like polycrystalline or granular samples . the bulk manganites la@xmath0a@xmath1mno@xmath2 ( @xmath11 ) have nearly cubic symmetry and therefore should not have any marked mr anisotropy . in contrast , the cmr films possess a pronounced mr anisotropy in low magnetic fields @xcite . due to the above - mentioned transport - magnetism correlation it should be thought that the mr anisotropy in cmr films is in fact some reflection of @xmath24 behavior . two main sources of mr anisotropy in ferromagnetic films are : ( 1 ) the existence of preferential directions of magnetization ( due to strains stemming from the lattice film - substrate mismatch or other sources ) , and ( 2 ) dependence of resistance on the angle between current and magnetization , which is inherent in ferromagnets ( the so called anisotropic magnetoresistance ( amr ) effect)@xcite . it was found@xcite for the cmr films , which are subject to compressive strain in the film plane , that if the easy magnetization axis is parallel to the film plane , an unusual positive mr appears when the magnetic field is perpendicular to the film plane , while for a parallel field the mr is negative . this behavior can be associated with concurrent influence of the above - mentioned anisotropy sources@xcite . in this article , we report that mr anisotropy in cmr films can also manifest itself in far more complex and puzzling ways . the object of study was la@xmath0ca@xmath1mno@xmath2 ( @xmath3 ) films prepared by pulsed - laser deposition ( pld ) onto laalo@xmath2 substrates . we found non - monotonic and alternating dependences of the mr on @xmath4 for both the perpendicular and parallel directions of @xmath4 relative to the film plane with @xmath4 perpendicular to the current ( preliminary short report about this behavior was presented at lt22@xcite ) . only in the longitudinal geometry ( when @xmath4 is parallel to both the current and the film plane ) was the mr always negative , as expected for cmr manganites . this rather complex behavior of mr manifests itself in the ferromagnetic state and has not been reported in previous studies . we will show below that this behavior is determined by a peculiar structural disorder induced by a film - substrate interaction . from the transport and magnetic properties of the film studied , it can be concluded that the film crystal structure should be inhomogeneous in such a way that various parts of the film have non - identical ( and quite distinct ) magnetic properties . this hypothesis is supported by x - ray diffraction which revealed that the film is inhomogeneous in strains , lattice parameters and lattice orientation . the possible reasons for the formation of such structure and its effect on mr anisotropy are considered . we note that although similar mr behavior was observed for several films prepared by this technique , the detailed measurements reported here were all taken on the same film , a representative specimen . la@xmath0ca@xmath1mno@xmath2 ( @xmath3 ) films were grown by pld on ( 100 ) oriented laalo@xmath25 substrate . a pld system from neocera inc . with a lambda physik krf excimer laser operating at 248 nm was used to ablate the target material with a nominal composition la@xmath26ca@xmath27mno@xmath2 . the main details of the target preparation and laser ablation technique are described in ref . . stoichiometric amounts of high purity la@xmath28o@xmath25 , cao and mnco@xmath25 were mixed and ball milled for several hours , reacted at 1100@xmath29c for 24 hours with intermediate grinding and mixing after 12 hours , and pressed with duramax b-1020 acrylic binder at 50 - 170 mpa to make the target pellet . the pellet was sintered at 600 1200@xmath29c for 12 hours in a box furnace in air to burn off the binder and strengthen the pellet . during deposition the pulse energy was 228 mj with a repetition rate of 8 hz . the target - substrate distance was about 7 cm . the film 80 nm thick@xcite described in this paper was ablated at a substrate temperature of 400@xmath29c in an oxygen atmosphere at pressure @xmath30 mtorr . time of deposition was about 20 min . immediately after deposition the film was annealed 30 min at @xmath31c in the same pld chamber at @xmath32 mtorr . the film was also post - annealed in flowing oxygen for 24 hours at 900@xmath29c . x - ray diffraction ( xrd ) study of crystal - structure of the film and the substrate was done using a dron-3 diffractometer with a ge(111 ) monochromator and cuk@xmath33 radiation . magnetization and ac susceptibility measurements were done by commercial squid magnetometer . resistance as a function of field and temperature was measured using a standard four - point probe technique . the available cryostat with a rotating electromagnet makes it possible to measure resistance in magnetic fields up to 20 koe with different directions of @xmath4 relative to the plane of the film and the transport current . the two methods of xrd study were used : ( i ) normal @xmath34 scanning , and ( ii ) diffractional reflection curve ( drc ) recording . drcs were recorded on symmetric and asymmetric reflections . the technique of sample rotation about the diffraction vector was used@xcite . that makes variations of the angle between the surface and incident beam or corresponding reflected one possible up to the critical angle of total external reflection which is about @xmath35 . the perfection of crystal structure was characterized by the drc half - width @xmath36 . the crystal lattice parameters were obtained by the bond technique@xcite . the substrate laalo@xmath25 ( from coating & crystal technology ( cct ) , kittaning , pa 16201 ) was ( 001 ) oriented . for determination of lattice parameters @xmath37 ( of the pseudocubic cell ) the reflections ( 400 ) , ( 330 ) and ( 003 ) were used . it was found that @xmath38 nm , @xmath39 nm , and @xmath40 nm . one can compare these values with cct data ( @xmath41 nm ) or with that from one of the special studies of laalo@xmath25 single crystals@xcite ( @xmath42 nm ) . the substrate is characterized by the availability of mosaic crystal blocks and twin structure which are common to laalo@xmath25@xcite . the angles of misalignment of fragments ( estimated with use of the asymmetric reflections ) range up to 0.2@xmath29 . the magnitudes of @xmath36 for these fragments are dispersed between 15 and 120 arcsec for different parts of the crystal . using the asymmetric reflection ( 101 ) enables us to conclude that drc does not experience any significant broadening even at minimum angles ( @xmath43 ) of reflected beam . this demonstrates that the pld process in our case does not involve a formation of a damage layer on the substrate surface in contrast to the study of ref . where the value of @xmath36 for the ndgao@xmath25 substrate was increased by two orders of magnitude , and the damage layer was 1.2 @xmath44 m thick . the film lattice parameters have been determined from the lines ( 040 ) , ( 220 ) and ( 022 ) . the in - plane lattice parameters are found to be @xmath45 nm and @xmath46 nm ; whereas , the out - of - plane one @xmath47 nm . hence the film has a tetragonal lattice . the ratio of the out - of - plane lattice parameter to the in - plane ones is about 1.009 . bulk la@xmath0ca@xmath1mno@xmath2 ( @xmath3 ) has a cubic lattice with lattice parameter @xmath48 in the range 0.3850.386 nm@xcite . it is thus apparent that the film is in a strained state . in the plane of growth the film is under biaxial compression , but it is under uniaxial tension in the direction perpendicular to the film plane . such a strain state was observed previously in la@xmath0ca@xmath1mno@xmath2 ( @xmath490.3 ) films on laalo@xmath25 substrates@xcite . the effects connected with it will be discussed below . it follows from the xrd data that the crystallographic substrate plane ( 100)@xmath50 ( which corresponds to the deposition surface ) is parallel to the plane ( 010)@xmath51 of the film . the fine - structure parameters were obtained from the analysis of several reflections taking into account the drc broadening . the @xmath36 values for the reflections of ( 101 ) , ( 202 ) and ( 303 ) are 0.375@xmath29 , 0.42@xmath29 , 0.56@xmath29 , correspondingly . the values of microblock angular misalignment , @xmath52 , dimensions of coherent scattering areas , @xmath53 , and microdeformation , @xmath54 , are 0.32@xmath29 , 60 nm , and @xmath55 , correspondingly . the drcs at angles in the range 16@xmath29 were asymmetric . there are reasons to believe that this is caused by superposition of reflections from parts of the film with different lattice parameters , since orientations @xmath56_{f}\parallel [ 010]_{s}$ ] and @xmath56_{f}\parallel [ 001]_{s}$ ] are quite possible due to the twin structure of substrate . these xrd data will be used below for explanation of the transport and magnetic anisotropy in the film . the transport properties of the film in fig.1 correspond well to the expected behavior of cmr films @xcite . namely , the temperature dependence of the resistance @xmath57 has a maximum ( peak ) at @xmath58 k. below @xmath13 a quite sharp resistance drop takes place which corresponds to paramagnetic - ferromagnetic transition that occurs approximately simultaneously with the insulator - metal transition . the resistivity @xmath59 at @xmath60 k was about 375 @xmath61 cm . the ratio of the resistances at @xmath62 and 4.2 k , @xmath63 , is about 31.4 . this fairly large variation of resistance with temperature for a rather disordered doped manganite should be attributed mainly to the strengthening of the magnetic order with decreasing temperature . the magnetic field @xmath4 produces a large decrease in resistance ( see the insert in fig.1 ) . for a measure of the mr we have taken @xmath64/r(0)$ ] it was found that @xmath65 has its maximum absolute value ( about 0.43 at @xmath66 koe ) at a temperature @xmath67 k. the temperature dependences of the magnetization @xmath22 and of the ac susceptibility @xmath68 for different directions of magnetic field relative to the film plane are shown in figs . 2 and 3 . these enable the value of @xmath5 to be estimated . since the magnetic moment is the order parameter at a paramagnetic - ferromagnetic transition , it is quite natural to define @xmath5 as the temperature where @xmath22 or @xmath68 starts to increase , when going from high to lower temperatures . in this case ( as may be seen in figs . 2 and 3 ) the @xmath5 value of the film is approximately equal to the value of @xmath69 k. the nearly identical values of @xmath5 and @xmath62 are characteristic of films with good enough crystal perfection and fairly large grain size@xcite . if @xmath5 is defined as the temperature at which @xmath22 comes to a half of the saturation value , or as the temperature of the inflection point in @xmath70 , as is sometimes done , the value will appear to be somewhat lower ( 270280 k ) . in any case , however , the value of @xmath5 and , especially , @xmath62 ( which is determined quite unambiguously ) seem to be somewhat higher than the corresponding values ( 260 - 270 k ) for bulk ca - doped manganites of the same composition ( @xmath71 ) based on the accepted bulk phase diagram@xcite . an increase in @xmath5 and @xmath62 was found earlier in bulk manganites under hydrostatic pressure@xcite or in films with considerable compressive strains due to a film - substrate interaction@xcite . the maximal effect of hydrostatic pressure on bulk la@xmath0ca@xmath1mno@xmath2 ( @xmath3 ) causes the increase of @xmath62 from @xmath72 k to 290 k , and @xmath5 from @xmath73 k to @xmath74 k@xcite . the total bulk strain in this film is highly compressive . indeed , in this film the volume of the unit cell , @xmath75 , is equal to @xmath76 nm@xmath77 which is less than @xmath75 for bulk la@xmath26ca@xmath27mno@xmath2 ( about @xmath78 nm@xmath77 provided @xmath79 nm ) . moreover , this is also less than previously reported values of @xmath75 for la@xmath0ca@xmath1mno@xmath2 films@xcite . it was found in preceding studies@xcite that @xmath62 ( and @xmath5 ) increase with decreasing @xmath75 in la@xmath0ca@xmath1mno@xmath2 films . for example , it was reported@xcite that for a film with x=0.3 and @xmath80 nm@xmath77 , @xmath62 is about 275 k . for the film of figs.13 the unit cell volume is less , explaining its higher @xmath5 ( 296 k ) . the strain state of the film is however inhomogeneous . in this case the influence of the jahn - teller part of the strain tensor on @xmath5 can be taken into account@xcite , in principle . the equilibrium lattice parameter in bulk la@xmath26ca@xmath27mno@xmath2 is not known , however , with necessary accuracy for quantitative consideration of this effect . according to early studies @xcite the mr value in manganites decreases profoundly ( but _ remains negative _ ) below @xmath14 as the temperature is reduced . more recently@xcite , it was shown that mr in strained manganite films can be positive at low temperature . we present below a far more complex behavior of @xmath81 for this film ( figs . 46 ) which is determined by magnetic anisotropy induced by a film - substrate interaction . in these figures , as well as in the following text of the paper , we will use designations @xmath82 and @xmath83 for the cases of field @xmath4 applied perpendicular and parallel to the film plane , correspondingly . let us consider , first of all , the mr behavior for the cases that the magnetic field is perpendicular to the current . figs . 4 and 5 present the case for different field orientation ( @xmath82 and @xmath83 ) . we will look more closely at the mr behavior at helium temperature . it can be seen , that , for increasing @xmath82 , the mr is first negative ( at @xmath84 koe ) , then positive ( 4 koe @xmath85 koe ) , and then negative again ( fig.4 ) . for increasing @xmath83 , the mr is positive below @xmath86 koe and negative above it ( fig.5 ) . anisotropic behavior of this kind occurs only at low temperatures . at @xmath87 k the mr is negative for both directions of the magnetic field . if @xmath4 is parallel to both the current @xmath88 and the film plane , the mr is always negative ( fig . 6 ) . it can be seen from fig . 6 that mr values differ signicantly for the cases when @xmath4 is perpendicular and those parallel to the transport current . this is because of a pronounced dependence of the resistance on angle @xmath7 between @xmath4 and the current @xmath88 . one of the measured angular dependences is shown in fig . it is found that @xmath89 at high enough field ( more than 8 koe ) can be described fairly well by the relation @xmath90 where @xmath91 -1\}$ ] is some positive parameter . this is a manifestation of the amr effect , which should be inherent in ferromagnets@xcite . in manganite films this effect was previously reported in refs .. the magnetic field dependence of the amr parameter @xmath92 at @xmath60 k is shown in fig . it can be seen that @xmath92 increases with field in the range below @xmath93 koe and comes to some saturation value at higher field . this saturation , as will be shown below , proceeds approximately at the field @xmath94 where the magnetization comes to rotational saturation in a field perpendicular to the film plane . some clear inflection in the @xmath95 dependence at @xmath96 koe can be seen . this inflection should not take place for homogeneous systems ( compare , for example , with results of ref . ) and reflects , as will be shown below , the structural and magnetic inhomogeneity of the film studied . a temperature dependence of saturated values of @xmath92 is shown in fig . 9 . three important features of @xmath97 behavior can be derived from the figure . first , @xmath92 is nearly constant at low temperatures ( up to 150 k ) , where the saturation magnetization of the film does not depend practically on the temperature , as will be shown below . second , @xmath92 values go up in the temperature range of the ferromagnetic - paramagnetic transition in such way that the @xmath97 curve has a maximum at @xmath98 k which is close to @xmath5 and @xmath62 ( see figs . third , the magnitude of @xmath92 goes clearly to zero at the transition to the paramagnetic state . the latter is quite expected since the amr effect is unique to the ferromagnetic state . let us present now a behavior of the ratio of resistances in magnetic fields perpendicular and parallel to the film plane ( we denote these resistances as @xmath99 and @xmath100 ) in the case that both fields are perpendicular to the current ( fig . 10 ) . in this case the amr effect has no influence on mr . it can be seen that the ratio of @xmath99 and @xmath100 is less than unity in low field ( @xmath101 koe ) and more than unity at higher field with a tendency to saturation at high enough fields . this behavior should reflect the magnetization anisotropy behavior . in fig . 11 the temperature dependence of the ratio between mr in parallel and that in perpendicular fields , @xmath102 , is presented at @xmath103 koe ( this field is high enough to saturate the magnetization in any direction , therefore , both the @xmath104 and @xmath105 are negative ) . it is seen that mr anisotropy is high below @xmath106 k. then , with increasing temperature , @xmath102 decreases to some nearly constant value about 1.3 , which persists up to 220 k. it decreases further at higher temperatures , reaching @xmath107 at room temperature , _ i.e. _ near the curie temperature @xmath5 . therefore , this type of mr anisotropy is connected with ferromagnetic state as well . this point will be discussed in more detail below . the behavior of these mr anisotropy effects suggests that the anisotropy is caused not only by the amr effect , but that it also results from the magnetization anisotropy . this anisotropy exists in this film as shown in figs . 2 and 3 . it is obvious from these figures that the film is magnetized more easily in the field direction parallel to the film plane . to check more thoroughly , we have measured @xmath108 in the both directions at different temperatures in the range 4100 k in fields up to 20 koe . we have found that the saturation magnetization @xmath109 essentially does not depend on the temperature in this range ( @xmath109 is lowered only by a few percent after warming from 4 k to 100 k ) . the @xmath108 dependences are observed to be quite different for the field directions parallel and perpendicular to the film plane , but for the same field direction the recorded @xmath108 curves for different temperatures practically merge together . only in the low fields ( @xmath110 koe in the parallel direction and @xmath111 koe in the perpendicular direction ) can differences be found between the low and high temperature behavior . to consider this and other effects more properly , fig . 12 shows @xmath108 dependences for @xmath112 k and 100 k. these two graphs illustrate low temperature ( below 30 k ) and high temperature ( above 50 k ) peculiarities of @xmath108 behavior for this same film . first of all , it should be noted , that experimental points in these graphs present the data recorded for increasing and subsequently for decreasing applied magnetic field . no significant hysteresis in the @xmath108 curves at low temperature and only a rather weak one at high temperature in weak fields can be seen . second , there is a pronounced difference in the @xmath108 curves recorded for parallel and perpendicular directions of the magnetic field , which provides an unquestionable evidence of the magnetization anisotropy . the @xmath108 curves for both field directions are rather extended ( therefore , it can be argued that neither represents the easy magnetization axis ) , but it is , however , clear that , on the whole , the film is magnetized more easily in the parallel field direction . one more difference can be seen from comparing the two graphs in fig . namely , at low temperatures it appears as if the magnetization is nonzero and rather high at zero field for both field directions [ fig . 12 ( a ) ] . for increasing field , the magnetization increases , however , rather slowly . by contrast , at high temperatures the magnetization increases with the field more gradually , beginning from zero , without any peculiarities in low fields [ fig . 12 ( b ) ] . it should be recognized , however , that true zero magnetic field can not be set in the magnetometer . for experimental reasons there is always some stray magnetic field of the order of 1 oe . this weak field is sometimes quite enough to cause a significant magnetization at nominal @xmath113 in the case of low coercivity . the @xmath108 behavior shown in fig . 12 ( a ) suggests that , at low temperatures for both perpendicular and parallel field directions , some parts of the film have a very low coercivity ( this causes the jump - like increase in the magnetization in very low fields ) ; whereas , other parts of the film have a higher coercivity . in other words , it can be argued that some parts of the film have a substantial in - plane magnetization ; whereas , other parts have a substantial out - of - plane magnetization for a magnetic field which is very close to zero . this is an evidence that the demagnetization energy at low temperatures can not overcome entirely the spontaneous ( parallel and perpendicular ) magnetization in some ( different ) parts of the film . this effect is not pronounced at higher temperatures ( @xmath114 k ) [ fig . 12 ( b ) ] . it should be noted , that some features of the magnetic - anisotropy behavior of this film correlate well with its mr behavior . it follows from fig . 11 , that the absolute values of negative mr in parallel field are greater than those of in the perpendicular field ( @xmath115 ) . since the conductivity of manganites increases with an enhancement of the magnetic order , this behavior just reflects the point that the magnetization increases more easily in a magnetic field parallel to the film plane . this mr anisotropy is connected with the ferromagnetic state . for this reason it disappears when @xmath116 approaches @xmath5 ( fig . in this section we will discuss the different sources of mr anisotropy in ferromagnets ( fm ) and their possible effects in this film . we will not consider here the influence of ballistic mechanisms of the mr and mr anisotropy in fm @xcite , which are connected with the curving of electron trajectories in a magnetic field . these are important only if the electron mean - free path is fairly large , which is not the case in rather resistive manganites . the amr effect which is an intrinsic source of mr anisotropy in any fm will be considered as the first point . this effect plays a crucial part in the mr anisotropy of the films studied . as the last but not least point , the extrinsic or induced sources of mr anisotropy which are caused by the shape and strain state of the fm will be thoroughly discussed . these are especially important for films and small particles . it will be shown that the rather complicated mr anisotropy behavior found in this film can be explained by the concurrent influence of these intrinsic and extrinsic sources of mr anisotropy . this effect in ferromagnets is thought to be caused by the spin - orbit interaction ( see refs . and references therein ) . the known theoretical models are related to 3@xmath117 metals such as ni , co , fe and its alloys . some attempts to apply the similar model concepts to manganites were made in ref . . it can be said at the moment , however , that mechanisms for amr in manganites are not clearly understood . from the other side , the essential features of this effect in manganites are already established rather well . the temperature dependence of the amr parameter @xmath97 recorded in this study ( fig . 9 ) corresponds well to previous results@xcite . among other factors , the most important features of @xmath97 behavior such as the constancy the magnitude of @xmath97 at low temperatures ( @xmath118 ) , the maximum in @xmath97 at @xmath119 , and the approach to zero of @xmath97 at @xmath120 , correspond to the above - mentioned results . there is no clear notion among scientists at present as to which factors determine the magnitude of the amr effect in manganites . most of the authors usually refer to behavior of 3@xmath121 metals and the corresponding theories developed for these metals . but this approach does not appear to be very fruitful . indeed , it follows from the 3@xmath121-metal models@xcite that the amr parameter @xmath92 should depend somehow on the magnetization . really , the amr effect takes place only in ferromagnets with a spontaneous magnetization . after the transition to the paragmanetic state , and the disappearence of the spontaneous magnetization , @xmath92 goes to zero . but no clear correlation between the amr amplitude and the saturation magnetization was found for 3@xmath121 metals@xcite . thus this type of general explanation about the influence of magnetization is not very productive for understanding the amr effect in manganites . let us consider an other example . it was experimentally found for 3@xmath121 metals ( with corresponding theoretical support)@xcite that @xmath92 is proportional to @xmath122 on approaching the curie temperature from below . this means a linear decrease in @xmath92 to zero when @xmath116 approaches @xmath5 . but this is clearly not the case for manganites where @xmath97 has a maximal amplitude near @xmath5 ( see fig . 9 of this paper and corresponding figures in refs . ) . therefore , the amr behaviors of manganites and 3@xmath121 metals are drastically different and must be governed by different mechanisms . upon a closer view of results of this study together with these of ref . a clear correlation between the magnitudes of @xmath92 and mr in manganites is revealed . this correlation is rather apparent , but , surprisingly , was never mentioned in preceding papers . indeed , the temperature dependence @xmath97 is entirely analogous to the mr temperature dependence ( see insert in fig . 1 or similar graphs in the numerous cmr papers , for example , in refs . ) . the mr in the cmr manganites has some minimal magnitude at @xmath118 , goes to maximal value near @xmath5 , and approaches zero at @xmath123 in the same manner as the amr parameter @xmath97 ( fig . it should be noted that this type of temperature behavior of @xmath97 and mr is a feature of manganite samples with fairly good crystal perfection only . in disordered doped manganites ( for example , in polycrystalline or granular samples ) , the mr can rise with decreasing temperature@xcite . in that case , as shown in ref . , the parameter @xmath92 increases with decreasing temperature as well . as was mentioned above , the resistance @xmath21 of manganites in the ferromagnetic state is a function of magnetization . the conductivity increases with enhancement of ferromagnetic order . this is the source of the huge negative mr in manganites . the mr magnitude is determined by the ability of an external magnetic field to increase the magnetization . it is obvious that in good crystals at low temperature ( @xmath118 ) , when nearly all spins are already aligned by the exchange interaction , the ability to increase the magnetization in the magnetic field is minimal . for increasing temperature and , especially , at temperatures close to the curie temperature @xmath5 , the magnetic order becomes weaker ( the magnetization goes down ) due to thermal fluctuations . in this case the possibility to strengthen the magnetic order with an external magnetic field increases profoundly . this is the reason for maximal mr magnitude near @xmath5 . at last , above @xmath5 , the spin arrangement becomes essentially random , the magnetization is zero , and , the mr is close to zero as well . from the aforesaid , it might be assumed that the magnitude of the amr effect in cmr manganites is determined by the same factors , as that of the mr , namely , by the possibility to increase the magnetic order by an applied magnetic field . this feature must be important for further determination of the nature of the amr effect in cmr manganites . one further comment must be added . to see the amr effect properly , the magnetization should be parallel to the applied magnetic field . that is , the magnitude of the field must be high enough to saturate or rotate the magnetization to the selected direction . in other words , the applied field should be sufficient to overcome the different anisotropy energies of the film . actually , the @xmath95 behavior should reflect somehow the magnetization curve @xmath108 in the direction perpendicular to the film plane , and , hence , reflect the influence of the available anisotropy sources . as indicated above in sec . [ magani ] , the recorded @xmath108 curves at low temperature ( @xmath118 ) [ fig . 12 ( a ) ] indicate that some parts of the film have a substantial in - plane spontaneous magnetization , while other parts have a substantial out - of - plane one in a magnetic field which is very close to zero . we believe that the inflection in the @xmath95 dependence ( fig . 8) at weak fields reflects this magnetization inhomogeneity in this film . it is known that fm consists of magnetic domains which are regions of spontaneous magnetization . therefore , even in the absence of an external magnetic field the electrons in fm feel the internal magnetic field @xmath124 . for this reason , any measured resistance for a fm is , in fact , some measure of its mr as well . apart from the influence of the intrinsic field , there are additional specific mechanisms of electron scattering in fm metals . let us call them the `` magnetic '' mechanisms of electron scattering . these can give a considerable contribution to the resistivity and mr of fm metals@xcite . in the matthiessen - rule approximation it is possible to write down for the resistivity of fm metal @xmath125 where @xmath126 is the `` non - magnetic '' part of the resistivity , which is stemming from the usual electron - scattering mechanisms common to non - magnetic metals ( scattering on impurities , phonons and so on ) , and @xmath127 represents the `` magnetic '' part of the resistivity . it has been long inferred that the behavior of @xmath127 is some reflection of the temperature and magnetic field dependences of the magnetization . with a rise of magnetization at the transition into fm state , @xmath128 drops sharply . in the fm state the `` magnetic '' resistivity is quite small and , in the limit , goes to zero or to very low values for ideal magnetic order . the external magnetic field enhances the magnetic order , that leads to a decrease of resistivity . that is why fm metals are characterized by a negative mr . of course , at finite temperatures there are some thermal disturbances of the long - range magnetic order ( spin waves or magnons ) which can determine the power - law temperature dependence of @xmath128 at low temperatures : @xmath129 , where @xmath130 value depends on the specific mechanism of disturbance . the known behavior of @xmath128 in fm metals@xcite indicates , therefore , that the influence of magnetic order ( or a magnetic lattice of spins ) on electron transport is quite similar to that of the crystal lattice order . if the crystal lattice is ideal , the resistance is zero . the same is true ( at least , to some significant degree ) in respect to magnetic order : for an ideal spin alignment the `` magnetic '' resistivity may be thought to be equal to zero . any deviations from long - range magnetic order lead to electron scattering . just as for crystals , disorder may be static or dynamic disturbances ( defects ) in the spin lattice . it follows from all this that the `` magnetic '' part of the resistivity , @xmath128 , is a direct function of the magnetization , that is @xmath131 , \eqnum{3}\ ] ] the relevant experimental dependences ( and its theoretical justifications ) for mangnites can be found in refs . . lastly , the thermodynamic fluctuations of magnetic order should be mentioned . they are especially strong near the curie temperature , and , therefore , behavior of @xmath132 and sometimes that of resistivity ( as in the case of manganites ) has peculiarities at @xmath133 . it should be noted in this place that the `` magnetic '' and `` non - magnetic '' contributions to the resistivity can not be considered as entirely independent . the lattice defects and deviations from magnetic order can be coupled rather strongly and be interdependent . for example , crystal lattice defects , such as grain boundaries , surface regions of film and others , induce disturbances in the magnetic order as well . on the other hand , the changes in the magnetic order , such as development of spontaneous magnetization at the paramagnetic - ferromagnetic transition or moving and disappearence of domain walls in an external magnetic field , can cause a response of the crystal lattice ( for example , changes in the elastic stresses and strains ) . in explanation of the results of this study it is helpful , as a first step , to consider and keep in mind some known simple cases of the mr anisotropy in manganite films . an instructive example can be found in ref . for films la@xmath134ca@xmath135mno@xmath2 , grown by molecular beam epitaxy on srtio@xmath25 substrates . the films were between 50 and 150 nm thick . the authors of ref . have studied @xmath136 dependences for the cases where the magnetic field was applied parallel or perpendicular to the film plane . for the perpendicular field direction a positive mr was observed at low fields , which changed to negative one at higher fields . the in - plane mr was only negative and depended on the angle , @xmath7 , between transport current and field according to eq . ( 1 ) , which has been attributed to the amr effect@xcite . the experimental dependences of @xmath108 revealed anisotropy , which is favorable for the in - plane magnetization . similar results were reported also for pr@xmath137sr@xmath138mno@xmath2 films@xcite deposited on srtio@xmath25 substrates . at first glance a positive mr in manganites appears to be quite impossible . really , the external magnetic field can only strengthen the long - range magnetic order , and , therefore , should lead to decreasing resistance . nevertheless , it turns out that a concurrence of surface ( or shape ) anisotropy and the amr effect can cause the positive mr in a perpendicular field . a comprehensive explanation of this effect can be found in ref . , so we will restrict ourself only to the main points , which are necessary to understand the observations of this study . an essential prerequisite is that the mr of manganites be determined by the dependence of the magnetization on magnetic field . assume now that field @xmath139 is applied perpendicular to the film plane . at @xmath140 the magnetization vector @xmath141 has an in - plane orientation ( due to the influence of surface anisotropy and the demagnetization energy ) . at low values of @xmath139 the applied field is actually perpendicular to @xmath141 . for increasing @xmath139 the magnetization begins to rotate so that a component @xmath142 appears which is perpendicular to the film plane@xcite . as this proceeds the absolute magnitude of the magnetization remains unchanged up to the moment when @xmath139 reaches some field @xmath143 at which rotational saturation of the magnetization of the film in perpendicular direction takes place . the constancy of the absolute value of the magnetization during the rotation means that the `` magnetic '' part of resistivity , which depends on @xmath22 , remains constant during the rotation as well . in this case it should be expected that the mr would be zero up to the field @xmath143 . above this field a further increase in @xmath139 results in increasing @xmath22 and , therefore , in a decreasing resistivity . however , instead of this , a positive mr was observed in low fields , and the negative one in higher fields . what is the reason for this behavior ? the point is that the mr is affected also by the amr effect ( that is , by dependence of mr on the angle between the current and magnetization ) . at @xmath140 the magnetization vector lies in the film plane . in the explanation of ref . it was implied that the current at @xmath140 is parallel to the magnetization . maybe this is not rigorously correct , but it is not so important , if an effective angle between @xmath141 and @xmath144 is fairly small . for increasing @xmath139 , the magnetization vector rotates , that is the angle between @xmath141 and the current increases . this leads to the resistivity rise [ see eq . ( 1 ) ] and is the cause of the positive mr . the magnetization @xmath141 becomes perpendicular to the film plane at @xmath145 . at @xmath146 the magnetization begins to increase and this leads to the resistance decrease . as a result of this kind of concurrence the dependence @xmath136 with a maximum takes place . in the parallel field the influence of amr effect can be thought as absent , therefore only the decrease in resistance in magnetic field inherent for cmr manganites is observed . the results presented in this paper are quite different from these of refs . . first of all , in the range of low magnetic field ( figs . 46 ) , for a perpendicular field , @xmath82 , we found the negative mr in low field , before going to positive mr at higher fields ( fig . 4 ) . at the same time , in a parallel field , @xmath83 , a positive mr is seen before becoming negative at higher fields ( fig . this behavior is quite challenging and puzzling . one of the most reasonable explanations for this is an inhomogeneous strain state of the film , that leads ( due to the magnetoelastic interaction ) to a difference in the magnetic properties ( for example , in directions of easy magnetization ) in different parts of the film . this turns out to be a basis for understanding of results of this study . the sources of internal strains and stresses in pld films are quite common and sometimes inevitable due to the lattice film - substrate mismatch . it is just these strains which can be the primary source of inhomogenenous magnetic state of the film studied . it follows from the known studies@xcite that magnetostriction in manganite films ( at least with composition la - ca - mn - o and la - sr - mn - o ) is positive . in this case , the magnetization orients parallel to the tensile stresses and perpendicular to the compressive stresses . for fairly smooth substrates , which make possible coherent epitaxy , the in - plane film strain depends on the lattice film - substrate mismatch . if the mismatch is not too large , it can be expected that the in - plane lattice parameters will match those of a substrate and the out - of - plane parameter will be elastically modified according to the poisson ratio . in this case , a biaxial strain is induced in the film plane which can be tensile or compressive depending on the ratio the substrate lattice parameters and those of the bulk target from which the film is deposited . this is true up to some critical thickness , above which misfit dislocations appear at the film - substrate interface to relax the strain . in that case , film lattice parameters become closer to those of bulk sample . the above - described scenario is just a generally accepted model which allows some crude predictions to be made . xrd and other studies are required for any specific film - substrate system to know exactly its strain state . such studies have been done in some studies of the mr anisotropy in doped manganite films . for example , it was found that the la@xmath134ca@xmath135mno@xmath2 films on srtio@xmath25 substrate studied in ref . have in - plane tensile strain@xcite , that causes the in - plane magnetization and the appearence of positive mr in the field perpendicular to the film plane . in that case the perfect matching of the in - plane lattice parameters of film and substrate was found . a different example ( compressive strain pr@xmath137sr@xmath138mno@xmath2 films ) is described in ref . . in this case the compressive strains cause the easy magnetization axis to be perpendicular to the film plane . it looks like that in the films , studied in refs . , the strains , which are induced by the film - substrate interaction , were extended over the most of the film thickness . consequently , the films in those studies can be considered as nearly ( or to a great extent ) homogeneous . this is especially true in respect to ref . where the film thicknesses were between 7.5 and 15 nm . for the la@xmath0ca@xmath1mno@xmath2 ( @xmath3 ) film in this study , grown on laalo@xmath25 , the substrate lattice parameters are less than those of the corresponding bulk sample . therefore , a compressive biaxial strain in the film plane , and a corresponding tensile uniaxial strain in direction perpendicular to the film plane should be expected . this strain state has been actually observed ( see sec . [ xray ] ) . a perfect matching of the in - plane lattice parameters of film and substrate was not found , which is in agreement with the study of la - ca - mn - o films on laalo@xmath25 in ref . . this is because of partial relaxation of the strain emposed by the substrate . additionally , the xrd data indicate that the crystall structure of the film is inhomogenenous . it consists of the regions with different crystallographic orientations ( see sec . [ xray ] ) . the origin of the inhomogeneous structure is probably connected with the twin structure of the substrate . as a result , in both the in - plane and out - of - plane directions tensile and compressive regions can be found . the inhomogeneous film structure reveals itself in the behavior of @xmath108 curves [ see ( fig . 12 ) and the discussion in sec . [ magani ] ] as well as in the magnetic field dependence of the amr parameter ( fig . 8) ( in the last case we mean the inflection in the @xmath95 curve which should not occur for a homogeneous system ) . the film inhomogeneity is a key to understanding the observed mr anisotropy ( fig . 4 and 5 ) as well . it can be assumed that the film is some mixture of regions with different strains . the size of the regions ( according to the xrd data ) is comparable with the film thickness . the regions with the in - plane compressive strains favor the out - of - plane magnetization in zero field ; whereas , the ones with the in - plane tensile strains favor the in - plane magnetization . in the following text we can conveniently speak about the `` compressive '' and `` tensile '' parts of the film . in spite of the fact that some part of the film is prone to the out - of - plane magnetization , the influence of the shape anisotropy may cause the total film magnetization to be in the film plane in zero magnetic field or to be canted , as was found , for example , in la - sr - mn - o films under compressive stress@xcite . in any case , however , it is possible to understand the appearence of a negative mr in low perpendicular fields , @xmath82 , ( fig . 4 ) for an inhomogenenous film of this type . the `` compressive '' part of the film can be magnetized in the perpendicular direction at very low magnitude of @xmath82 up to the rotation saturation value , and after this the absolute value of magnetization begins to increase with increasing magnetic field . this leads to a resistance decrease , that is , to a negative mr . at the same time , the magnetization of the `` tensile '' part of the film is in the film plane at zero field and can not be rotated so easily in the field direction . considerably higher fields are needed for it . therefore , at higher fields the explanantion given in ref . , which takes into accont the amr effect , is quite applicable to justify the positive mr at higher fields . all these combined effects can produce the observed rather complicated @xmath147 dependence ( fig . 4 ) . let us turn now to fig . 5 which represents the behavior of the resistance in a magnetic field , @xmath83 , parallel to the plane of the film . in this case the resistance goes up initially for increasing field , but then goes down in higher field . that is , the change in the mr sign ( from positive to negative ) takes place . to understand this , it should be recalled that both , @xmath147 and @xmath148 , dependencies ( figs . 4 and 5 ) were registered in magnetic fields perpendicular to the current . at low magnitudes of @xmath83 , however , the film magnetization is definitely not perpendicular to the current . at least in the `` tensile '' regions of the film with the magnetization easy axis parallel to the film plane , the magnetization is by no means perpendicular to the current . the vectors @xmath141 in these regions should have some spread in directions due to the demagnetization energy . only for increasing @xmath83 do these vectors become strictly perpendicular to the current . since mr is maximal when the magnetization is perpendicular to the current due to the amr effect ( fig . 7 ) , the in - plane rotation of magnetization in low fields @xmath83 leads to an increase in resistance . this explains the positive mr for small parallel fields . after aligning the spins in the `` tensile '' regions parallel to magnetic field , the magnetization begin to increase which causes as usual the resistance decrease . it should be mentioned that a similar effect was observed in ref . for a la@xmath137sr@xmath138mno@xmath25 film for magnetic fields applied in the film plane . they found a positive mr for @xmath149 as opposed to the negative one with @xmath150 . @xmath136 curves were measured , however , only in low fields ( @xmath151 koe ) , which thus excludes a comparison with the data of this paper ( registered in fields up to about 20 koe ) . in fields parallel to both , the current @xmath88 and the film plane , only the negative mr is found in this study ( fig . this is quite expected , since there are no mechanisms for positive mr in this case . the mr behavior in the pld manganite films can differ dramatically from that of bulk samples . this is due to a film - substrate interaction , which determines the structural and magnetic state of the films . some film - substrate combinations can lead to rather complicated and puzzling mr behavior for different directions of the magnetic field relative to the film plane and the transport current . to understand such cases properly , one needs to have enough data about the structural and magnetic properties of the films . the mr and mr anisotropy of such film systems depends on the existence of preferential directions of magnetization ( due to the strains arising from the lattice film - substrate mismatch and other sources ) , and from the amr effect . we have presented in this paper an example of complicated behavior of the low - field mr and mr anisotropy for la@xmath0ca@xmath1mno@xmath2 films on laalo@xmath2 substrates , and demonstrated how it can be understood . based on the results of this study together with the known results of other authors , we have indicated fairly conlusively ( and for the first time ) that a clear correlation exists between the magnitudes of the amr effect and mr in manganites . this suggests that the amr effect in manganites is determined by the ability of the magnetization to increase in an external magnetic field . this important correlation can be helpful in further disclosing the nature of the amr effect in cmr manganites . the authors are indebted to dr . n. v. dalakova for help in measuring @xmath57 dependence in zero magnetic field and dr . j. h. ross for his efforts in measuring of film thickness by afm . support at tamu was provided by the robert a. welch foundation ( grant a-0514 ) and thecb arp 010366 - 003 . bib and dgn acknowledge support by nato scientific division ( collaborative research grant no . 972112 ) . s. r. surthi , s. bhat , r. k. pandey , k. d. d. rathnayaka , a. parasiris , a. c. du mar and d. g. naugle , in _ integrated thin films and applications _ ( ceramic transactions , vol . 86 ) , edited by r. k. pandey , david e. witter , usha varshney ( the american ceramic society , westerville , ohio , 1998 ) pp . 109 - 118 . in ref . we have estimated the film thickness to be @xmath152 nm . after that we have measured the film thickness with afm and find out the mean thickness is about 80 nm , but the surface is rather rough , so the highest points can reach about 100 nm . r. b. praus , g. m. gross , f. s. razavi , and h .- u . habermeier , j. magn . mater * 211 , * 41 ( 2000 ) ; r. laiho , k. g. lisunov , e. lhderanta , p. petrenko , v. n. stamov , and v. s. zakhvalinskii , _ ibid . _ * 213 , * 271 ( 2000 ) . r. shreekala , m. rajeswari , r. c. srivastava , k. ghosh , a. goyal , v. v. srinivasu , s. e. lofland , s. m. bhagat , m. downes , r. p. sarma , s. b. ogale , r. l. greene , r. ramesh , t. venkatesan , r. a. rao , and c. b. eom , appl . * 74 , * 1886 ( 1999 ) .

we present a study of anisotropy of transport and magnetic properties in a la@xmath0ca@xmath1mno@xmath2 ( @xmath3 ) film prepared by pulsed - laser deposition onto a laalo@xmath2 substrate . we found a non - monotonic dependence of magnetoresistance ( mr ) on magnetic field @xmath4 for both @xmath4 perpendicular and parallel to the film plane but perpendicular to the current . in the longitudinal geometry ( when @xmath4 is parallel to both the current and the film plane ) the mr was negative at all fields below 20 koe , as expected for colossal - magnetoresistance manganites . this rather complex behavior of mr manifests itself at rather low temperatures , far below the curie temperature @xmath5 , which was close to room temperature . two main sources of mr anisotropy in the film have been considered in the explanation of the results : ( 1 ) the existence of preferential directions of magnetization ( due to strains stemming from the lattice film - substrate mismatch or other reasons ) ; ( 2 ) dependence of resistance on the angle between current and the magnetization , which is inherent in ferromagnets . the transport and magnetic properties of the film correspond well to this view . in particular , the following angle dependence of mr is found : @xmath6 ( where @xmath7 is the angle between the field and current directions in the plane normal to the film but parallel to the current ) . the temperature and magnetic field dependences of @xmath8 were recorded and analyzed . a clear magnetization anisotropy , that generally favors the magnetization in the film plane is also found . at the same time the recorded magnetization curves ( as well as the mr data ) indicate , that the film crystal structure should be inhomogeneous in such a way that various parts of the films have non - identical magnetic properties ( with different directions of spontaneous magnetization ) . this hypothesis is supported by x - ray diffraction which revealed that the film is inhomogeneous in strain , lattice parameter and lattice orientation . this peculiar macroscopic - scale disorder is caused by a film - substrate interaction . the possible reasons for formation of such structure and its effect on mr anisotropy are considered . # 1@xmath9#1 2

the ferromagnetic ( fm ) to paramagnetic ( pm ) transition temperature of the elemental ferromagnet ni is known to be suppressible to absolute zero on alloying it with a critical concentration ( @xmath4 ) of a non - magnetic @xmath7-element , like pd,@xcite pt,@xcite rh,@xcite or v.@xcite the ground state of such an alloy system , thus , undergoes an fm - pm quantum phase transition ( qpt ) across @xmath4 , known as the quantum critical point ( qcp ) . among the ni binary alloys , ni@xmath1pd@xmath0 and ni@xmath0v@xmath1 have experimentally been shown to undergo a qpt.@xcite a qpt is driven by quantum fluctuations with an energy scale e@xmath8 , which compete with thermal fluctuations of energy scale @xmath9 t at finite temperatures , and dominate the system s properties over the latter for e@xmath10t.@xcite as a consequence , the system exhibits unconventional physical behaviour , like a non - fermi liquid ( nfl ) phase , characterized by non - universal power - law temperature dependences of physical observables , around qcp.@xcite in case of ni@xmath1pd@xmath0 , the compositional disorder at the qcp ( @xmath11 ) is small , and hence the nfl behaviour is observable in this system.@xcite the ni@xmath0v@xmath1 system , however , is associated with a considerable compositional disorder at the qcp ( @xmath12)@xcite because of the larger @xmath4 . the non - universal behaviour in such disordered systems is not limited just to the qcp region ; rather , it extends in a finite temperature range , identifiable as a quantum griffiths phase ( qgp ) , on the paramagnetic side of the transition.@xcite at very low temperatures on the paramagnetic side , however , the non - universal behaviour is masked by the appearance of a cluster glass phase . fundamentally , a qpt in metals is proposed to be associated with a qualitative change in the fermi surface ( fs ) in the vicinity of the qcp.@xcite this proposition can have important consequences in case of nanomaterials , wherein the quantum confinement effects lead to properties different from their bulk counterparts . further , the fs of a nanoparticle is also supposedly different from the corresponding bulk fs,@xcite and may modify the quantum critical behaviour . this had led the authors earlier to investigate the occurrence of qpt in ni@xmath1pd@xmath0 nanoalloys,@xcite wherein the nanoalloys were found to exhibit a qpt , in spite of not showing any nfl behaviour . along the same line , it is quite intriguing to investigate whether the qgp , the characteristic feature of the ni@xmath0v@xmath1 bulk alloys , appears also in the nanoparticles of this alloy system , although the magnetic phase diagram of a nanoparticle system , which may include superparamgnetic ( spm ) , blocked fm , spin - glass , etc . phases , is more complex than the corresponding bulk,@xcite and may render the investigation difficult . ni nanoparticles@xcite and ni - v alloy microparticles@xcite have earlier been synthesized and shown to possess fm,@xcite spm,@xcite photocatalytic@xcite and h - storage properties.@xcite nanoparticles of ni - v alloys , on the other hand , have hitherto not been synthesized or studied for magnetic properties , or even for any application , to the best of the authors knowledge . so , any kind of investigation , including the exploration of any signature of a qgp , on this nanoalloy system requires , as a pre - step , finding a method to prepare these nanoalloys . in this work , we aim at exploring the signature of qgp in ni@xmath0v@xmath1 nanoalloy system . for preparation of the nanoalloys , the chemical reflux method used to synthesize ni@xmath1pd@xmath0 nanoalloys in our previous work,@xcite but with an appropriately modified set of chemicals , was adopted . nanoparticles of ni , v and ni@xmath0v@xmath1 , with @xmath3 in the vicinity of @xmath4 , were prepared this way for the investigations . after determining the sizes , phases and compositions by different microscopic and spectroscopic techniques , the existence of qgp was explored using dc magnetization and electrical resistivity measurements . the nanoparticles of ni , v and ni@xmath0v@xmath1 ( 0.05 @xmath13 0.20 ) were synthesized basically by reduction of metal precursor salts vanadium ( iii ) chloride ( vcl@xmath14.h@xmath15o ) and nickel ( ii ) chloride ( nicl@xmath16 ) , either separately for the elemental cases or simultaneously with appropriate stoichiometry for the nanoalloys , by hydrazine hydrate in the presence of the surfactant diethanolamine in a conventional reflux apparatus.@xcite in the cases of elemental nanoparticles , typically 0.5 mmol of vcl@xmath14 ( nicl@xmath16 ) was dissolved in 30 ml distilled water to yield complexes of v@xmath17 ( ni@xmath17 ) ions in the solution ; for nanoalloys , proportionately appropriate amounts of the two salts were dissolved sequentially in distilled water . subsequently , 5 ml of diethanolamine was added as a surfactant to the above solution , followed by 6 ml of hydrazine hydrate as the common reducing agent . finally , 40 ml distilled water was added to this , and the resulting solution was refluxed for 8 h at 110 @xmath18c in an oil bath . the black - colored precipitate , i.e. , the alloy , was then washed with warm distilled water , centrifuged at 3500 rpm and dried in vacuum for 48 h. the morphologies of the nanoalloys were investigated using ( i ) a zeiss supra 40 field - emission scanning electron microscope and ( ii ) a jeol jem-2100 high resolution transmission electron microscope operated at 200 kv . a drop of the colloidal nanoparticles , pre - sonicated in acetone , was placed on a small quartz peace to prepare the sample for field - emission scanning electron microscopy ( fesem ) ; the drops were placed on a carbon supported cu transmission electron microscope grid for high resolution transmission electron microscopy ( hrtem ) and selected area electron diffraction ( saed ) . energy dispersive x - ray analyses ( edax ) of the nanoalloys were performed using a jeol scanning electron microscope to determine the final synthesized composition @xmath19 . the phases were determined by x - ray diffraction ( xrd ) on a philips xpert mrd system using cu k@xmath20 radiation operated at 45 kv and 40 ma . the stoichiometries of the samples were further studied using x - ray photoelectron spectroscopy ( xps ) . xps spectra were recorded on a phi 5000 versaprobe ii system using a micro - focused monochromatic al @xmath21 source ( @xmath22 = 1486.6 ev ) , a hemispherical analyser , and a multichannel detector . charge neutralization in each measurement was achieved using a combination of low energy ar@xmath23 ions and electrons . the binding energy scale was charge referenced to c 1s peak at 284.5 ev . high - resolution xps spectra were acquired at 58.7 ev analyzer pass energy in steps of 0.25 ev . further , the temperature dependences of sample resistivities in the temperature range 15 k - 300 k were measured on pelletized nanoalloys by four - probe technique using a lakeshore resistivity 7500 set - up with the help of a nanovoltmeter as a current source . finally , dc magnetizations versus temperature ( 5 k @xmath24 t @xmath24 300 k ) at 500 oe and versus field ( -5 t @xmath24 h @xmath24 5 t ) at 2 k were measured using either a cryogen - free 9 t cryogenic physical property measurement system or a quantum design mpms squid vsm evercool system . determined from edax and the initial composition @xmath25,scaledwidth=50.0% ] the variation of edax determined composition @xmath19 with the initial composition @xmath25 is plotted in fig . as is clearly evident from the figure , @xmath19 shows a linear variation ( @xmath19 = 0.0034 + 0.997 @xmath25 ) with @xmath25 and confirms that the stoichiometries taken during the syntheses are essentially the same as in the finally synthesized samples . henceforth , the value of @xmath3 in ni@xmath0v@xmath1 will be taken as @xmath19 . figure 2 shows the fesem images , along with the corresponding size distributions , of pure ni and ni@xmath0v@xmath1 alloy samples . as can be seen from the figure , the particle sizes range between 18@xmath266 nm to 33@xmath2612 nm for the various samples . to analyze the particles further , hrtem images and corresponding saed patterns were taken for three representative samples with @xmath3 = 0 ( pure ni ) , 0.098 and 0.11 . the images and patterns are shown in fig . the clearly visible nanoparticle agglomeration , which is not present in the corresponding fesem images , indicates that the particles are magnetic.@xcite the occurrence of dots and concentric rings in each saed pattern further elucidates that the particles are crystalline in nature . figure 4(a ) shows the xrd patterns of all the ni@xmath0v@xmath1 ( 0 @xmath24 x @xmath24 1 ) samples studied . the pattern for the pure ni nanoparticles consists of three clean peaks at 44.50@xmath18 , 51.84@xmath18 and 76.36@xmath18 . according to the joint committee on powder diffraction standards ( jcpds ) data , these peaks correspond to ( 111 ) , ( 200 ) and ( 222 ) reflections of fcc ni , but relatively displaced towards 2@xmath27 values higher than the corresponding bulk . this observation reveals that the ni nanoparticles are pure in phase and are of a lattice constant ( 3.499 ) smaller than the corresponding bulk ( 3.524 ) value . such a reduction of lattice constants in nanoparticles has earlier been predicted and demonstrated.@xcite the addition of v atoms to ni up to @xmath3 = 0.17 does not alter the three - peak structure , except that these peaks progressively shift towards lower 2@xmath27 values . further , essentially no additional peak(s ) appear on v incorporation . this suggests that the alloy nanoparticles also are in the fcc phase and hence are in the form of ni - v solid solutions . the alloy lattice constants @xmath28 , as determined from the ( 111 ) peak positions , are plotted as a function of @xmath3 in fig . 4(b ) , and seem to vary linearly with @xmath3 . a linear fit yields @xmath28 = 3.4964 + 0.1898 @xmath3 . assuming a close - packed accommodation of the impurities , and hence that @xmath28 is proportional to the weighted average of the atomic radii @xmath29 and @xmath30 of ni and v , respectively , the ratio @xmath29/@xmath30 from the fitting parameters comes out to be 1.054 . this value is close to 1.033 , the covalent radius ratio for ni and v , as available easily on the internet . this once again confirms the above inference that the ni - v alloys are basically ni - v solid solutions . notably , the solid solubility of v ( 17 @xmath31 ) in the ni nanoparticles is more than the solubility limit ( @xmath24 14 @xmath31 ) for the bulk according to the ni - v phase diagram.@xcite solid solubilities in nanophase have earlier also been reported to be enhanced with respect to the bulk,@xcite and favour our results . the highly oxidized v nanoparticles , as evidenced by the dominant oxide peaks at 36.06@xmath18 corresponding to v@xmath15o@xmath14 ( 111 ) and at 51.06@xmath18 corresponding to v@xmath15o@xmath32 ( 200 ) , were not studied further anyway , but helped confirm the absence of any oxide of vanadium in all other samples . the survey xps spectra of all the studied samples are shown in fig . the presence only of ni and v peaks , apart from the adventitious c 1@xmath33 peak at 284.5 ev and one o ( 1@xmath33 ) peak at 531 ev , shown later not to participate in any oxide formation except in the pure v nanoparticle case , corroborates the xrd results on the high purity of the pure ni and alloy nanoparticles . further , the high - resolution xps spectra in ni and v binding energy ( be ) regions are shown in figs . 5(b ) and 5(c ) , respectively . the non - deconvolutable single spin - orbit split peaks in each case confirm the absence of any oxide in the samples . further , the peaks shift towards higher be with respect to the pure ni nanoparticle values on increasing @xmath3 in both the sets . the peak shifts ( @xmath34 ) are then plotted as a function of @xmath3 in fig . 5(d ) . both the ni and v peaks can be seen to vary monotonically and essentially concurrently with @xmath3 , confirming the ni - v alloy formation with different v concentrations , as also reported earlier.@xcite the residual resistivity ratio ( rrr ) , defined as rrr = [ @xmath35 ( t ) - @xmath35 ( 15 k)]/@xmath35 ( 15 k ) , has been plotted as a function of temperature for four representative samples in fig . 6 ; @xmath35 ( t ) here is the resistivity at temperature t. from the figure , it can be seen that apart from the monotonic increase of resistivities confirming the metallic nature of the alloy nanoparticles and ruling out their oxidation , each curve shows a peak in 40 k - 60 k temperature range . this peak is indicative of the presence of a small nanoparticle volume with uncorrelated pm - like spins , which start getting gradually aligned in the field direction at this temperature and on lowering the temperature further , in line with the observations reported earlier.@xcite = 0.000 , 0.085,0.098 , and 0.11.,scaledwidth=45.0% ] the zero - field cooled ( zfc ) and field cooled ( fc ) dc magnetization versus temperature curves in the range 5 k @xmath24 t @xmath24 300 k are plotted in fig . 7 for all the samples . the curves are suitably shifted to coincide roughly at the same point at 300 k. the zfc - fc splitting in all the cases is characteristic of spm nature of nanoparticles,@xcite and indicates that the nanoparticles with v concentration as high as 17 @xmath31 are all magnetic . this is in agreement with the hrtem observation of particle agglomeration as pointed out above . on the other hand , the fact that the ni@xmath36v@xmath37 nanoparticles are also magnetic , is in discordance with the pm bulk behaviour of the alloy of this composition.@xcite this discordance , however , is acceptable since it is well known that in many cases pm materials transform to magnetic phases in nanodimensions . despite the zfc peak in every case being broad because of the wide particle size distribution@xcite as found in fesem images , the blocking temperature t@xmath38 can still be determined by estimating the inflection point of the dm / dt versus t curve ( see fig . below t@xmath38 , the nanoparticles are in a blocked fm state , which is also signalled by a kind of saturation of fc magnetization in this region.@xcite t@xmath38 can be plotted as a function of @xmath3 to separate the spm and blocked fm phases above and below it , respectively . a striking feature of both the zfc and fc curves for each composition is the occurrence of a pm - like increase in magnetization below an @xmath3-dependent temperature t@xmath39 ( see fig . 7 ) . this temperature can be estimated in the following manner : first , the initial part of the fc curve below t@xmath38 is extrapolated to the lowest measured temperature , and then the extrapolated curve is subtracted from the original fc curve . the difference would start rising up from zero at t@xmath39 when going down in temperature , as shown schematically in fig . incidentally , the t@xmath39 estimated in this manner is quite close to the peak position in the corresponding resistivity curve . figure 8(b ) displays a relation between the two temperatures . certainly , this temperature rise below t@xmath39 is indicative of the emergence of a pm - like phase inside the blocked fm nanoparticles . it is this phenomenon which had resulted in the occurrence of the peak in the corresponding resistivity curve ( fig . 6 ) . for an estimation , the t@xmath39 and the corresponding resistivity peak position can be averaged to get a modified t@xmath39 , which can then be plotted as a function of @xmath3 to draw a boundary in the phase diagram below which the pm - like phase coexists with the blocked fm phase . the phase diagram estimated this way is drawn in fig . this coexistence of the two phases at lower temperatures is also supported by the presence of a small hysteresis loop along with an unsaturated magnetization in the m - h curve of each sample at 2 k , as shown in fig . although the inference above on the coexistence of the blocked fm and pm - like phases below t@xmath39 is enough as an interpretation to the limited amount of data presented in this work , it would perhaps still be inequitable to ignore examining the shapes of the m - t curves below t@xmath39 further to some extent . for this purpose , it would be sufficient to investigate just the fc magnetization curves . as can be seen from fig . 7 , there are two kinds of patterns of the fc magnetization below t@xmath39 : it either saturates below an even lower temperature as in the case for @xmath3 = 0 , or keeps on increasing down to the lowest measured temperature , as is observable for all other compositions . let us start from @xmath3 = 0 ( pure ni ) case . the second near - saturation of magnetization below @xmath40 10 k suggests that a small volume in each nanoparticle is still fm but with a weak exchange interaction ( @xmath41 ) . a curie - weiss fit of the low - temperature magnetization results in the curie temperature ( t@xmath42 ) and curie constant ( c ) values of 15 k and 0.14 k , respectively . further , the derived weiss field constant @xmath43 = 115 , which is a measure of @xmath41,@xcite is much lower than its value @xmath40 5000 for fm bulk fe , and substantiates the argument that the exchange interaction in this volume is indeed weak . this observation is in line with a report by qin _ et al._,@xcite according to which a superparamagnetic nanoparticle may have a core region with strong @xmath41 surrounded by spins with weaker @xmath41 at the surface . with decreasing temperature , the fluctuations of the surface spins slow down and a short range correlation grows between them , giving rise to a sharp increase in magnetization , which saturates below t@xmath42 when all the surface spins are aligned along the field direction . a schematic of the magnetic structure of pure ni nanoparticles is shown in fig . = 0.000 , 0.085 , 0.098 , 0.11 and 0.17 . the region containing essentially the zfc curves is hatched for visual separation between fc and zfc curves.,scaledwidth=50.0% ] = 0.000 , 0.085 , 0.098 , 0.17 and at 5 k for @xmath3 = 0.11 . inset : low - field region of the m - h curves showing hystereses.,scaledwidth=50.0% ] on introducing the first increment of v in ni ( @xmath3 = 0.085 in this report ) , the v atoms statistically occupy ni sites either as single - atom impurities or as clusters . according to friedel,@xcite a v impurity creates a spin reduction on the neighbouring ni sites . the effect of v substitution on the surface would then be to reduce @xmath41 further and make the surface essentially pm . the same process happens in the bulk . additionally , the v clusters are proposed to create isolated pm zones in the bulk , as shown schematically in fig . 9 . in this scenario , the low temperature susceptibility , which does not saturate till the lowest measured temperature ( 5 k ) , must follow either a curie - weiss law @xmath44 ( 1.25 @xmath45 1.3 ) of ferromagnetism with t@xmath46 5 k , or the curie law @xmath47 t@xmath48 ( @xmath49 = 1 ) of paramagnetism.@xcite assuming t@xmath50 0 in the former case , @xmath49 can be obtained from the slope of the log @xmath51 versus log t curve at the lowest temperatures and must lie between 1 and 1.33 . the curve and its linear fit are shown in fig . the @xmath49 value is 1.12 , which is in agreement with the model of the pure ni nanoalloy as discussed above . a further increase in v concentration , then , is supposed to make the nanoalloys more and more pm and hence one should get the @xmath49 value in the low temperature region @xmath40 1 for all other compositions . however , and as can be seen from the fig . 11 , @xmath52s obtained this way for @xmath3 = 0.098 and 0.11 are 0.83 and 0.41 , respectively , deviating considerably and clearly from the universal power - law ( 1 @xmath45 1.33 ) behaviour . this is characteristic of a qgp , wherein a pm phase possesses a magnetic cluster inclusion , as reported by ubaid - kassis _ et al_. for bulk ni@xmath0v@xmath1 alloys.@xcite the more the @xmath3 deviates from @xmath4 in the pm region , the less the value of @xmath49 is and the stronger the qgp nature becomes.@xcite in the studied nanoparticles , an increased v concentration would result in bigger pm zones in the vicinity of bigger v clusters . statistically , there is a finite probability of much smaller ni clusters to be enclosed within these larger pm clusters , as shown schematically in the fig . 9 , giving rise to the observed quantum griffiths behaviour . on augmenting the v content further , the pm zone would expand . the simultaneous depletion of ni content would then reduce the possibility of ni clusters to get included in the pm zone and the nanoparticle would now comprise only of the pm zones in addition to the rest of the fm volume . a schematic of this structure is also shown in fig . 9 . this way , one would expect the @xmath49 value to enhance back to @xmath53 1 . this is indeed the case for @xmath3 = 0.17 with @xmath49 = 1.23 , as is apparent from the fig . figure 9 also includes a plot between @xmath49 and @xmath3 , and the region in the range @xmath6 having a signature of qgp in the phase diagram is identified . versus log t curves ( symbols ) at low t range for various @xmath3 values and their linear fits ( solid lines ) . shown also is the @xmath49 value for each curve.,scaledwidth=60.0% ] ni@xmath0v@xmath1 ( 0 @xmath24 x @xmath24 0.17 ) nanoalloys of mean diameters 18 - 33 nm were prepared by a chemical reflux method with hydrazine hydrate as the reducing agent and diethanolamine as the surfactant . the compositions of the finally synthesized nanoalloys were determined using edax . the particle sizes were calculated from the fesem images , while hrtem images and saed patterns displayed magnetic and crystalline structure of the nanoparticles , respectively . the magnetic nature was later confirmed from m - h and m - t measurements . further , xrd and xps spectra confirmed that the nanoalloys were indeed a solid solution of ni and v without any trace of oxides . the temperature dependence of resistivity , apart from ascertaining the metallic nature of the nanoparticles , revealed a pm phase coexisting with the fm phase in the nanoparticles in the form of a peak in 40 k - 60 k temperature range . the m - t curves of all the samples exhibited spm nature of the nanoparticles . however , each of these curves was associated with a pm - like increase below an @xmath3-dependent temperature t@xmath39 , coinciding roughly with the peak position in the corresponding resistivity curve . while on the one hand this increase suggests the coexistence of a weak fm phase with the blocked fm below t@xmath39 in the case of pure ni nanoparticles , it is found to be associated with a pm - 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metallic ni@xmath0v@xmath1 alloys are known to exhibit a ferromagnetic to paramagnetic disordered quantum phase transition ( qpt ) at the critical concentration @xmath2 0.114 in bulk . such a qpt is accompanied by a quantum griffiths phase ( qgp ) , the physical observables in which follow non - universal power - law temperature dependences , in a finite temperature range on the paramagnetic side of the transition . in the present work , we explore the occurrence of qgp in nanoparticles of this alloy system . nanoalloys with @xmath3 in the neighbourhood of @xmath4 and mean diameter 18 - 33 nm were prepared by a chemical reflux method . following a few microscopic and spectroscopic studies to determine the sizes , compositions and phases , dc magnetization measurements were also performed to seek out any signature of qgp in the nanoalloys . a paramagnetic - like increase of magnetization is observed to emerge below an @xmath3-dependent transition temperature @xmath5 within the blocked ferromagnetic state of the nanoparticles , and is corroborated by a peak at @xmath5 in the temperature dependence of resistivity . the magnetic susceptibility in this emergent phase follows a non - curie power - law temperature dependence below 10 k for @xmath6 , indicating the presence of a qgp in the nanoparticles within these temperature and composition ranges .

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poor and charge - rich environments coexist at the nanoscale . hence , following ref . @xcite , the nqr results show that the whole set of samples displays intrinsic electronic inhomogeneity , as pointed out for @xmath54=0 - 0.2 in ref.@xcite . the intensity of the @xmath55f nmr signal measured at room temperature shows that the fluorine content is constant throughout all the samples within an accuracy of @xmath56 = 2% . the mssbauer measurements were performed in transmission geometry in the 2.3 k - 300k temperature range using a cryovac konti it cryostat on the lafe@xmath1mn@xmath2aso@xmath8f@xmath9 for x=0.5% . as the @xmath28 source , a @xmath57co in rhodium matrix was used . we used ferrocen powder to measure the influence of the experiment on the line width . the data was analyzed using the transmission integral . mssbauer data for representative temperatures above and below t@xmath11 are shown in fig . [ fig : moss ] . at all investigated temperatures a three peak structure is observed . the two outer peaks at @xmath58 - 0.80 and 1.65mm / s correspond to the ferrocen reference absorber while the inner peak is identified with the lafe@xmath1mn@xmath2aso@xmath8f@xmath9 . at 296k a non - resolved doublet structure is observed due to the interaction of the nucleus with an electric field gradient ( efg ) . in the principal - axis system , the efg is fully determined by its _ z _ component @xmath59 and the asymmetry parameter @xmath60 . we obtained a center shift of 0.451(1)mm / s and a value of @xmath59=4.5v / @xmath61 . @xmath62 was found to be zero at all temperatures . the center shift increases upon cooling due to the temperature - dependent second - order doppler effect to a value of 0.586(8)mm / s at 4.2k . @xmath59 increases with decreasing temperature to a value of 8.6v / @xmath61 at 82k . at the magnetic phase transition @xmath59 increases to 12(2 ) v / @xmath61 and remains constant within error bars down to lowest temperatures . this change of the the electric field gradient probed by fe around the magnetic transition reflects the structural transition in agreement with the one probed by as in the nqr measurements discussed in the main text . below @xmath63 a broad peak is observed which is identified with a non - resolved sextet structure due to a field distribution . this magnetic field distribution was modeled using a gaussian distribution . the temperature dependence of the first moment is shown in fig . 1c of the main part of the paper . in order to carry out the nmr experiments the polycrystalline lafe@xmath1mn@xmath2aso@xmath8f@xmath9 samples were crushed to a fine powder to improve radio frequency penetration . since @xmath0as is a spin i = 3/2 nucleus , above t@xmath11 , the nqr spectrum is characterized by a single line at a frequency @xmath64 with @xmath65 the electric quadrupole moment of the @xmath0as nucleus , @xmath59 the main component of the electric field gradient ( efg ) tensor at the as site generated by the surrounding charge distribution and @xmath62 its asymmetry . since lafeaso is tetragonal @xmath66 and @xmath23 is the quantization axis . thus , the broadening of the line is mainly due to the disorder present in the system since the efg strongly depends on the local charge distribution . below t@xmath11 , in case of a stripe magnetic order , an internal field h@xmath67 is present at the as nuclei and we can perform standard nmr experiments with the only difference that the magnetic field is not provided by an external magnet but by the magnetic ordering of the fe moments ( see also the main part of the paper ) . even if below t@xmath11 the unit cell is orthorhombic the asymmetry @xmath62 is still small ( @xmath68 ) as reported in ref . @xcite . @xmath0as nqr and zf - nmr spectra were derived by recording the integral of the echo signal after a @xmath69 pulse sequence as a function of the irradiation frequency . all the zf - nmr spectra were measured with exactly the same set - up and coil filling factor in order to compare the relative intensity of the lines . it must also be noted that the length of the pulses was optimized for the @xmath20 line and kept constant for the whole spectrum . however the resulting distortion in the spectrum amplitude is not sample dependent and does not modify the position of the lines . the resulting spectra both for undoped lafeaso and for lafe@xmath1mn@xmath2aso@xmath8f@xmath9 samples are reported in fig . 1a of the main part of the paper . since , the stripe magnetic order is also present in lafe@xmath1mn@xmath2aso@xmath8f@xmath9 ( see next section ) we can derive the intensity of the hyperfine field at the as site from the frequency @xmath70 transition ( see main part ) . the results are reported in fig . [ fig : fields ] . the @xmath0as nmr experiments were performed by a homemade nmr spectrometer and a home - assembled probehead placed in the variable - temperature insert of a field - sweeping cold - bore cryomagnet . the large capacitance span of the variable capacitor in the probehead ( approx . 2 - 100 pf ) provides a tuning range of more than two octaves . this allowed us to cover the entire spectral range of the @xmath0as and @xmath71la resonances at 8 t with a single coil . the usage of the same coil for all the resonances , along with a careful calibration of the frequency response of the spectrometer , ensures a reliable quantitative comparison of the amplitudes from different spectral features . the spectra were recorded by tuning the probehead at discrete frequency steps by means of a software - controlled servomechanism featured by the spectrometer itself , and exciting a spin echo . the spin - echo sequence was a standard @xmath72 one , with equal rf pulses @xmath73 of duration @xmath74 and intensity suitably adjusted to optimize the signal . the delays @xmath75 was kept as short as possible with respect to the dead time of the resonant probehead ( @xmath7620 - 35@xmath77s depending on the working frequency ) . the fraction of nuclei participating in the majority and minority @xmath0as signals , respectively , was estimated from the integral of the the normalized spectral amplitude ( i.e. the amplitude divided by the frequency dependent sensitivity @xmath78 ) . in the case of the minority signal , characterized by a hyperfine field estimated in the order of 8 - 10 t from the nmr spectra in lower external fields ( not shown ) , a further correction factor is given by the rf enhancement @xmath79 originating from the hyperfine coupling between electronic and nuclear spins . such a coupling , on one hand , amplifies the driving rf field at the nucleus , so that the resonance can be excited by a rf field reduced by @xmath79 ; on the other , it enhances the e.m.f . induced in the pick - up by the same factor . in a strong external field @xmath80 as in the present case , a factor of the order of @xmath81 ( a formula strictly valid for a uniaxial ferromagnet , indeed ) , where @xmath82 is the effective field at the nucleus , is expected , @xcite hence @xmath83 . a similar value of @xmath84 is obtained by comparing the excitation conditions for the two signals , as the minority signal was optimally excited with 6db extra - attenuation . after correcting ( i.e. dividing ) the minority signal amplitude by @xmath85 , we obtained our estimate for the volume fraction of the minority estimate @xmath0as nuclei in the order of 3% of total . in order to interpret the zf - nmr results we performed simulations of the internal field at the as site for different types of long range magnetic order and for various mn concentrations . both the long range dipolar interaction and the short range transferred hyperfine interaction between the as nucleus and magnetic moments on the four nearest neighbor fe ions ( see fig . [ hyperfine ] ) have been considered in the calculations . the internal field can be written as the sum of the contributions from each one of the fe sites : @xmath86 where @xmath87 is the ordered electron moment at the i - th fe site and @xmath88 is the nuclear - electron coupling tensor between the as nucleus and i - th fe site . we considered only the contributions due to the fe sites in the same plane of the as nucleus since the contribution to the internal field from the other fe - as layers is vanishingly small due to the @xmath89 scaling of the dipolar coupling . the diagonal components of the symmetric transferred hyperfine interaction tensor for the four nearest neighbor fe sites ( fig . [ fig : nnfe ] ) was derived from knight shift measurements while two of the three off diagonal components can be derived from the strength of the internal field in the stripe order configuration , as reported in ref . . since we are only interested in understanding which types of order give rise to spectra in qualitative agreement with the measured ones , we used the values of the transferred hyperfine tensor components reported in ref . and chose @xmath90 for the fe magnetic moment and @xmath91 for the mn magnetic moment . the third off - diagonal component of the transferred hyperfine coupling is relevant only in case of neel order and was chosen equal to the stripe one . the distribution of the internal fields for each type of magnetic order was calculated by randomly substituting mn ( for @xmath10 % ) on the fe site in a 24 @xmath92 24 size mesh and repeating the calculation @xmath93 times . the spectra were then obtained by diagonalizing the zeeman - quadrupole hamiltonian : @xmath94\end{aligned}\ ] ] for each value of the magnetic field and applying magnetic dipole selection rules . the results for orthomagnetic and stripe order are reported in fig . [ fig : int - field ] . for the orthomagnetic order we found @xmath95 . this value is incompatible with the resonance frequency measured by @xmath0 zf - nmr for x=0.5 % and 0.75 % ( main part fig . 1a ) , which displays a sizable internal field with about 80 % of the value found for pure lafeaso . in case of nel order the internal field is found to be h@xmath96 t. however in this magnetic structure the field is parallel to @xmath97 and the splitting between the nqr lines is expected to be half of the observed one . another possible type of magnetic structure is the spin - charge order @xcite which should give rise to two inequivalent iron sites . but this is in contrast with the mssbauer measurements which reveal only one iron site . therefore , the only magnetic structure compatible with the observed experimental results is the ( @xmath98,0 ) or ( 0,@xmath98 ) stripe ordering . in addition , one must notice that the linewidth induced by the magnetic disorder is three orders of magnitude smaller than the one observed . this implies that the observed line broadening of lafe@xmath1mn@xmath2aso@xmath8f@xmath9 for x=0.5 % and 0.75 % with respect to lafeaso is mostly due to the disorder induced by f doping . a proper modelling of lafe@xmath1mn@xmath99aso@xmath8f@xmath9 includes both a realistic ( five - band ) model of the kinetic energy @xmath100 with tight - binding parameters determined in ref . , and inclusion of electronic interactions given by the multi - orbital hubbard hamiltonian @xmath101 here @xmath102 are orbital indexes , @xmath103 denotes lattice sites , and @xmath104 is the spin . the interaction includes intraorbital ( interorbital ) repulsion @xmath105 ( @xmath106 ) , the hund s coupling @xmath107 , and the pair hopping energy @xmath108 . we assume @xmath109 , @xmath110 , and choose @xmath111 . magnetic disorder modeling the mn moments is included by @xmath112 , where @xmath113 is magnetic moment in orbital @xmath12 at the disorder sites given by the set @xmath114 coupled to the spin density of the itinerant electrons . for the results in fig . 5 , we have used @xmath115ev , @xmath116ev , @xmath117ev . a comprehensive description of the band structure and all details of the self - consistent solutions of the mean - field decoupled hamiltonian in real - space can be found in ref . and its associated supp . material . f. hammerath , p. bonf , s. sanna , g. prando , r. de renzi , y. kobayashi , m. sato , and p. carretta , phys . rev . b * 89 * , 134503 ( 2014 ) . g. lang et al . , phys . lett . * 104 * , 097001 ( 2010 ) . y. kobayashi , e. satomi , s. c. lee , m. sato , j. phys . japan * 79 * , 093709 ( 2010 ) . m.fu et al . , phys . lett . * 109 * , 247001 ( 2012 ) . s. kitagawa , y. nakai , t. iye , k. ishida , y. kamihara , m. hirano , h. hosono , phys . b , * 81 * , 212502 ( 2010 ) . k. kitagawa , n. katayama , k. ohgushi , m. yoshida , m. takigawa , j. phys . . jpn . * 77 * , 114709 ( 2008 ) . g. giovannetti , c. ortix , m. marsman , m. capone , j. van den brink , and j. lorenzana , nat . commun . * 2 * , 398 ( 2011 ) .

@xmath0as nuclear magnetic ( nmr ) and quadrupolar ( nqr ) resonance were used , together with mssbauer spectroscopy , to investigate the magnetic state induced by mn for fe substitutions in f - doped lafe@xmath1mn@xmath2aso superconductors . the results show that @xmath3% of mn doping is enough to suppress the superconducting transition temperature @xmath4 from 27 k to zero and to recover the magnetic structure observed in the parent undoped lafeaso . also the tetragonal to orthorhombic transition of the parent compound is recovered by introducing mn , as evidenced by a sharp drop of the nqr frequency . the nqr spectra also show that a charge localization process is at play in the system . theoretical calculations using a realistic five - band model show that correlation - enhanced rkky exchange interactions between nearby mn ions stabilize the observed magnetic order , dominated by @xmath5 and @xmath6 ordering vectors . these results give compelling evidence that f - doped lafeaso is a strongly correlated electron system at the verge of an electronic instability . in the cuprates and in the electron - doped iron - based superconductors the effects of the electronic correlations become progressively weaker as the charge doping level increases towards optimal doping @xcite . accordingly , a fermi liquid description can account for different experimental findings and density functional theory ( dft ) calculations can still provide reliable results @xcite . these superconductors , as the ones belonging to the lnfeaso family ( ln1111 , ln = lanthanides ) , have been typically studied as a function of electron doping and of the external pressure @xcite , both factors which tend to decrease the electronic correlations and to enhance the metallic character . another tuning parameter which allows to span the phase diagram of these superconductors and to probe the relevance of the electronic correlations is the impurity doping level . impurities induce a local perturbation of the electron system @xcite and , depending on the strength of the electronic correlations , the response function can be significantly enhanced , as it happens close to a quantum critical point ( qcp ) . mn doping at the fe sites was reported to have one of the strongest effects @xcite . about 0.2 - 0.3% of mn impurities suppress superconductivity from the optimal t@xmath7 value of 27 - 28 k and then , at higher mn doping levels static magnetism appears ( see fig . [ fig : zf - nmr]b ) @xcite . such a dramatic effect suggests that significant electronic correlations must be present in lafeaso@xmath8f@xmath9 superconductors . the understanding of why such a dramatic effect is present , what type of magnetic order is developing and how to describe these materials at the microscopic level are presently subject of debate . here we show , by means of zero - field ( zf ) nmr , nuclear quadrupole resonance ( nqr ) and mssbauer spectroscopy that the introduction of 0.5 % of mn induces the recovery of the magnetic order and of the tetragonal to orthorhombic ( t - o ) structural transitions observed in lafeaso , the parent compound of la1111 superconductors . moreover the decrease of the transfer energy and the enhanced electron correlations lead to the electron localization . in addition , we present theoretical calculations showing that correlation - enhanced rkky exchange couplings between neighboring mn ions stabilize the magnetic order characterized by @xmath5 and @xmath6 domains . the polycrystalline samples of lafe@xmath1mn@xmath2aso@xmath8f@xmath9 under investigation are the same ones studied in ref . , which were prepared from the starting f concentration of 11% . further details on the sample preparation and their characterization are reported in the suppl . materials @xcite . as zero - field nmr spectra between the lafeaso parent compound and lafe@xmath1mn@xmath2aso@xmath8f@xmath9 for @xmath10 % ( green ) and 0.75 % ( red ) . for the sake of comparison the intensity of the spectra for the mn - substituted compounds are multiplied by 4 . inset : sketch of the magnetic unit cell for the stripe order ( the red arrows represent the fe magnetic moments directions while the magenta arrow corresponds to the orientation of the internal field at the as site ) b ) electronic phase diagram of lafe@xmath1mn@xmath2aso@xmath8f@xmath9 . t@xmath7 and t@xmath11 were determined from magnetization ( squid ) and zero field @xmath12sr measurements , respectively ( see ref . ) . c ) temperature dependence of the hyperfine magnetic field , @xmath13 , for @xmath10 % as derived from mssbauer spectra . the red solid line tracking the order parameter is a phenomenological fit of @xmath13 with @xmath14 where t@xmath15 k and @xmath16.,width=325 ] @xmath0as zero - field ( zf ) nmr spectra were obtained by recording the echo amplitude as a function of the irradiating frequency in the 6 - 26 mhz range for @xmath17 k ( see fig . [ fig : zf - nmr]a ) . as for lafeaso , also for the @xmath18 % sample the spectrum is characterized by two peaks , which in the latter compound are rigidly shifted to lower frequencies and broadened . these two peaks are associated with the @xmath19 and @xmath20 transitions , with @xmath21 the component of the nuclear spin @xmath22 along the quantization axis , which in the case of a stripe magnetic order , as it is the case for lafeaso , is along the @xmath23 axis . the frequency shift between the two peaks is given by the nuclear quadrupole frequency , which at 8 k is @xmath24 mhz both for lafeaso and for the @xmath18 % sample . the frequency of the low - frequency peak ( @xmath25 ) , associated with the @xmath26 transition , is determined by the magnitude of the hyperfine field at @xmath0as , and one can write that @xmath27 , with @xmath28 the @xmath0as gyromagnetic ratio , @xmath29 the hyperfine coupling tensor and @xmath30 the average electron spin , corresponding to the order parameter of the magnetic phase . accordingly , the low - frequency shift of the two peaks in the sample with @xmath18 % would indicate a reduction of the order parameter to about 80% of the value found for lafeaso . the sample with @xmath10 % displays a very similar behavior with a slight increase in the low - temperature order parameter , following its slightly higher magnetic transition temperature ( t@xmath11 ) @xcite . from the magnetic point of view the two samples @xmath10 % and 0.75 % are almost equivalent , as already shown from previous muon spin relaxation experiments @xcite . in order to further study the magnetic order parameter we measured the temperature dependence of mssbauer spectra for the @xmath10 % sample . fig.[fig : zf - nmr]c shows that in the low temperature limit the internal field at the fe site is of about 3.5 t , i.e. the magnitude of the order parameter is reduced to about 70% of the value found in pure lafeaso @xcite , in reasonable agreement with what we derived above from zf - nmr . now , one has first to consider if also other types of magnetic orders could give rise to a similar zf - nmr spectrum , taking into account the reduction in the fe moment to about 80% of the value found in lafeaso . the other possible magnetic orders considered for this compound are the nel and orthomagnetic type @xcite . calculations of the hyperfine magnetic field at the as site ( see the supplementary material @xcite ) show that both these magnetic orders would give rise to zf - nmr lines significantly shifted from the ones reported in fig . [ fig : zf - nmr]a . hence , one can be quite confident that the observed zf - nmr spectrum is associated with a stripe magnetic order . on the other hand , one could argue that the collinear order could coexist with other types of order developing close to mn impurities and that we are actually detecting the signal from a fraction of all @xmath0as nuclei only . thus , we have performed a quantitative estimate of the amount of nuclei contributing to the @xmath18 % sample zf - nmr spectrum by comparing its integral with that of the lafeaso sample , where the sample is 100 % in the stripe collinear phase . we found that @xmath31 % of the @xmath18% sample is in the stripe order . in order to further check if there is a small ( @xmath32 % ) fraction of @xmath0as nuclei that we are missing , we have performed @xmath0as nmr measurements at an applied magnetic field @xmath33h= 8 t. in fig.[fig : nmrspectrum ] the powder nmr spectrum displays a large fraction of nuclei with a spectrum broadened by the internal field developing in the stripe phase ( cyan diamonds ) as well as a small fraction of about @xmath34 % of @xmath0as nuclei with a significant nmr shift ( yellow circles ) . these latter nuclei are likely the ones close to mn impurities where a large hyperfine field is expected . in particular , for the @xmath10% sample there are 2% of @xmath0as nuclei which are nearest neighbors of a mn impurities , a value very close to the one we found . hence , the introduction of mn suppresses superconductivity and leads to the recovery of the stripe magnetic order found in the parent lafeaso compound . more interestingly , the nuclear quadrupole frequency ( fig . [ fig : structransition ] ) shows a jump on passing from just above t@xmath11 ( @xmath0as nqr ) to below t@xmath11 ( @xmath0as zf - nmr ) which is very similar to the one detected @xcite in lafeaso . this abrupt change in @xmath35 is associated with the t - o distortion . therefore , the observation of a similar change in @xmath35 for the @xmath10 % compound indicates that when the stripe magnetic order is recovered by mn doping also the t - o structural transition is recovered , confirming that this transition is driven by the onset of large stripe magnetic correlations . we further remark that the t - o transition causes also a change in the electric field gradient probed by fe nuclei , as detected by mssbauer spectroscopy ( see suppl.materials ) . another relevant aspect can be grasped by looking at the @xmath0as nqr spectra @xcite in fig . [ fig : nqrspectra ] , measured at @xmath36 k for @xmath37 % and at @xmath38 k ( above t@xmath11 ) for @xmath39 % . the data show a clear shift of the nqr spectrum towards lower frequency with increasing mn content ( see fig . [ fig : nqrspectra]b ) and a rapid change in the intensity of the low - frequency peak for @xmath39 % . noteworthy for @xmath39 % the frequency of the dominant low frequency peak perfectly matches that of the paramagnetic phase of lafeaso ( see fig . [ fig : structransition ] and [ fig : nqrspectra]a ) , indicating a similar electronic ground state . according to lang et al . @xcite , the low and high - frequency nqr peaks should be associated with nanoscopically segregated regions with different electron doping levels . in particular , the low - frequency peak should be associated with a lower electronic concentration of weakly itinerant electrons . hence , the increase in the magnitude of the low - frequency peak above @xmath40 % indicates a tendency towards electron localization . this finding is also corroborated by the rapid increase of the electric resistivity as a function of mn content previously observed @xcite across this metal - insulator crossover taking place around @xmath41% . a similar rise in resistivity was also observed @xcite in lafeaso@xmath1f@xmath2 with decreasing f content . we add here that the increase of the resistivity and the slight increase of the lattice constant @xmath23 follow the suppression of the superconductivity ( see fig . 12 of ref . @xcite for details ) . in fact , the @xmath23 axis value appears to correlate with the t@xmath7 value , irrespective of the microscopic mechanism of the suppression . in order to further clarify the origin of the dramatic effect of mn doping in lafeaso@xmath8f@xmath9 , we carried out real space theoretical calculations using a realistic five band hamiltonian . in addition to a kinetic energy term appropriate for ln1111 materials , the model includes multi - orbital hubbard interactions and an impurity term which takes into account the interaction between the mn moments and the spins of the itinerant electrons ( see the supplementary material for further details ) . in ref . @xcite it was demonstrated that in this framework the enhanced spin correlations developing around mn severely speed up the reduction of @xmath4 driven by the magnetic disorder , and may quench the entire superconducting phase already at mn concentrations below 1 % . the mn moments , while substituting random fe positions , orient their moments favorably to generate a long - range ordered sdw phase which minimizes the total free energy of systems at the brink of a sdw instability.@xcite for 0.55% mn moments and ( b ) its fourier transform @xmath42 . ( c ) local phase map @xmath43 showing the ordering vectors on the lattice : @xmath44 and @xmath45 for single - q @xmath46 ( blue ) and @xmath47 ( red ) domains respectively , and @xmath48 for double - q regions ( white ) . the coefficient associated with @xmath46 ( @xmath47 ) , @xmath49 $ ] , is calculated by a filtered fourier transform with the @xmath50 wave vectors contained inside the blue and red squares shown in panel ( d ) . the inset illustrates the definition of the local phase @xmath51.,height=264 ] in fig . 5(a ) we show the total magnetization for a collection of 0.55% mn ions randomly placed in the square fe lattice . this concentration of mn is able to fully suppress @xmath4 and spin polarize all fe sites ( which were all non - magnetic without mn impurity ions ) . the mn - induced magnetic order existing in the inter - impurity regions is long - ranged as reproduced by the sharp peaks in fig . 5(b ) . a small fraction of the sites , roughly corresponding to the mn sites and to their nearest - neighbors ( amounting to @xmath52 of the lattice ) exhibits a significantly larger moment , in overall agreement with the above discussion of the @xmath0as nmr data ( see fig . [ fig : nmrspectrum ] ) . the magnetic order generated by mn doping is efficiently stabilized due to correlation - enhanced rkky exchange couplings between neighboring mn ions . the structure of the induced order is thus dictated by the susceptibility of the bulk itinerant system which , in the present case , is peaked at @xmath5 and @xmath6 regions . in fig . 5(c ) we provide a real - space map of the dominant momentum structure by utilizing a filtered fourier transform illustrated in fig . 5(d ) and the associated caption . as seen , the system breaks up into regions of single - q domains , i.e. either @xmath46- or @xmath47-dominated regions , and does not exhibit substantial volume fraction of double - q order.@xcite this is consistent with the presence of a ( reduced ) orthorhombic transition associated with the mn - induced magnetic order , as found by @xmath0as nqr ( see fig . [ fig : structransition ] ) . all these theoretical results match with the experimental outcomes . overall the above scenario is a clear indication that in lafeaso@xmath8f@xmath9 superconductivity emerges from a strongly correlated electron system close to a metal - insulator transition . the electron correlations are so strong that , owing to the enhanced spin susceptibility at @xmath53 , the effect of a tiny amount of impurities extends over many lattice sites , giving rise to a sizable rkky coupling among them able to abruptly destroy superconductivity and to restore the stripe magnetic order . remarkably , the onset of the magnetic order is intimately related with the charge localization @xcite and hence to the t@xmath7 suppression . this situation is very much akin to the one observed in heavy fermion compounds at a qcp when the rkky coupling overcomes the kondo coupling . the abrupt suppression of the superconducting phase and the recovery of the pristine magnetic order and of the structural t - o transition give compelling evidence that the optimally f doped lafeaso is at the verge of an electronic instability , very close to a qcp.@xcite previous experimental results have shown that this system can be progressively driven away from the qcp via the total substitution of la with nd or by the partial substitution with y,@xcite which shrink the structure and cause a reduction of the electronic correlations @xcite . hence , the ln1111 compound can be considered as a formidable example in physics of how the electronic properties of strongly correlated systems can be significantly affected by finely and selectively tuning the correlation strength with impurities and chemical pressure .

x - ray binaries show x - ray flux variations in a broad range of time scales ( fig.[fig : pds_cygx2 ] ) . their power density spectra often reveal a number of periodic and quasi - periodic phenomena , corresponding to the orbital frequency of the binary system , the spin frequency of the neutron star and quasi - periodic oscillation , which nature is still poorly understood . in addition , aperiodic variability is observed , giving rise to the broad band continuum component in power density spectra , extending from @xmath15 msec to the longest time scales accessible for monitoring instruments . the aperiodic variability is often characterized by the fractional rms values of @xmath16 tens of per cent , indicating that variations of the mass accretion rate @xmath2 in the broad range of time scales are present in the innermost region of the accretion flow , where x - ray emission is produced . from the point of view of characteristic time scales , the high frequency variations could potentially be produced in the vicinity of the compact object . longer time scales , on the contrary , exceed by many orders of magnitude the characteristic time scales in the region of the main energy release . hence , the low frequency @xmath2 variations have to be generated in the outer parts of the accretion flow and to be propagated to the region of the main energy release . due to the diffusive nature of the standard shakura - sunyaev disk @xcite , it plays the role of the low - pass filter , at any given radius @xmath17 suppressing @xmath2 variations on the time scales shorter than the local viscous time @xmath18 . the viscous time is , on the other hand , the longest time scale of the accretion flow . hence , it is plausible to suppose that any given radius @xmath17 contributes to @xmath2 and x - ray flux variations predominantly at the frequency @xmath19 @xcite . @xcite demonstrated that in this picture @xmath20 power density spectra can naturally appear with slope @xmath21 , in good agreement with observations . owing to the finite size of the accretion disk , the longest time scale which can appear in the disk is restricted by the viscous time on its outer boundary @xmath22 . below this frequency the x - ray flux variations are uncorrelated , therefore the power density spectrum of the accretion disk should become flat at @xmath23 . due to the mechanism suggested by @xcite are discussed in section [ sec : qpo ] ] if there are several components in the accretion flow each having a different viscous time scale ( e.g. geometrically thin disk and diffuse corona above it ) , several breaks can appear on the power spectra at the frequencies , corresponding to the inverse viscous time scale of each component . yet another source of @xmath2 variations might be the variability of the mass transfer rate from the donor star . as these variations also have to be propagated through the accretion disk , only low frequency perturbations , with @xmath24 , will reach the region of the main energy release . this might give rise to an independent continuum component in the power density spectrum at @xmath24 . the shape of the final power density spectrum of x - ray binary will depend on the relative amplitudes of the @xmath2 variations due to different components of the accretion flow and the donor star . as these are independent from each other , a distinct features should appear in the power density spectrum of x - ray binary at frequencies @xmath25 . power law.,scaledwidth=51.0% ] the viscous time scale of the accretion disk is @xcite @xmath26 where @xmath27 is the disk half thickness and @xmath28 is the keplerian frequency at the outer disk boundary @xmath29 . for a geometrically thin disk , @xmath30 , the viscous time scale is significantly longer than the keplerian time at the outer disk boundary . if the disk fills a large fraction of the roche - lobe of the primary , as is the case for steady disks in persistent lmxbs , @xmath31 , ( section [ sec : tvisc_h2r ] ) , the viscous time at the outer boundary @xmath22 will exceed the orbital period of the binary system @xmath32 as well . therefore , one might expect to find features associated with the viscous time of the accretion disk at frequencies below the orbital frequency of the binary system . detection of such features for a sample of lmxbs would give an opportunity to probe the accretion disk parameters , such as viscosity parameter @xmath33 and the disk thickness at its outer boundary . in this paper we study a sample of persistent low mass x - ray binaries aiming to detect such features in their power density spectra . longevity of typical orbital periods in lmxbs requires long monitoring observations to achieve this goal . this has become possible thanks to the long operations of the all sky monitor aboard rossi x - ray timing explorer , which provided long term light curves of hundreds of galactic sources covering period of @xmath34 years . some sources with rather short orbital period , @xmath35 few hours may be studied based on the exosat data . lcccccccc source & @xmath36 & @xmath37 & @xmath38 & @xmath39 & q & @xmath40 & d & ref . + & @xmath41 & @xmath41 & hrs & @xmath42 cm & & @xmath43 & kpc & + & ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) & ( 6 ) & ( 7 ) & ( 8) + + grs 1915 + 105 & @xmath44 & @xmath45 & @xmath46 & @xmath47 & @xmath48 & @xmath49 & @xmath50 & 1,2 + gx 13 + 1 & 1.4 & @xmath51 @xmath52 & 592.8 & @xmath53 & @xmath54 & & @xmath55 & 3 + cir x-1 & 1.4 & @xmath56 & 398.4 & @xmath57 & @xmath58 & & @xmath59 & 4 + cyg x-2 & @xmath60 & @xmath61 & 236.3 & @xmath62 & @xmath63 & @xmath64 & @xmath65 & 5 + gx 349 + 2 & 1.4 & @xmath66 & 22.5@xmath670.1 & @xmath68 & @xmath69 & & @xmath70 & 6 + sco x-1 & 1.4 & 0.42 & 18.92 & @xmath71 & @xmath72 & @xmath73 & @xmath74 & 7 + + exo 0748 - 676 & 1.4 & @xmath75 & @xmath76 & @xmath77 & @xmath78 & @xmath79 & @xmath80 & 8 + 4u 1636 - 536 & @xmath81 & @xmath82 & @xmath83 & @xmath84 & @xmath85 & & 5.9 & 9 + 4u 1323 - 619 & 1.4 & 0.26 & 2.93 & @xmath86 & @xmath87 & @xmath88 & @xmath89 & 9 + 4u 1916 - 053 & 1.4 & @xmath90 & 0.83 & @xmath91 & @xmath92 & @xmath93 & 10.0 & 10 + 1h 1627 - 673 & @xmath81 & @xmath94 & @xmath95 & @xmath96 & @xmath97 & @xmath98 & 8.0 11,12 + 4u 1820 - 303 & @xmath81 & @xmath99 & @xmath100 & @xmath101 & @xmath102 & & 8.0 & 13 + + the paper is structured as follows . in section [ sec : data ] we describe the selection criteria and sample of lmxbs . power density spectra and their approximations are presented in the section [ sec : pds ] . the viscous time scale of the accretion disks as obtained from the power spectra analysis and constrains on parameters of the outer disk are discussed in the section [ sec : tvisc ] . in section [ sec : discussion ] we consider the vertical structure of the semi - thick accretion flow , derive parameters of the disk corona and compare these with other evidence of the existence of diffuse coronal flow in lmxbs . our results are summarized in section [ sec : summary ] . in the appendix we describe the method used to calculate power density spectra from the asm light curves . we used data of the asm instrument of rxte observatory ( swank , 1999 ) , covering the period from 19962004 ( mjd range from @xmath103 ) . the asm instrument operates in the 212 kev energy range and performs flux measurements for over @xmath104 x - ray sources from the asm source catalog once per satellite orbit , i.e. every @xmath105 min . each flux measurement ( dwell ) has duration of @xmath105 sec . due to navigational constrains and appearance of very bright transient sources , the asm light curves for individual sources sometimes have gaps of duration of up to few months . the dwell - by - dwell light curves were retrieved from the public rxte / asm archive at heasarc . for sources with short orbital periods , @xmath106 hours , we used the data of the medium energy ( me ) detector of exosat satellite @xcite . it provided @xmath16 several tens of ksec long light curves with typical time resolution of @xmath107 sec in the 0.98.9 kev energy range . the exosat data were also retrieved from heasarc . we have selected low mass x - ray binaries , satisfying the following criteria : 1 . _ persistent sources _ , which light curves do not show off - states or x - ray novae - like outbursts . this ensures that the accretion disk was in the same state during the analyzed period . 2 . the _ orbital period _ @xmath32 is known . orbital parameters of x - ray sources were taken from the catalog of @xcite . + the nyquist frequency in the power density spectra obtained from the asm light curves , @xmath108 hz , correspond to period of @xmath109 hours . in practice , due to specifics of the asm light curves , the high frequency part of the power spectra can sometimes be distorted by the aliasing effect . to minimize the contribution of this effect we selected source with sufficiently long orbital periods , @xmath110 hours . + although the aliasing is not an issue for exosat light curves , their duration , typically limited by @xmath111 ksec , imposes a different constrains on the @xmath38 . as we are looking for features at the time scales larger than @xmath38 , we required that the light curve covers at least several @xmath38 . this leads to the constrain for the sources which can be studied using exosat light curves , @xmath112 hours . source brightness . _ we selected sources with the average count rate exceeding @xmath113 cnts / s for asm and @xmath114 cnts / s for exosat . the _ noise level _ , calculated for asm power spectra , although approximately correct , is not accurate enough , due to existence of unaccounted systematic errors in the flux measurements ( e.g. * ? ? ? * ) . to avoid additional uncertainties in the power spectra we considered only sources with sufficiently high signal at the orbital frequency , @xmath115 at @xmath116 for exosat light curves the noise level is not an issue . there are 6 sources from the asm catalog and 6 sources observed by exosat , satisfying the selection criteria of the section [ sec : selection ] . the final asm sample includes the following sources : grs 1915 + 105 , gx 13 + 1 , cir x-1 , cyg x-2 , gx 349 + 2 , sco x-1 the exosat sample includes : 4u 1323619 , 4u 1636536 , 4u 1820303 , 4u 1916053 , 1h 1627673 , exo 0748676 the binary system parameters for selected sources are listed in table [ tab : general ] and their x - ray light curves obtained by asm are shown in fig . [ fig : lc_asm ] . power density spectra were computed in the 2 - 12 kev ( asm ) and 0.98.9 kev ( exosat me ) energy range . the pds of the sources from asm sample were obtained using the method based on the autocorrelation function calculation as described in the appendix [ sec : method ] . the exosat light curves were analyzed with the aid of the _ powspec _ task from ftools 5.1 . in analyzing the exosat data we averaged the power spectra obtained in ( nearly ) all individual observations with adequate time resolution available in the public archive at heasarc . all but one source in the exosat sample are x - ray bursters . the presence of x - ray bursts in their light curves can results in appearance of an additional component in the pds , having no relation to the power spectrum of @xmath2 variations due to the accretion disk . to avoid this contamination , we screened the exosat light curves to exclude the time intervals corresponding to x - ray bursts . in the case of exo 0748676 we also excluded the time intervals corresponding to the x - ray eclipses . no attempt to screen out the x - ray dips has been done , due to ambiguity of their identification . the power spectra based on the asm data are not subject to a significant contamination due to x - ray bursts and dips / eclipses . [ fig : pds_asm ] [ fig : pds_exosat ] the power spectra of selected sources are shown in figs.[fig : pds_asm ] and [ fig : pds_exosat ] . for reference , we plot in each panel a power law spectrum @xmath117 . flattening at frequencies @xmath118 hz can be seen in the pds of grs 1915 + 105 . this can potentially be related to the specifics of asm light curves , namely , it can be caused by the aliasing effect leading to the leakage of higher frequency power below the nyquist frequency . the assessment of the reality of this flattening is beyond the scope of the present paper . also seen for some sources ( most prominently for cir x-1 and 4u1820303 ) are the peaks due to the orbital modulation . in fig.[fig : pds_broadband ] we show combined power spectra based on the asm and exosat light curves . these spectra cover @xmath119 hz frequency range . for several faint sources from the exosat sample no meaningful power spectra can be obtained from the asm light curves due to too low count rate . for grs1915 + 105 , obviously , no exosat data exist . the power spectra of almost all sources show a clear low frequency break , below which they are nearly flat . for some sources the second very low frequency component is present at lower frequencies . the most clearly this component can be seen in the case of cir x-1 , sco x-1 and 4u1916536 . some sources also show broad qpo like features near or below the break frequency . at high frequencies , above the break , the power spectra follow the @xmath120 power law . remarkably , the slope of the power law appears to be similar in all 12 sources , @xmath121 . the same is true for the normalization . there is at least one exception from the above behavior . no evidence for break was found in the pds of 4u 1636 - 536 . although the power spectrum of this source might have several weak features , its overall behavior in an extremely broad frequency range from @xmath122 hz is well represented by the power law with slope @xmath123 and without any obvious break ( fig.[fig : pds_broadband ] ) . the second possible exception is 4u 1820 - 303 . in addition to clearly visible peak due to the orbital modulations , it appears to have a shoulder at @xmath124 hz followed by a power law at lower frequencies . the presence of this shoulder is more obvious in the data of individual exosat observations , shown in fig.[fig : pds_4u1820 ] . in the formal statistical sense the existence of the low frequency break is highly significant . for example , approximation of the aug . 1920 , 1985 data in the @xmath125 hz frequency range gave the following values : @xmath126 ( 11 d.o.f . ) and @xmath127 ( 12 d.o.f . ) for the power law model with and without low frequency break respectively , resulting in @xmath128 for one additional parameter . however , considering the overall shape of the power spectrum ( e.g. fig.[fig : pds_broadband ] ) , this source presents a less obvious case than the others in our sample . @xcite analyzing long term x - ray variability of 4u 1820 - 303 confirmed earlier suggestion that it is a hierarchical triple system , where ultra - close x - ray binary is orbited by companion with orbital period about 1.1 day . due to influence of the third star the well - known 176 day x - ray modulation of 4u 1820 - 303 is generated . influence of the third star can also give rise to very low frequency modulation of the mass transfer rate in this system , unrelated to the @xmath2 variations intrinsic to the accretion disk . this could , in principle , explain strong very low frequency power law component at @xmath129 hz . in the following we associate the shoulder at @xmath130 with the low frequency break similar to the ones observed in other lmxbs in our sample . we emphasize that this interpretation is not unique and 4u 1820 - 303 , like 4u 1636 - 536 , can show the behavior different from other sources in our sample . power law . the vertical dashed line marks the orbital frequency.,scaledwidth=50.0% ] the power density spectra were approximated by a model of a power law with low frequency break : @xmath131 an additional lorentzian component was added to the model for the sources with strong orbital modulation and/or broad low frequency qpo peaks . the frequency range was chosen for each source individually , depending on the duration of the available time series and the presence of the additional noise component below @xmath132 . the adopted frequency ranges and best fit values of the model parameters are listed in table 2 , along with the count rates and x - ray luminosities of the sources . the large width of the qpo component present in the power spectra of several sources close to the low frequency end makes the determination of the break frequency somewhat ambiguous . for these sources ( grs1915 + 105 , cyg x-2 and 4u 1323619 ) we list the values of the break frequency obtained with and without the lorentzian component in the model . in the following the average of the frequencies of the two models the statistical uncertainty of the break frequency in these sources are obtained from combined confidence intervals of the models with and without lorentzian component . at least three harmonics of the orbital frequency are present in the power spectrum of 4u1323619 . the corresponding bins of the power spectrum were excluded from the fitting procedure . [ cols="<,^,^,^,^,^,^,^ " , ] + for the standard accretion disk ( shakura & sunyaev , 1973 ) the viscous time scale at the outer edge of the disk @xmath29 can be estimated as : @xmath133 where @xmath29 is the disk outer radius , @xmath134 is the keplerian frequency , @xmath27 is the disk half - thickness at the outer edge , @xmath36 is the mass of the primary and @xmath33 is the dimensionless viscosity parameter . combining the eq.([eq : tvisc ] ) with the third kepler law : @xmath135^{-1/2 } , \label{eq : kepler3}\ ] ] and assuming that the binary has a circular orbit one finds : @xmath136 where @xmath137 is the mass ratio , @xmath39 is binary separation , @xmath138 and @xmath139 . from eq.([eq : fvisc2forb ] ) it follows that for fixed viscosity parameter and relative disk thickness and size , the viscous time scale is directly proportional to the orbital frequency of the binary system , as intuitively expected . to estimate the expected range of values of @xmath140 one needs to know the disk size and thickness . the thickness of the standard shakura sunyaev disk is @xmath141 where @xmath142 is the mass of the compact object in solar units , @xmath143 cm is defined so that @xmath144 , @xmath145 erg / s is x - ray luminosity , and @xmath146 cm is the outer disk radius . the disk thickness predicted by this relation varies from @xmath147 for the most compact and low luminosity systems to @xmath148 for sources in the upper part of the table [ tab : general ] . irradiation of the disk by the x - rays from the vicinity of the compact object , although does increase significantly the surface temperature of the outer disk , has a little effect on its mid - plane temperature , due to large optical depth of the shakura - sunyaev disk @xcite . correspondingly , account for irradiation effects does not change the above estimates significantly . with plausible values of two other parameters in eq.([eq : fvisc2forb ] ) , @xmath149 and @xmath150 , the standard theory predicts : @xmath151 power density spectra of all but one lmxbs from our sample have the low frequency break predicted in the simple qualitative picture outlined in the introduction . [ fig : fbreak_forb ] shows the dependence of the best fit value of the break frequency on the orbital frequency of the binary system . there is an obvious positive correlation between these two frequencies broadly consistent with eq.([eq : fvisc2forb ] ) . remarkably , this correlation holds over 3 order of magnitudes in the orbital frequency , from compact binaries 4u 1820 - 303 and 1h1627673 with @xmath152 and @xmath153 min to extremely wide system grs 1915 + 105 with @xmath154 days . the value of @xmath6 ranges from @xmath155 for long period binaries to @xmath156 for the more compact ones . the broad band spectra shown in fig.[fig : pds_broadband ] confirm that the breaks are indeed unique and well defined features in the power spectra . in the following we assume that the break frequency corresponds to the viscous time scale of the accretion disk , and they are related by a simple linear relation : @xmath157 note that this simplified picture does not take into account complexity of the power density distribution near the viscous time scale . with this definition the break frequency is related to the orbital frequency via : @xmath158 cir x-1 appears to deviate from the general trend in fig.[fig : fbreak_forb ] , having the break / viscous time scale by a factor @xmath12 shorter than expected given its orbital period . the significant orbital modulation of the x - ray activity suggests a highly eccentric orbit in this binary system @xcite . @xcite proposed a model , where the system consists of a neutron star on a highly eccentric , @xmath159 orbit around a @xmath160 @xmath161 sub - giant companion . in this model the donor star fills its roche lobe and the mass transfer occurs only during the periastron passage . outside the periastron the donor star is detached from its roche lobe surface and the mass transfer in the binary stops . in such a system the disk radius would be defined , to first approximation , by the minimal separation between the components , @xmath162 . as the eq.([eq : fvisc2forb ] ) was derived for a circular orbit , the following substitution should be made in interpreting the cir x-1 data : @xmath163 plotted in fig.[fig : fbreak_forb ] are two point for cir x-1 . the open circle corresponds to the observed orbital period of the source , the filled circle corresponds to the orbital period corrected for the eccentricity of the orbit assuming @xmath164 . with this substitution the consistency with other sources is restored . although the break frequency does increase linearly with the binary orbital frequency , the ratio @xmath165 is notably larger than predicted for the shakura - sunyaev disk , @xmath166 . the latter range of values is shown as the hatched area in the fig . [ fig : fbreak_forb ] . obviously , larger values of @xmath140 imply that the disk viscous time as traced by the position of the break on the pds is by a factor of @xmath167 shorter than predicted by the theory . as it follows from eq.([eq : fvisc2forb ] ) and ( [ eq : fbr2forb ] ) , there are several parameters affecting the disk viscous time , of which the strongest dependence is upon the disk size and thickness . the theoretical and observational constrains on the disk size are summarized in fig.[fig : rdisk ] . it has two well known theoretical limits . due to the angular momentum conservation , the disk radius can not be smaller than the circularization radius . the results of the numerical calculations of @xcite can be approximated by @xmath168 this approximation is accurate to @xmath169 in the range of the mass ratios @xmath170 . the upper limit on the disk size is given by the tidal truncation radius @xcite . in the case of small pressure and viscosity its value is close to the radius of the largest non - intersecting periodic orbit @xcite . the latter can be approximated by : @xmath171 accurate to @xmath169 in the range of @xmath172 . these two limits on the disk radius are shown as thick dash - dotted lines in fig.[fig : rdisk ] . on the observational side , there is a number of spectroscopic and photometric determinations of the disk radius in cvs @xcite and fewer in lmxbs @xcite . these data indicate ( fig.[fig : rdisk ] ) that for a steady disk the disk radius is close to the tidal truncation radius as defined by @xcite , with possible exception of the small - q systems ( section [ sec : tvisc_lindblad ] ) . with these values of the disk size and accepting @xmath173 as a plausible value of the viscosity parameter , the disk thickness of @xmath174 is required in order explain observed values of the break frequency . this conclusion relies on the association of the break in the power density spectra with the viscous time scale of the disk and as long as the assumption @xmath175 is valid , it is very robust . indeed , in order to describe the data with @xmath176 , as predicted by the standard theory , one would need to increase dramatically the @xmath33-parameter , up to implausibly high values of @xmath177 . alternatively , short viscous time of the disk could be achieved by reducing the roche - lobe filling factor of the disk , down to @xmath178 . this is significantly smaller than the circularization radius and contradicts to both theoretical expectations and observations of the disks in cvs and lmxbs ( fig.[fig : rdisk ] ) . the latter possibility can be also interpreted that only the very inner part of the accretion disk , @xmath179 , contributes to the observed variability of lmxbs in x - rays . this can not be excluded a priori . we note however , that in this case the sharpness of the breaks observed in the power spectra of at least some sources ( e.g. sco x-1 , exo 0748676 , etc ) implies that the transition from the ( outer ) region of negligibly small amplitude of the @xmath2 perturbation to the ( inner ) region of relatively large ones should occur in a rather narrow range of radii . as is evident from fig.[fig : fbreak_forb ] , there is a significant dichotomy in the value of @xmath6 ratio between long- and short - period binaries . this is further illustrated by fig.[fig : fbr2forb ] where the ratio @xmath6 is plotted against the binary orbital period and the mass ratio . the average values of @xmath6 are @xmath180 and @xmath181 for the wide and compact systems correspondingly . the sense of the difference is that the compact systems have systematically shorter , by a factor of @xmath182 , viscous time expressed in the orbital periods than the wide ones , the boundary lying at @xmath183 hours or @xmath184 . the fact that this bimodality occurs between compact and wide systems or , equivalently , ( with the exception of grs1915 + 106 ) between small- and large - q binaries suggests that it may be caused by the excitation of tidal resonances in the accretion disk . the phenomenon of tidal resonances in accretion disks is well studied in the context of dwarf novae @xcite . the resonance occurs if the angular frequency of the orbital motion of the particle in the disk is commensurate with the angular frequency of the orbital motion of the secondary . for a keplerian disk , the location of the resonant orbits is given by @xmath185 where @xmath186 are integers ( e.g. * ? ? ? an obvious condition for a resonance being excited is that this radius lies within the accretion disk . from analytical studies and numerical simulations it is known that the strongest resonance occurs at the lowest order commensurability , @xmath187 ( e.g. * ? ? ? * ) . however the 2:1 resonant radius lies within the accretion disk only at extreme values of the mass ratio , @xmath188 . the next strongest resonance , most important in the context of binaries with not too extreme values of the mass ratio lies at the 3:1 commensurability . the higher order resonances are probably not excited in the accretion disks . the dependence of the 3:1 resonance radius on the mass ratio @xmath14 is shown by the dashed line in fig.[fig : rdisk ] , confirming the well known fact that the @xmath189 resonance can be excited in the systems with the mass ration @xmath190 . note that the precise value of the threshold @xmath14 depends on the definition of the tidal truncation radius , due to small angle between the two curves in fig.[fig : rdisk ] . under the assumption that @xmath191 it is @xmath192 . with the definition of @xmath193 used in fig.[fig : rdisk ] the threshold value is slightly larger . numerical simulations of the non - linear stage of the instability have shown that excitation of tidal resonances results in a significant asymmetry of the accretion disk and causes its precession in the reference frame of the binary @xcite . additionally , the accretion disk is truncated near the resonant radius . this explains the values of the disk radii smaller than the tidal truncation radius observed in cvs and lmxbs with small mass ratios ( fig.[fig : rdisk ] ) . to conclude , with exception of grs1915 + 105 , discussed in section [ sec : grs1915 ] , the excitation of the 3:1 inner lindblad resonance provides a natural explanation of the dichotomy in the viscous time scale between wide and compact systems . the values of the viscous time scale of the disk inferred by the position of the low frequency break in the power spectra of lmxbs require rather large disk thickness , @xmath194 , by a factor of @xmath195 exceeding the prediction of the standard theory of the accretion disks . taken at the face value , @xmath7 appears to agree with the statistics of the eclipsing systems among lmxbs ( e.g. * ? ? ? * ; * ? ? ? . however , large values of @xmath196 result in small optical depth of the outer disk , which is inconsistent with numerous observations of the optical emission from accretion disks , as discussed below . we consider the structure of an accretion disk with some value of the vertical scale - height @xmath197 , without specifying the source of the additional energy dissipation . under the assumption of vertical hydrostatic equilibrium and stationarity , both the mid - plane disk temperature and the density are steep functions of the disk thickness : @xmath198 , @xmath199 . due to strong dependence of the rosseland mean opacity on the density and temperature , typically as @xmath200 in the parameters range of interest , the optical depth of the disk @xmath201 is also steeply decreasing function of the disk thickness , @xmath202 with the power law index @xmath203 changing from @xmath204 to @xmath109 as the @xmath196 increases ( fig.[fig : tau ] ) . therefore , even a moderate increase of the disk thickness would lead to dramatic decrease of its optical depth . as a result , for @xmath197 , the accretion disk in the majority of lmxbs from our sample would be optically thin . this would contradict to a number of observational facts , mostly from the optical band , proving existence of the optically thick outer disk in lmxbs . indeed , the free - free optical depth in the optical v - band equals @xmath205 where @xmath206 is the disk geometrical thickness normalized to 0.1 . the above approximation is accurate to @xmath207 in the @xmath208 k temperature range , the logarithmic term due to the gaunt factor varies from 2.9 to 7.5 for the temperature in the range @xmath209 k. for the parameter of binaries from our sample the optical depth in the v - band would be @xmath210 , with the exception of sources with the largest @xmath2 ( sco x-1 , gx349 + 2 and 4u1820 ) where @xmath211 . the low optical depth in the v - band would conflict with numerous spectroscopic observations of the optical emission from the lmxbs , suggesting existence of the continuum emission originating from the optically thick outer accretion disk ( e.g. * ? ? ? * ; * ? ? ? on this basis the possibility of the semi - thick , @xmath197 outer disk in lmxbs can be excluded . cm on the @xmath196 ratio for @xmath212 g / s , @xmath213 @xmath161 , @xmath214 . the solid line shows the optical depth based on the opal opacities @xcite , long dashed line kramer law ( @xmath215 @xmath216/g ) , short dashed free - free absorption in the optical v - band , dash - dotted thompson optical depth . ] instead , one can envisage a two phase accretion flow with a shakura - sunyaev - like disk in the mid - plane surrounded by the tenuous optically thin coronal flow with the aspect ratio @xmath217 and temperature of @xmath218 . in order to explain the prominence of the low frequency break in the power spectrum , a significant fraction of the accretion has to take place in this corona @xmath219 the temperature and density in the coronal flow are : @xmath220}\end{aligned}\ ] ] @xmath221}\end{aligned}\ ] ] where @xmath222 . as the surface mass density depends on the thickness of the disk as @xmath223 , such a corona will contain a small fraction of the mass of the accreting matter : @xmath224 assuming @xmath225 . the vertical and radial column density of the coronal flow are @xmath226 $ ] : @xmath227 the plausible mechanism of formation of the coronal flow is the disk evaporation process originally proposed by @xcite for dwarf novae . in x - ray binaries the physics of the disk evaporation is significantly affected by the illumination and compton heating and cooling of the disk - corona system by the x - ray emission produced in the vicinity of the relativistic object . although the the self - consistent treatment of the problem is yet to be done , the likely net effect of the x - ray irradiation is to increase the fraction of @xmath2 in the coronal flow ( * ? ? ? * ; * ? ? ? * f.meyer , private communication ) . an independent evidence of the existence of a diffuse ionized gas above the accretion disk plane is provided by observations of lmxbs with high inclination the adc sources and dippers . based on observations of partial x - ray eclipses in 4u1822371 , @xcite concluded that the compact x - ray source in this system is diffused by a large moderately compton thick highly ionized corona located above and below the accretion disk . detailed modeling of the eclipse light curves in the adc sources 4u1822371 and 4u2129 + 47 @xcite yielded estimates of the radial extent of the accretion disk corona in these sources , @xmath228 cm and @xmath229 . these numbers are comparable with the value of the tidal radius in the systems , @xmath230 cm . the partial nature of the eclipses in these sources , with the eclipsed flux at the level of @xmath231 of the uneclipsed value indicated that the corona had non - negligible radial optical depth , @xmath232 , in agreement with the estimate of eq.([eq : corona - nh ] ) . high resolution spectroscopic observations of lmxbs with chandra and xmm - newton gratings are revealing complex absorption / emission features in their x - ray spectra @xcite . these features are mostly pronounced in the systems with high binary inclination angle and suggest the presence of tenuous photo - ionized plasma orbiting the compact object above the orbital plane of the binary . from the analysis of the recombination emission lines of h- and he - like ions of o , ne and mg in exo0748676 @xcite constrained the density and radial extend of the photoionized plasma , @xmath233 @xmath10 , @xmath234 cm and the temperature @xmath235 k. this range of the parameters is consistent with the numbers computed from the above formulae for exo0748676 and assuming the aspect ratio of the coronal flow @xmath236 : the tidal truncation radius @xmath237 cm , density in the coronal flow @xmath238 @xmath10 , and temperature of @xmath239 k. based on the observations of the narrow absorption lines of the h- and he - like fe in the spectrum of 4u1916053 , @xcite estimated the ionization parameter and the column density of the absorbing gas : @xmath240 , @xmath241 @xmath242 . from these numbers one can estimate the radial extent and the density of the absorbing gas , @xmath243 cm , @xmath244 @xmath10 . the tidal radius in this system is @xmath245 cm , in agreement with the value derived from the xmm - newton data . the density in the coronal flow with @xmath236 is @xmath246 @xmath10 . the latter number is somewhat larger than derived from the xmm - newton data . this can be explained by smaller inclination angle in this source resulting in lower density along the line of sight . indeed , unlike exo0748676 showing deep x - ray eclipses , 4u1916053 shows only x - ray dips . in this picture , the @xmath2 variations in the geometrically thin shakura sunyaev disk should give rise to the second very low frequency power law component , revealing itself below the break frequency . such power law component is indeed observed in the power spectra of some of the sources . most obvious its presence is in cir x1 and 4u1916053 and , to less extent , in sco x1 ( fig.[fig : pds_broadband ] ) . remarkably , the slope of the power spectrum before and after the break are very similar , suggesting similar nature of the processes causing the variability furthermore , the second break should be expected at the low frequencies , corresponding to the ( larger ) viscous time of the geometrically thin disk . the frequency of the second break is described by eq.([eq : fbr2forb ] ) with the @xmath196 given by eq.([eq : h2r_ss73 ] ) . an evidence of the second break may be seen in the power spectra of sco x-1 and , possibly , of 4u1916053 . in the former case the break frequency equals @xmath247 hz and the ratio @xmath248 . this value is within the range predicted for standard shakura sunyaev disk with the aspect ratio of @xmath249 , eq.([eq : fvisc2vorb_std ] ) . the value of the break frequency in the case of 4u1916053 is probably @xmath250 hz , although this number is less secure than that for sco x-1 . if correct , this is significantly , by @xmath251 dex smaller that expected in the above scenario . in the conclusion of this section we note that due to power law dependence of the viscous time scale on the disk radius , @xmath252 , the radial drift velocity of the matter in the accretion flow should be increased throughout the most ( @xmath253 ) of the radial extent of the accretion disk . therefore the bulge at the outer edge of the disk , resulting from the disk stream interaction ( e.g. * ? ? ? * ) is insufficient as it is located at the outer edge of the disk . for the same reason this analysis based on the viscous time of the disk probes the outer disk and is insensitive to the conditions in its innermost parts , e.g. @xmath254 , which contribution to the total viscous time is negligibly small . the dichotomy in the viscous time between long- and short - period systems is naturally explained by excitation of the 3:1 inner lindblad resonance . this suggestion is supported by detection of superhumps in several lmxbs with small mass ratios ( e.g. * ? ? ? * ; * ? ? ? importantly , the superhumps are detected not only in transient sources but also in one or two persistent lmxbs , including one source from our sample , 4u1916053 @xcite . the specific mechanism by which the tidal instability is affecting the viscous time of the disk is , however , unclear . it is also unclear if the coronal flow in the small - q system has indeed @xmath255 and @xmath256 as it is formally required by the observed @xmath257 . several possibilities can be mentioned : 1 . mass transfer in tidal waves excited in the disk by the tidal forces . it has been shown theoretically and confirmed in numerical simulations , that in the non - linear regime strong tidal waves are excited in the accretion disk @xcite . these waves , propagating in the disk with the speed significantly exceeding the radial drift velocity of the matter , can decrease the effective viscous time scale . if correct , the aspect ratio and coronal temperature in the small - q systems may be not significantly different from those in the large - q ones . 2 . heating of the outer disk due to resonant tidal interaction @xcite can increase the temperature of the disk and/or coronal flow and , correspondingly , decrease the viscous time scale . as was mentioned above , in order to explain the observed viscous time scale the coronal flow temperature of @xmath256 is required . non - trivial definition of the viscous time for an eccentric and precessing disk . in the fully developed instability the particles in the disk move on non - circular trajectories with large eccentricity . therefore , the average velocity of the radial drift of the disk particles can be significantly different from the value predicted by the standard disk theory for a circular keplerian disk . we note , that the truncation of the accretion disk at the 3:1 resonance radius ( fig.[fig : rdisk ] ) can be excluded as the cause of the shorter viscous time scale in the small-@xmath14 systems , because the change of the outer disk radius by a @xmath258 is insufficient to explain the observed decrease of the viscous time by the factor of @xmath12 . grs1915 + 105 clearly stands out among other small - q systems ( fig.[fig : fbr2forb ] , right panel ) . although it has small mass ratio , @xmath259 @xcite , its ratio @xmath6 is close to the value found in the binaries with @xmath260 . the most plausible explanation of such behavior is related to the transient nature of the source . we outline two possibilities : 1 . the tidal instability is known to have a rather long growth time , of the order of @xmath261 binary orbital periods ( e.g. * ? ? ? * ) . for grs1915 + 105 @xmath154 days and @xmath262 corresponds to @xmath12 years . therefore the time passed since the onset of the outburst ( @xmath12 years ) was insufficient for the tidal instability to fully develop . in this scenario one would expect an increase of the break frequency in the course of the outburst , unless the instability growth time is significantly longer than @xmath12 years . however , no statistically significant difference in the break frequency has been found between the first and second halves of the data . this fact speaks against the above suggestion . 2 . due to the large size of the system ( @xmath263 cm ) , the irradiation effects are insufficient to ionize the outer disk in grs1915 + 105 which is in the cold , low - viscosity state ( h.ritter , private communication ) . therefore its outer boundary is located near the circularization radius rather than has expanded to the tidal radius , as is the case for the persistent sources . therefore there is no accretion disk at the 3:1 resonance radius ( fig.[fig : rdisk ] ) and the tidal instability is not excited . in this respect the present outburst of grs1915 + 105 might be similar to normal outbursts in dwarf novae . as well known , for the same reason the superhumps in dwarf novae are observed ( tidal instability is excited ) only in during the superoutbursts but not during the normal outbursts ( e.g. * ? ? ? * ) . for several sources broad qpo - like features are obvious in the power density spectra . most apparent these features are in the case of cyg x-2 and grs1915 + 105 ( fig.[fig : pds_asm ] , [ fig : pds_exosat ] ) . among plausible mechanisms of appearance of quasi periodic variability in x - ray binaries on these time scales are the radiation - driven warping and precession of the accretion disk ( e.g. * ? ? ? * ) and mass - flow oscillations caused by diffusion instability @xcite . both types of oscillations are caused by the irradiation of the accretion disk by the primary and their characteristic time scale is of the order of the viscous time scale of the disk . accordingly , these features should appear near the break frequency in the power density spectrum . another possible mechanism of appearance of broad low frequency qpos near the viscous frequency of the disk was suggested by @xcite . this mechanism is related to the anti - correlation of the @xmath2 perturbations in the accretion flow on the time scales @xmath264 . the anti - correlation appears in the presence of the steady mass supply through the outer boundary of the accretion flow and is a direct consequence of the energy conservation law . in this respect the broad qpo features observed near the viscous time scale of the outer disk might be similar to the broad qpos detected near the high frequency break of power density spectrum of cyg x-1 in the hard state , at frequency of @xmath265 hz @xcite . in the latter case the values of the break frequency and of the qpo centroid frequency are defined by the viscous time scale of the inner hot flow , @xmath266 r@xmath267 , responsible for the production of the hard comptonized emission in the low spectral state of black hole candidates . we studied the low frequency variability of low mass x - ray binaries in order to search for the signatures in their power density spectra related to the viscous time scale of the disk . our results can be summarized as follows : 1 . as the viscous time is the longest time scale of the accretion disk , the @xmath2 variations produced in the disk due to various instabilities should become independent of each other on the time scales longer than the viscous time , i.e. @xmath268 at @xmath269 correspondingly , the power density of @xmath2 perturbations produced in the disk and of the observed x - ray flux variations should become independent of the frequency at @xmath270 . 2 . using archival data of rxte / asm and exosat / me we studied x - ray variability of persistent lmxbs in the @xmath271 hz frequency range . in power density spectra of 11 sources out of 12 satisfying our selection criteria , we found a very low frequency break . the spectra approximately follow @xmath4 power law above the break and are nearly flat below the break ( figs . [ fig : pds_asm][fig : pds_broadband ] ) . in some cases ( sco x-1 , cir x-1 and 4u1916053 ) a second power law component with the same slope appears at @xmath272 . 3 . in a very broad range of binary periods , from @xmath273 min in ultracompact binary 4u1820 - 303 to @xmath274 days in grs1915 + 105 , the break frequency correlates with the orbital frequency @xmath138 of the binary system ( fig.[fig : fbreak_forb ] ) , in good agreement with the theoretical prediction for the viscous frequency @xmath275 ( eq.[eq : fvisc2forb ] ) 4 . assuming that the low frequency break is associated with the viscous time scale of the disk , @xmath276 , the measured values of the break frequency imply that the viscous time of the disk is by a factor of @xmath277 shorter than predicted by the standard theory ( section [ sec : tvisc_h2r ] , fig.[fig : fbreak_forb ] ) . taken at the face value , this requires the relative height of the outer disk @xmath7 . motivated by the low vertical optical depth of such semi - thick accretion flow we propose instead , that significant fraction of the accretion occurs through the coronal flow above the standard geometrically thin shakura - sunyaev disk ( section [ sec : corona ] , fig.[fig : tau ] ) . the aspect ratio of the coronal flow , @xmath7 , corresponds to the gas temperature of @xmath278 . the corona has moderate optical depth in the radial direction , @xmath232 , and contains @xmath279 of the total mass of the accreting matter . the estimates of the temperature and density of the corona are in quantitative agreement with the parameters inferred by the x - ray spectroscopic observations by chandra and xmm - newton of complex absorption / emission features in the lmxbs with large inclination angle . we found a clear dichotomy in the @xmath6 between wide and compact system , the accretion flow in the compact systems having @xmath12 times shorter viscous time expressed in the orbital periods of the binary , than in the wide ones ( fig.[fig : fbr2forb ] ) . the jump in the @xmath11 ratio occurs at the binary system mass ratio of @xmath184 . this strongly suggests that the dichotomy between small - q and large - q systems is caused by the excitation of the 3:1 lindblad resonance in the small - q systems ( section [ sec : tvisc_lindblad ] ) . we are grateful to friedrich meyer and hans ritter for numerous discussions of the physics of the accretion disks and superhumps phenomenon in dwarf novae . va would like to acknowledge the partial support from the president of rf grant ns-2083.2003.2 and from the rbrf grant 03 - 02 - 17286 . this research has made use of data obtained through the high energy astrophysics science archive research center online service , provided by the nasa / goddard space flight center . bandyopadhyay r. , shabaz t. , charles p. , naylor t. , 1999 , mnras , 306 , 417 boirin l. et al . , 2004 , a&a , 418 , 1061 callanan p.j . , grindlay j.e . & cool a.m. , 1995 , pasj , 47 , 153 chakrabarty d. , 1998 , apj , 492 , 342 chaty s. , haswell c.a . , malzac j. , hynes r.i . , shrader c.r . , cui w. , 2003 , mnras , 346 , 689 chou y. , & grindlay j. , 2001 , apj , 563 , 934 churazov e , gilfanov m. , revnivtsev m , 2001 , mnras , 321 , 759 cottam j. , kahn s.m . , brinkman a.c . , den herder j.w . , erd c. , a&a , 2001 , 365 , 277 edelson r.a . & krolik j.h . , 1988 , apj , 333 , 646 fender r. et al . , 1999 , mnras , 304 , 865 grimm h .- j . , gilfanov m. , & sunyaev r. , 2002 , a&a , 391 , 923 harlaftis e.t . & marsh t.r . , 1996 , a&a , 308 , 97 harlaftis e. & greiner j. , 2004 , a&a , 414 , l13 harrop - 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correlation function : @xmath295 the eq.[eq : acf_power ] expresses the well known fact that power equals the cosine transform of the auto - correlation function . the noise level in the power density spectrum equals : @xmath296 where @xmath297 is the error of @xmath280 . for poisson errors in @xmath280 , @xmath298 and @xmath299 , in agreement with the conventional form ( e.g. * ? ? ? the asm time series consists of the count rate measurements @xmath300 , [ cnts / sec ] , at unevenly spaced moments @xmath301 , @xmath302 . the values of @xmath300 are averaged over the time interval ( @xmath105 sec ) much shorter than the distance between adjacent bins ( @xmath105 min ) . as before , the auto - correlation function @xmath303 is defined on the grid of @xmath304 time bins , with the @xmath281-th bin corresponding to the time interval @xmath305 , where index @xmath281 runs from 0 to @xmath283 . the @xmath294 is computed as @xmath306 where the averaging is performed over all pairs @xmath307 , @xmath308 , for which @xmath309 falls in the @xmath310-th time bin of the grid . the time interval @xmath311 and the duration of the time series @xmath291 are parameters of the transformation and define the frequency range covered by the output power density spectrum . their choice depends on the effective nyquist frequency of the time series and its duration . appropriate for the parameters of the asm light - curves are the following values : @xmath312 sec and @xmath313 sec . advantage of this method is that the correctness of the chosen values of @xmath311 and @xmath291 can be easily checked using the number of pairs @xmath307 used to compute the autocorrelation value at the given time scale @xmath314 and the uniformity of these numbers across the full range of the time scales , from @xmath311 to @xmath291 . with the auto - correlation computed according to eq.[eq : acf_asm ] , the power density spectrum can be easily computed using eq.[eq : acf_power ] . as usual in the time series analysis , the entire time series is divided into @xmath318 segments and the power spectrum is computed for each segment separately . their average and dispersion give estimates of the average power density spectrum and its uncertainty ( under the assumption of stationarity ) . if the errors for @xmath300 are known , the noise level in the output power spectrum can be computed as @xmath319 where @xmath297 is the error of @xmath300 ( in units of cnts / sec ) and @xmath320 is the number of asm measurements used to compute one power spectrum . in practice , the flux measurement errors given in the asm light - curves are slightly underestimated , therefore the noise level computed with eq.[eq : asm_noise ] is somewhat imprecise @xcite . the performance of the method described above was verified in simulations . the initial light curve with the time resolution of @xmath321 sec was computed assuming the power law spectrum with slope of 1.3 and low frequency break at @xmath322 hz and @xmath323 hz and random fourier phases . from this light curve the asm time series was simulated using the measurements times @xmath301 of the real asm light curve of cyg x-2 . the obtained light curve was randomized assuming gaussian errors with the standard deviation equal the real asm measurement error for the given measurement @xmath300 . the obtained time series was analyzed with the same code as used for the analysis of the real asm data with parameters @xmath324 sec and @xmath325 sec . the noise level estimated using eq.[eq : asm_noise ] was subtracted from the power density spectra . the results of simulations are shown in fig.[fig : pds_fake ] along with the model power density spectra used to generate the initial high resolution light curves .

based on rxte / asm and exosat / me data we studied x - ray variability of persistent lmxbs in the @xmath0 hz frequency range , aiming to detect features in their power density spectra ( pds ) associated with the viscous time scale of the accretion disk @xmath1 . as this is the longest intrinsic time scale of the disk , the power density of its @xmath2 variations is expected to be independent on the frequency at @xmath3 . indeed , in the pds of 11 sources out of 12 we found very low frequency break , below which the spectra are nearly flat . at higher frequencies they approximately follow the @xmath4 law . the break frequency correlates very well with the binary orbital frequency in a broad range of binary periods @xmath5 , in accord with theoretical expectations for the viscous time scale of the disk . however , the value of @xmath6 is at least by an order of magnitude larger than predicted by the standard disk theory . this suggests that a significant fraction of the accretion @xmath2 occurs through the optically thin and hot coronal flow with the aspect ratio of @xmath7 . the predicted parameters of this flow , @xmath8 k and @xmath9 @xmath10 are in qualitative agreement with recent chandra and xmm - newton observations of complex absorption / emission features in the spectra of lmxbs with high inclination angle . we find a clear dichotomy in the value of @xmath11 between wide and compact systems , the compact systems having @xmath12 times shorter viscous time . the boundary lies at the mass ration @xmath13 , suggesting that this dichotomy is caused by the excitation of the 3:1 inner lindblad resonance in low-@xmath14 lmxbs . accretion , accretion disks x - rays : binaries

the o(3 ) nonlinear sigma model has long been the subject of intense research due to its theoretical and phenomenological basis . this theory describes classical ( anti ) ferromagnetic spin systems at their critical points in euclidean space , while in the minkowski one it delineates the long wavelength limit of quantum antiferromagnets . the model exhibits solitons , hopf instantons and novel spin and statistics in 2 + 1 space - time dimensions with inclusion of the chern - simons term . the soliton solutions of the model exhibit scale invariance which poses difficulty in the particle interpretation on quantization . a popular means of breaking this scale invariance is to gauge a u(1 ) subgroup of the o(3 ) symmetry of the model by coupling the sigma model fields with a gauge field through the corresponding u(1 ) current . this class of gauged o(3 ) sigma models in three dimensions have been studied over a long time . initially the gauge field dynamics was assumed to be dictated by the maxwell term @xcite . later the extension of the model with the chern - simons coupling was investigated @xcite . a particular form of self - interaction was required to be included in these models in order to saturate the bogomolnyi bounds @xcite . the form of the assumed self - interaction potential is of crucial importance . the minima of the potential determine the vaccum structure of the theory . the solutions change remarkably when the vaccum structure exhibits spontaneous breaking of the symmetry of the gauge group . thus it was demonstrated that the observed degeneracy of the solutions of @xcite is lifted when potentials with symmetry breaking minima were incorporated @xcite . the studies of the gauged o(3 ) sigma model is important due to their intrinsic interest and also due to the fact that the soliton solutions of the gauged o(3 ) chern - simons model may be relevant in planar condensed matter systems @xcite . recently gauged nonlinear sigma model was considered in order to obtain self - dual cosmic string solutions @xcite . this explains the continuing interest in such models in the literature . a particular aspect of the gauged o(3 ) sigma models where the gauge field dynamics is governed by the maxwell term can be identified by comparing the results of @xcite and @xcite . in @xcite the vaccum is symmetric and the @xmath0 soliton solution does not exist , @xmath1 being the topological charge . here , solutions exists for @xmath2 onwards . moreover , these soliton solutions have arbitrary magnetic flux . when we achieve symmetrybreaking vaccum by chosing the potential appropriately @xcite soliton solutions are obtained for @xmath0 . soliton has been shown to follow from general analytical method in @xcite.]these solutions have quntized magnetic flux and qualify as magnetic vortices . it will be interesting to follow the solutions from the symmetrybreaking to the symmetric phase . this is the motivation of the present paper . we will consider a generalisation of the models of @xcite and @xcite with an adjustable real parameter @xmath3 in the expression of the self - interaction potential which interpolates between the symmetric and the symmetrybreaking vaccua . this will in particular allow us to investigate the soliton solutions in the entire regime of the symmetrybreaking vacuum structures and also to follow the collapse of the @xmath0 soliton as we move from the assymmetric to the symmetric phase . the solitons of the model are obtained as the solutions of the self dual equation obtained by saturating the bogomolnyi bounds . unfortunately , these equations fall outside the liouville class even after assuming a rotationally symmetric ansatz . thus exact analytical solutions are not obtainable and numerical methods are to be invoked . the organisation of the paper is as follows . in the following section we present a brief review of the o(3 ) nonlinear sigma model this will be helpful in presenting our work in the proper context . in section 3 our model is introduced . general topological classifications of the soliton solutions of the model has been discussed here . in section 4 the saturation of the self - dual limits has been examined and the bogomolnyi equations have been written down . also the analytical form of the bogomolnyi equations has been worked out assuming a rotationally symmetyric ansatz . these equuations , even in the rotationally symmetric scenario , are not exactly integrable . a numerical solution has been performed to understand the details of the solution . a fourth order runge kutta algorithm is adopted with provision of tuning the potential appropriately . in section 5 the numerical method and some results are presented . we conclude in section 6 . it will be useful to start with a brief review of the nonlinear o(3 ) sigma model @xcite . the lagrangian of the model is given by , @xmath4 here @xmath5 is a triplet of scalar fields constituting a vector in the internal space with unit norm @xmath6 the vectors @xmath7 constitute a basis of unit orthogonal vectors in the internal space . we work in the minkowskian space - time with the metric tensor diagonal , @xmath8 . the finite energy solutions of the model ( [ lo3 ] ) satisfies the boundary condition @xmath9 at physiacal infinity . the condition ( [ bn ] ) corresponds to one point compactification of the physical infinity . the physical space @xmath10 becomes topologically equivalent to @xmath11 due to this compactification . the static finite energy solutions of the model are then maps from this sphere to the internal sphere . such solutions are classified by the homotopy @xcite @xmath12 we can construct a current @xmath13 which is conserved irrespective of the equation of motion . the corresponding charge @xmath14 gives the winding number of the mapping ( [ homo ] ) @xcite . in the class of gauged models of our interest here a u(1 ) subgroup of the rotation symmetry of the model ( [ lo3 ] ) is gauged . we chose this to be the so(2 ) [ u(1 ) ] subgroup of rotations about the 3 - axis in the internal space . the lagrangian of our model is given by @xmath15 @xmath16 is the covariant derivative given by @xmath17 the so(2 ) ( u(1 ) ) subgroup is gauged by the vector potential @xmath18 whose dynamics is dictated by the maxwell term . here @xmath19 are the electromagnetic field tensor , @xmath20 @xmath21 is the self - interaction potential required for saturating the self - dual limits . we chose @xmath22 where @xmath3 is a real parameter . substituting @xmath23 we get back the model of @xcite whereas @xmath24 gives the model of @xcite . we observe that the minima of the potential arise when , @xmath25 which is equivalent to the condition @xmath26 on account of the constraint ( [ const ] ) . the values of v must be restricted to @xmath27 the condition ( [ nb ] ) denotes a latitudinal circle ( i.e. circle with fixed latitude ) on the unit sphere in the internal space . by varying v from -1 to + 1 we span the sphere from the south pole to the north pole . it is clear that the finite energy solutions of the model must satisfy ( [ nb ] ) at physical infinity . for @xmath28 this boundary condition corresponds to the spontaneous breaking of the symmetry of the gauge group and in the limit @xmath29 the asymmetric phase changes to the symmetric phase . we call the potential ( [ pot ] ) interpolating in this sense . in the asymmetric phase the soliton solutions are classified according to the homotopy @xmath30 instead of ( [ homo ] ) . in the symmetric phase , however , this new topology disappears and the solitons are classified according to ( [ homo ] ) as in the usual sigma model ( [ lo3 ] ) . a remarkable fallout of this change of topology is the disappearence of the soliton with unit charge . the fundamental solitonic mode @xmath0 ( @xmath1 being the vorticity ) ceases to exist in the symmetric phase . the modes corresponding to n = 2 onwards still persist but the magnetic flux associated with them ceases to remain quantised . in the asymmetric phase the vorticity is the winding number i.e. the number of times by which the infinite physical circle winds over the latitudinal circle ( [ nb ] ) . associated with this is a uniqe mapping of the internal sphere where the degree of mapping is usually fractional . by inspection we construct a current @xmath31\label{tcur1}\ ] ] generalising the topological current ( [ tcur ] ) . the current ( [ tcur1 ] ) is manifestly gauge invariant and differs from ( [ tcur ] ) by the curl of a vector field . the conservation principle @xmath32 thus automatically follows from the conservation of ( [ tcur ] ) . the corresponding conserved charge is @xmath33 using ( [ tcur1 ] ) and ( [ tch ] ) we can write @xmath34\nonumber\\ & - & { \frac{1}{4\pi}}\int_{boundary}(v -\phi_3 ) a_\theta r d\theta\label{tnw}\end{aligned}\ ] ] where r,@xmath35 are polar coordinates in the physical space and @xmath36 . using the boundary condition ( [ n3b ] ) we find that t is equal to the degree of the mapping of the internal sphere . note that this situation is different from @xcite where the topological charge usually differs from the degree of the mapping . in this context it is interesting to observe that the current ( [ tcur1 ] ) is not unique because we can always add an arbitrary multiple of @xmath37 with it without affecting its conservation . we chose ( [ tcur1 ] ) because it generates proper topological charge . in the previous section we have discussed the general topological classification of the solutions of the equations of motion following from ( [ lgo3 ] ) . in the present section we will discuss the solution of the equations of motion . the euler - lagrange equations of the system ( [ lgo3 ] ) is derived subject to the constraint ( [ const ] ) by the lagrange multiplier technique @xmath38 { \bphi } + { \bf n}_3(v - \phi_3)\nonumber\\ & + & ( v - \phi_3)\phi_3{\bphi}\label{elphi}\\ \partial_\nu f^{\nu\mu } = j^\mu\label{ela}\end{aligned}\ ] ] where @xmath39 using ( [ elphi ] ) we get @xmath40 from ( [ ela ] ) we find , for static configurations @xmath41 from the last equation it is evident that we can chose @xmath42 as a consequence we find that the excitations of the model are electrically neutral . the equations ( [ elphi ] ) and ( [ ela ] ) are second order differential equations in time . as is well known , first order equations which are the solutions of the equations of motion can be derived by minimizing the energy functional in the static limit . keeping this goal in mind we now construct the energy functional from the symmetric energy - momentum tensor following from ( [ lgo3 ] ) . the energy @xmath43.\label{e}\ ] ] for static configuration and the choice @xmath44 = 0 , @xmath45 becomes @xmath46\label{est}\ ] ] several observations about the finite energy solutions can be made at this stage from ( [ est ] ) . by defining @xmath47 we get @xmath48 the boundary condition ( [ nb ] ) dictates that @xmath49 at infinity . from ( [ est ] ) we observe that for finite energy configurations we require @xmath50 on the boundary . this scenario is exactly identical with the observations of @xcite and leads to the quantisation of the magnetic flux @xmath51 the basic mechanism leading to this quantisation remains operative so far as @xmath3 is less than 1 . at @xmath24 , however , the gauge field @xmath52 becomes arbitrary on the boundary except for the requirement that the magnetic field b should vanish on the boundary . remember that not all the vortices present in the broaken phase survives this demand . specifically , the @xmath0 vortex becomes inadmissible . now the search for the self - dual conditions proceed in the usual way . we rearrange the energy functional as @xmath53 \pm 4\pi t\label{ebog}\ ] ] equation ( [ ebog ] ) gives the bogomolnyi conditions @xmath54 which minimize the energy functional in a particular topological sector , the upper sign corresponds to + ve and the lower sign corresponds to -ve value of the topological charge . we will now turn towards the analysis of the self - dual equations using the rotationally symmetric ansatz @xcite @xmath55 from ( [ n3b ] ) we observe that we require the boundary condition @xmath56 and equation ( [ ab ] ) dictates that @xmath57 remember that equation ( [ ab ] ) was obtained so as the solutions have finite energy . again for the fields to be well defined at the origin we require @xmath58 substituting the ansatz([ans ] ) into ( [ sdphi ] ) and ( [ sda ] ) we find that @xmath59 where the upper sign holds for + ve t and the lower sign corresponds to -ve t.equations ( [ eqg ] ) and ( [ eqa ] ) are not exactly integrable . in the following section we will discuss the numerical solution of the boundary value problem defined by ( [ eqg ] ) and ( [ eqa ] ) with ( [ gb ] ) to ( [ ag0 ] ) . using the ansatz ( [ ans ] ) we can explicitly compute the topological charge t by performing the integration in ( [ tch]).the result is @xmath60-{\frac{1}{2}}[v - \cos g(\infty)]\label{t}\ ] ] the second term of ( [ t ] ) vanishes due to the boundary condition ( [ gb ] ) . also , when g(0 ) = 0 , @xmath61 and , when g(0 ) = @xmath62 , @xmath63 it is evident that t is in general fractional . due to ( [ tnw ] ) it is equal to the degree of mapping of the internal sphere . this can also be checked explicitly . from the above analysis we find that g(0 ) = 0 corresponds to + ve t and g(0 ) = @xmath62 corresponds to -ve t. we shall restrict our attention on negetive t which will be useful for comparision of results with those available in the literature . the boundary value problem of interest is then @xmath64 with @xmath65 in addition we require @xmath66 @xmath67 0 as @xmath68 . this condition follows from ( [ eqg1 ] ) , ( [ eqa1 ] ) and ( [ boun ] ) and should be considered as a consistency condition to be satisfied by their soloutions . the simultaneous equations ( [ eqg1 ] ) and ( [ eqa1 ] ) subject to the boundary conditions ( [ boun ] ) are not amenable to exact solution . they can however be integrated numerically . we have already mentioned the quenching of the @xmath0 solution in the limit @xmath69 . this is connected with the transition from the symmetry breaking to the symmetric phase . the numerical solution is thus interesting because it will enable us to see how the solutions change as we follow them from the deep assymetric phase @xmath70 to the symmetric phase @xmath71 . in the following we provide the results of numerical solution to highlight these issues . let us note some details of the numerical method . a fourth order runge kutta method was employed . the point @xmath72 is a regular singular point of the equation . so it was not possible to start the code from @xmath72 . instead , we start it from a small value of @xmath73 . the behaviour of the functions near r = 0 can be easily derived from ( [ eqg1 ] ) and([eqa1 ] ) @xmath74 @xmath75 here @xmath76 is an arbitrary constant which fixes the values of g and a at infinity . in the symmetrybreaking phase the numerical solution depends sensitively on the value of @xmath76 . may have arbitrary values.]there is a critical value of @xmath76 , @xmath77 for which the boundary conditions are satisfied . if the value of @xmath76 is larger than @xmath78 the conditions at infinity are overshooted , whereas , if the value is smaller than the critical value g(r ) vanishes asymptotically after reaching a maximum . the situation is comparable with similar findings elsewhere @xcite . the values of @xmath79 and @xmath80 were calulated at a small value of @xmath73 using ( [ bc1 ] ) and ( [ bc2 ] ) . the parameter @xmath76 was tuned to match boundary conditions at the other end . interestingly , this matching is not obtainable when @xmath81 and @xmath82 . this is consistent with the quenching of the @xmath0 mode in the symmetric vacuum situation . after the brief discussion of the numerical technique we will present a summary of the results . as may be recalled , the purpose of the paper is to study the solutions throughout the asymmetric phase with an eye to the disappearence of the @xmath81 mode . accordingly profiles of @xmath79 and @xmath80 will be given for @xmath0 , for different values of @xmath3 . in figures 1 and 2 these profiles are shown for @xmath83 . the corresponding magnetic field distributions are given in figure 3 . another interesting issue is the change of the matter and the gauge profiles with the topological charge . in figures 4 and 5 this is demonstrated for different @xmath1 values for a constant @xmath3 . _ the matter profile . _ the function @xmath84 is plotted against @xmath73 for @xmath0 and different @xmath3 values , indicated on the right hand top corner . , width=207 ] _ the gauge field profile . _ the function @xmath85 is plotted against @xmath73 for @xmath0 and different @xmath3 values . , width=207 ] _ the magnetic field profiles . _ plot of the function @xmath86 against @xmath73 . , width=207 ] _ the matter profiles for different n ( v = .4 ) . _ , width=207 ] _ the gauge field profiles for different n ( v = .4 ) . _ , width=207 ] the o(3 ) sigma model in ( 2 + 1 ) dimensional space time with its u(1 ) subgroup gauged was mooted @xcite as a possible mechanism to break the scale invariance of the soutions of the original 3 - dimensional o(3 ) sigma model . the model finds possible applications in such diverse areas such as planar condensed matter physics @xcite , gravitating cosmic strings @xcite and as such is being continuously explored in the literature @xcite . an interesting aspect of the gauged o(3 ) sigma models is the qualitative change of the soliton modes in the symmetric and symmetrybreaking vacuum scenario , as can be appreciated by a comparision of solutions given in @xcite . in this paper we have considered a gauged o(3 ) sigma model with the gauge field dynamics determined by the maxwell term as in @xcite . an interpolating potential was included to invesigate the solutions in the entire symmetrybreaking regime . this potential depends on a free parameter , the variation of which effects transition from the asymmetric to symmetric phase . we have discussed the transition of the associated topology of the soliton solutions . the bogomolnyi bound was saturated to give the self dual solutions of the equation of motion . the self - dual equations are , however , not exactly solvable . they were studied numerically to trace out the solutions in the entire asymmetric phase with particular emphasis on the @xmath0 mode . our analysis may be interesting from the point of view of applications , particularly in condensed matter physics . the author likes to thank muktish acharyya for his assistance in the mumerical solution . schroers , phys . * b 356 * ( 1995 ) 291 . ghosh and s.k . ghosh , phys . lett . * b366 * ( 1996 ) 199 . k. kimm , k. lee and t. lee , phys . rev . * d 53 * ( 1996 ) 4436 . p. mukherjee , phys . * b 403 * ( 1997 ) 70 . p. mukherjee , phys . d 58 * ( 1998 ) 105025 . k. c. mendes , r. r. landim , c. a. s. almeida , mod.phys.lett . * a20 * ( 2005 ) 1005 m. s. cunha , r. r. landim , c. a. s. almeida phys.rev . * d74 * ( 2006 ) 067701 . bogomolnyi , sov . j. nucl * 24 * ( 1976 ) 449 . c han , phys . rev . * d47 * ( 1993 ) 5521 . y. verbin , s. madsen , a.l . larsen , phys.rev . * d67 * ( 2003 ) 085019 . h. r. vanaie , n. riazi , int.j.mod.phys . * a19 * ( 2004 ) 3595 . a.a . belavin and a.m. polyakov , jetp lett . * 22 * ( 1975 ) 245 . f. j. hilton , an introduction to homotopy theory ( cambridge university press , cambridge , england , 1953 ) . r.rajaraman , solitons and instantons ( north holland publishing company , amsterdam , 1989 ) . schroers , nucl.phys . * b475 * ( 1996 ) 440 . y.s . wu and a. zee , * b147 * ( 1984 ) 325 . r. jackiw , k. lee and e. weinberg , phys . rev . * d 42 * ( 1990 ) 3488 . f. wilczek and a. zee , phys . * 51 * ( 1983 ) 2250 ; m. bowick , d. karabali and l. c. r. wijewardhana , nucl . * b 271 * ( 1986 ) 417 ; r. banerjee , nucl . phys . * b 419 * ( 1994 ) 611 .

soliton modes in a gauged sigma model with interpolating potential have been investigated . numerical solutions using a fourth order runge kutta method are discussed . by tuning the interpolation parameter the transition from symmetrybreaking to the symmetric phase is highlighted .

it is well known that quantized vortices play an important role in understanding many phenomena of superfluid , superconductivity and even astrophysics . in the latest several years a remarkable experimental development has been made in the rapidly rotating bose - einstein condensate(bec ) . a large amount of angular momentum is imparted into the system and its rotation frequency becomes faster and faster . the ground state of the fast rotating bec was observed to be a regular triangular vortex lattice at nearly zero temperature . this provides a new stage of studying both the equilibrium and the dynamic properties of the vortex matter . recently two novel experiments were reported by cornell s group in jila.@xcite@xcite atoms in the hyperfine spin @xmath0 state , were rotated rapidly by the evaporative spin - up mechanism . about @xmath1 vortices were observed in the cloud and they were arranged into a regular lattice structure . and then , an atom removal laser , whose frequency was tuned to the transition energy between the @xmath0 state and the @xmath2 state , was applied to the center of the condensate , and the recoil from a spontaneously scatted photon blasts atoms out of the condensate . in this way , the atoms with lower angular momentum were selectively removed from the system . many exotic features were observed when this technique is applied . after the atom removal laser was applied continuously , the vortex number contained in the cloud increased form @xmath1 to about @xmath3 , and the vortices began to aggregate to the center of the condensate , eventually they formed a giant vortex . if the laser was applied for a short limited time , it was reported in ref.@xcite that a periodic formation and disappearance of the giant vortex core was observed after applying a strong atom removal pulse , and in ref.@xcite that the tkachenko oscillation of the vortex lattice would be found after a weak pulse . both kinds oscillation were found damped . @xcite@xcite a number of questions are raised from these experiments , such as how to understand the vortex number increase and the vortices aggregating to the center of condensate , why the strong and weak laser strength will cause two different kinds of oscillations , how to understand the relation between the oscillation frequency and the initial conditions of the condensate , and which mechanism leads to the damping . we notice that recently there are lots of theoretical efforts around these experiments , some of them try to understand these observations numerically@xcite , and some of them focus on the tkachenko wave of the vortex lattice state@xcite@xcite@xcite@xcite . however , in this paper , we will present an explanation to these questions from another point of view , namely using a dual @xmath4-dimensional maxwell electrodynamic ( med ) description , and we will show that the main features observed can be understood naturally in this framework , although qualitatively in some aspects . this dual electromagnetic description of the interacting bosons model was obtained by the duality transformation early in the study of helium superfluidity,@xcite@xcite and then in the past decades it has been discussed in the contents of superconductivity and superfluid film by various authors.@xcite@xcite compared to the model discussed before , we would like to emphasize some important differences of the system studied here . first , a large amount of angular momentum have been imparted into the atoms and the rotational invariance is restored before the formation of condensate , therefore the total angular momentum of the condensate should be conserved to a non - zero value . the second , the atom removal laser violates the current conservation condition . and the third , the system is dilute and the interaction between atoms is weak , the sound velocity is consequently relatively small . we will show that these differences lead to these exotic phenomena in these experiments , in other words , these observed phenomena reflect some intrinsic properties of the rotating bec . in this section we will firstly make a brief description of the dual picture for the interacting bosons system , for the details we refer to the ref.@xcite@xcite@xcite . beginning with the coherent state path integral formulism,@xcite we write down the quantum mechanical propagator @xmath5\exp\{i\int dt d^2\vec{r } \mathcal{l}\ } } , \label{partition}\ ] ] where the lagrangian @xmath6 @xmath7 is the interaction strength and equals to @xmath8 for ultracold atomic system , where @xmath9 is the @xmath10-wave scattering length between atoms . here @xmath11 is a complex field describing the bosonic many - body system , and @xmath12 is its complex conjecture . we can factor @xmath13 as @xmath14 . the field @xmath15 describes the density of bosons . the phase of @xmath11 contains two parts , @xmath16 is the smooth part free from singularity and @xmath17 is that caused by the presence of vortices . first of all , we can introduce an auxiliary field @xmath18 by performing a hubbard - stratanovich ( h - s ) transformation , the physical meaning of which is the superfluid current . by defining the three vectors @xmath19 and @xmath20 , in the low frequency and long wavelength limits one can neglect the second order gradient term @xmath21 , and obtain that @xmath22d[\rho ] d[\theta ] d[\theta_{v } ] \exp\left\{i\int dt d^2\vec{r } \left(-\tilde{j}\cdot\tilde{\nabla}\theta-\tilde{j}\cdot\tilde{\nabla}\theta_{v } + \frac{m}{2\rho}|\vec{j}|^2 -\frac{g}{2}\rho^2\right)\right\}.\nonumber\\\label{hstransformation}\end{aligned}\ ] ] integrating over @xmath16 , one can naturally obtain a @xmath23-function constraint @xmath24 , which is just the conventional current conservation condition @xmath25 owing to this constraint , we can introduce a three - component gauge field @xmath26 defined as @xmath27 , which satisfies @xmath28 the density and the current of the bosons are thereby determined by the gauge field . hence , @xmath29 is compared to a magnetic field perpendicular to the 2-dimensional plane . we can expand @xmath15 as @xmath30 , @xmath31 is the average density which is compared to an external magnetic field , while @xmath32 is the density fluctuation which is compared to the electromagnetic wave . the electric field @xmath33 is defined as @xmath34 , and it is perpendicular to the boson current @xmath18 . therefore the eq.([currentconservation ] ) can be rewritten as @xmath35 which is just the maxwell equation in a monopole free space - time . with replacing the @xmath15 by @xmath31 in the denominator , the last two terms in eq.([hstransformation ] ) can also be reexpressed in terms of the gauge field @xmath26 , which is just the dynamic term @xmath36 of the gauge field as in the conventional med . it determines the dynamic properties of boson density fluctuation , i.e. phonon , and recovers the well - known result for the sound velocity @xmath37 , which plays the same role as the light velocity in @xmath4-dimensional med . the coupling between the vortices and bosons can be illustrated as @xmath38 here @xmath39 is defined as @xmath40 . it can be verified that the zero - component of @xmath39 , which is denoted by @xmath41 , represents the vortex density , and the other two spatial components represent the vortex current . the eq.([a14 ] ) indicates that the vortices can be viewed as charged particles , which are coupled to an electromagnetic field via gauge coupling . the coupling constant , that is the charge @xmath42 of the charged particles , is unit . in the coulomb gauge , the scalar potential @xmath43 is instantaneous , one can integrate it out and obtain the mass of vortex @xmath44 , which is equal to @xmath45 for a uniform condensate , and the mutual logarithmic interaction between vortices described by @xmath46 here @xmath47 is the radius of the condensate and @xmath48 is the radius of vortex core . thus we have obtained a dual electrodynamic description of quantized vortices , which is briefly summarized in the following table . in the dual picture the vortex lattice state corresponds to the wigner crystal state of 2-dimensional electrons in the external magnetic field . + [ cols="^,^ " , ] + + such a description can be directly applied to the confined bec when it is in the thomas - fermi regime . the average density here is therefore determined by the trapping potential , as well as the centrifugal repulsive force caused by rotating , and we will take it as inverted parabola in this regime . here we would like to point out that the main assumption leading to the electrodynamic description is that the vortices are treated as point - like particles , the physics inside the vortex core is not considered , as a consequence we take the mass of vortex as its electromagnetic mass throughout this paper@xcite . besides , in the following treatment we view the system as quasi - two dimensional and neglect the density inhomogeneity when estimating the vortex mass . in the quantum hall mean field regime@xcite , it is shown that the physics of vortex core will become important . however , before entering this regime , and when the thomas - fermi theory is still valid , these neglected effects will only slightly renormalize the vortex mass as well as the interaction between vortices . in jila experiments the rotational invariance is restored before condensation , therefore the condensate should obey the total angular momentum conservation constraint . such a system can be described by adding a lagrange multiplier term to the lagrangian . @xmath49 it is obvious that one will obtain a @xmath23-function which means that the total angular momentum must always equal to @xmath50 after integrating @xmath51 out . the physical meaning of @xmath51 is in fact the effective rotation frequency . following the procedure discussed in above section , it is found that only three additional terms , namely @xmath52 , will be added to the lagrangian . one can see that only the zeroth component of the gauge field responds to the rotation . integrating @xmath51 out and neglecting the density fluctuation , we obtain @xmath53 following the same step which leads to eq.([log - interaction ] ) , integrating out @xmath43 , we can obtain a local chemical potential term in the following lagrangian @xmath54 here @xmath55 is the green s function for the @xmath43 field acquiring a very small mass @xmath56 , which will vanish in the large atom number limit . the first term in the eq . ( [ chenmicalporential ] ) indicates that the vortex system can be viewed as an interacting coulomb gas system , and the second term defines a local chemical potential @xmath57 for vortices emerging from the angular momentum constraint , which is proportional to @xmath58 . the atom removal laser , which removes atoms with lower angular momentum , decreases the atom density @xmath15 with little change to the total angular momentum @xmath50 , leads to the increase of the vortex chemical potential . notice that the interaction energy @xmath59 depends quadratically dependence on vortex density @xmath41 , and the interaction between vortices are repulsive , so @xmath60 this shows the increase of vortex density . for simplicity we assume that the atom removal laser acts only at the center point . extension to the gaussian shape laser is straightforward and the qualitative results will not be changed . when the laser is applied , a centripetal atom current will be induced and the current conservation condition will be violated at the center . in the dual picture , a circular electric field will be generated and the maxwell equation should be modified in the following way @xmath61 here the value of @xmath62 equals to the number of atoms removed from the condensate per unit time . therefore it is modelled as a monopole current running through the plane . the magnitude of the electric field is approximately @xmath63 nearby the center . for the case of continuous laser the vortices as charges will be accelerated azimuthally by the electric field , and the lorentz force will drive the vortices to aggregate toward the center , and eventually to form a giant vortex . after a pulse laser which only acts for a short time @xmath64 , the vortices will acquire a velocity @xmath65 and leave the balance positions of the lattice . for the vortices closest to the center , the magnitudes of their velocities can be approximated by @xmath66 @xmath67 is the vortex lattice spacing . therefore , there are two forces which can cause the oscillation motion , one is the lorentz force @xmath68 , and the other is the mutual interaction force between vortices @xmath69 . it is easy to obtain the ratio of the two forces : @xmath70 for these experiments we can naturally assume that the number of atoms removed from the condensate @xmath62 is an increase function of the power of removal laser . thus eq.([radio ] ) indicates that the lorentz force is dominant for a strong laser pulse , such as the pulse used in ref.@xcite . in this limit , the vortices will execute circumnutation . when the vortices move very close to each other , they will merge into a giant vortex , and separate into several single vortices again when they oscillate away from each other . the mutual interaction force will be dominant for the weak pulse case . in this limit , it will cause a collective oscillation of vortex lattice known as tkachenko oscillation . the oscillation frequency @xmath71 depends on the initial conditions of the condensate , such as the number of atoms and the rotation frequency @xmath51 . in the strong pulse limit , for the motion is mostly determined by the lorentz force , @xmath72 . because the density @xmath15 is proportional to the number of atoms @xmath73 and is a decreasing function of the rotation frequency @xmath51 , the oscillation frequency is linearly dependent on @xmath73 and increases with @xmath51 decreasing . in the weak pulse limit , the oscillation frequency @xmath71 is proportional to @xmath74 . it is not obvious to tell whether @xmath71 will decrease or not when the condensate rotates faster , because both @xmath15 and @xmath67 will decrease with the rotation frequency @xmath51 increasing . we make a rough estimate as following : for a fast rotating bec confined in a harmonic trap with trapping frequency @xmath75 , the effective confining potential is @xmath76 . using the thomas - fermi approximation we can obtain that the condensate density nearby the center @xmath31 is proportional to @xmath77 , and then using the formula @xmath78 , we can obtain that @xmath79 here @xmath80 is a complicated constant , which is related to the atom mass , vortex mass , trap frequency and the density distribution along the third spatial dimension . the eq.([fomega ] ) shows that @xmath71 is a decreasing function of @xmath51 when @xmath81 , which is shown in the left side of fig.([fig : f - omega])@xcite . as a function of rotational frequency of condensate @xmath51(in the unit of @xmath75 ) . [ fig : f - omega],width=268 ] from above discussion we can see the atomic current toward the center is of essential importance for causing oscillations . hence if there is any other ways to generate a similar atomic current , such as using optical dipole force to draw atoms into the middle of the condensate in the ref.@xcite , the similar oscillation of vortex lattice was observed . ( in the unit of oscillation period @xmath82 ) [ fig : damping],width=268 ] one feature of these oscillation motions shown in these experiments is that the oscillation amplitudes are heavily damped , especially for the tkachenko oscillation reported in ref.@xcite . it is quite natural in the view of the dual picture . because the coupling between the vortex current and the phonon field is a gauge coupling , an accelerated vortex will radiate sound wave in the same way as an accelerated electron radiating electromagnetic wave . in the view of hydrodynamic picture , when a vortex is accelerating , the circular current @xmath18 , as well as the singularities of field @xmath18 , can not be stationary in any inertial frame . due to the current conservation condition eq.([currentconservation ] ) , a time dependent density fluctuation @xmath32 will be generated . hence the oscillating vortices will induce sound wave , lose their energy and result in the damping of amplitude . in the dual picture this process can be calculated through a standard med calculation . the retarded green s function for @xmath83 med is @xmath84 and the gauge potential @xmath85 for the radiation field is @xmath86 here @xmath87 is the vortex @xmath88-current . unlike the @xmath89d case , the green s function contains the step function @xmath90 instead of dirac s @xmath23 function , so one can not easily find a general expression for the radiation energy of a charge with arbitrary acceleration.@xcite@xcite here we only consider two special cases , the linear oscillation @xmath91 , and circular motion @xmath92 . following the approximation made in ref.@xcite , we replace @xmath93 with @xmath47 and calculate the energy radiation along large circle @xmath94 . then we can perform an analytical calculation for the average energy radiated over one period , @xmath95 where @xmath96 is the poynting vector . for these two special cases , the result is @xmath97 the radiation rate is inverse proportional to the interaction strength @xmath7 , so the damping effect is remarkable for the weak interacting boson gases . recalling the approximate expression for the vortex mass @xmath98 , eq.([radiationpower ] ) can be reexpressed as @xmath99 hence @xmath100 this indicates that the oscillation energy exponentially decays as @xmath101 , and the ratio of the exponential damping time @xmath102 to the oscillation period @xmath82 is equal to @xmath103 . a typical oscillation curve in the current experiment conditions is schematically shown in the right side of fig.[fig : damping ] , one can see that the oscillation amplitude will decay rapidly within one period , and this is qualitatively coincident with the experimental observation@xcite@xcite . however , there are also a little quantitatively disagreement between the theoretical damping curve and the experiments , in ref.@xcite the amplitude vanishes within about three periods and in ref.@xcite it is about one and a half periods . this disagreement occurs because in the real case the vortex mass should be modified due to the spatial density inhomogeneity , the third spatial dimension effect and the edge effect . the mass of vortex , which is proportional to the self - energy , is therefore changed with the position of vortex.@xcite it is expected that the damping curve will agree better with the experiments when these subtle effects are carefully considered . according to the effective maxwell electrodynamic model the vortices in the bec can be viewed as an ensemble of classical charges moving in an external magnetic field . so far we have applied this description to understand the observed phenomena of vortices induced by selective removal atoms from a fast rotating bec . the atom removal laser applied to the center of the condensate will cause two leading effects in the view of dual picture . one is the increase of the vortex chemical potential and then subsequently the increase of vortex density . the other is that it will induce a circular electric field , and each vortex will acquire an azimuthal velocity . we clarified that there are two forces that can cause oscillation motion , and discussed the the relation between the oscillation frequencies and the number of atoms together with the effective rotation frequency . we showed that the dispersion mechanism for the vortices oscillation comes from the emission of phonon , which is the same as the radiation of the electromagnetic wave of an accelerated charge . and the damping effect is more remarkable in these experiments because the interaction strength is weak . in this paper the dual description of vortices has been used to explain current experiments . although it is not yet a detailed quantitative description , it provides a natural physics picture of most key experimental observations , which convinces us to believe that this description is valid to the extent that the bec is fast rotated but still not entering the quantum hall mean field regime . we hope this will start a new route to study the issues in the rapidly rotating bec system . _ acknowledgements _ : hz would like to thank professor c. n. yang for encouragement . and the authors would like to acknowledge z.y . weng , t.l . ho , l. chang , r. l@xmath104 and x.l . qi for helpful discussions . this work is supported by national natural science foundation of china ( grant no . 99 we have made a rough estimate that the frequency given by equation([fomega ] ) is the the same order of the experiment date . and we use the experiment date @xmath105 , @xmath106 reported in ref.@xcite to determine the constant @xmath80

it is known that the quantized vortices in a superfluid can be described by a dual electromagnetic model through the duality transformation . recently a new technique , which can selectively remove atoms from a bose - einstein condensate , was applied to the vortex lattice state of the rapid rotating boson gases . the increase of vortex number , dynamical formation of giant vortex , the oscillation and damping of vortex motion were observed in these experiments . in this paper we will discuss these observations in the framework of dual description and show how to understood these observations naturally .

in the unquenched qcd simulations using the hybrid monte carlo algorithm , the computational cost rapidly grows as the chiral limit of sea quark is approached . in practical simulations with wilson - type fermions the sea quark mass is limited to be heavier than @xmath0 . to obtain the physical results for @xmath1 and @xmath2 quarks , therefore , the chiral extrapolation is indispensable . in the chiral extrapolation the chiral perturbation theory ( chpt ) may be used to decide the functional form , as it is an effective theory valid for low energy qcd . once the available lattice data are confirmed to be consistent with chpt , the extrapolation to the @xmath1 and @xmath2 quark masses using the chpt formula is justified . the practical question is , then , whether the lattice results could reproduce the sea quark mass dependence , especially the chiral logarithm , predicted by chpt . in this talk , we present the sea quark mass dependence of the pseudoscalar meson decay constant obtained in unquenched qcd , and compare them with the one - loop chpt prediction . the unquenched simulations are done using the standard gauge and nonperturbatively @xmath3-improved fermion action at @xmath4 = 5.2 with sea quark masses corresponding to the pion mass in the range 5501000 mev . further details of the simulation are discussed in a separate talk @xcite . we also discuss the uncertainty associated with the chiral extrapolation taking light - light and heavy - light decay constants as examples . in full qcd the chpt predicts a specific functional dependence of physical quantities on the quark mass , _ i.e. _ the chiral logarithm , at the one - loop order . for @xmath5 flavors of degenerate quarks with a mass @xmath6 , the pseudoscalar meson decay constant @xmath7 is given by @xcite @xmath8,\ ] ] with @xmath9 . while the low energy constants @xmath10 are unknown parameters , the chiral log term @xmath11 appears with a definite coefficient depending only on the number of flavors . [ fig : fpi_vs_mpi2 ] figure [ fig : fpi_vs_mpi2 ] shows the lattice results together with fitting curves . if we leave the coefficient of the chiral log term as a free parameter , the fit result is consistent with zero ( solid line ) , while the fit with the fixed coefficient @xmath12 gives a bad @xmath13 ( dashed line ) . a similar observation is obtained for the pcac relation @xmath14 @xcite . partially quenched chpt @xcite may be used to explicitly test the presence of the chiral logarithm . for non - degenerate pions composed of quarks with mass @xmath6 and @xmath15 ( @xmath16 denotes a valence quark ) , the low energy constants cancel out in the double ratios @xmath17 with @xmath18 , and only the chiral log terms remain . a parameter @xmath19 obtained as a coefficient of @xmath20 in ( [ eq : ratio_test_pcac ] ) and in ( [ eq : ratio_test_decay_constant ] ) are plotted as a function of @xmath21 in figure [ fig : ratio_test ] . we find that the results are much smaller than the prediction of the partially quenched chpt shown by a steep dashed line . [ fig : ratio_test ] since the lattice data do not support the presence of the chiral logarithm for the sea quark masses used in the simulation , the chiral extrapolation using the one - loop chpt formula is not fully justified . instead , we consider several possible functional forms to approach the chiral limit and discuss their associated uncertainty . our observation suggests that the mass region where the chiral logarithm becomes important is around or below 500 mev , and the chpt ceases to converge above that scale . then , a possible way to extrapolate the data including the effect of the chiral logarithm is to use a polynomial fit ( quadratic fit , for example ) above some energy scale @xmath22 , and then switch to the one - loop chpt formula below @xmath22 . an example is shown in figure [ fig : fpi_chiral_log_below_m ] for @xmath22 = 300 and 500 mev . the limit of @xmath22 = 0 mev corresponds to the usual polynomial fit . since the scale @xmath22 is unknown , the variation of several fit curves , about @xmath23 10% in the chiral limit , should be taken as systematic uncertainty . another possible functional form suggested by the adelaide - mit group @xcite is the one - loop chpt with a hard momentum cutoff @xmath22 . it amounts to replace the chiral log term @xmath24 by @xmath25 . changing the unknown `` cutoff '' scale @xmath22 from 0 to 1 gev , we obtain the similar size of uncertainty in the chiral limit . [ fig : fpi_chiral_log_below_m ] the su(3 ) breaking ratio in the decay constant @xmath26 may be obtained with a fit to partially quenched lattice data . our result from the above fit functions changes from 1.167(3 ) ( quadratic fit , corresponding to @xmath22 = 0 mev ) or 1.190(3 ) ( adelaide - mit fit , @xmath22 = 500 mev ) to 1.276(7 ) ( chiral log plus quadratic , corresponding to @xmath27 mev ) . although it is clear that our two - flavor qcd result is significantly higher than the quenched result 1.081(5)(17 ) @xcite and closer to the physical value 1.22 , the uncertainty is still sizable . for the heavy - light decay constant the prediction of chpt is available in the heavy quark limit for quenched , partially quenched and full qcd @xcite . the chiral logarithm appears with a definite coefficient but including an additional coupling constant @xmath28 describing the @xmath29 interaction . the chiral extrapolation of @xmath30 and @xmath31 including the chiral logarithm is shown in figure [ fig : fsqrtm_strange ] with two representative values of @xmath28 . as in the pion decay constant , the uncertainty in the chiral limit is enhanced by the chiral logarithm . the ratio @xmath32 is needed in the extraction of the ckm matrix element @xmath33 . since the bulk of systematic errors cancels in the ratio , one may expect better accuracy than the determination of @xmath34 solely from @xmath35 . our preliminary result varies from 1.24 ( quadratic fit ) to 1.38 ( chiral log , @xmath28 = 0.59 ) . it suggests that the ratio and its error can be significanly larger than the previous world average 1.16(4 ) . similar discussion , but using quenched data , has been made in @xcite . [ fig : fsqrtm_strange ] the chiral logarithm expected from chpt is not observed in the unquenched lattice data with @xmath36 greater than about 500 mev , which suggests that chpt may only be applied in smaller mass regions . the estimate using model functions for the chiral extrapolation leads to the uncertainty as large as @xmath2310% for the decay constants . this work is supported by the kek supercomputer project no . 79 ( fy 2002 ) , and also in part by the grant - in - aid of the ministry of education ( nos . 11640294 , 12640253 , 12740133 , 13135204 , 13640259 , 13640260 , 14046202 , 14740173 ) . is supported by the jsps research fellowship . j. gasser and h. leutwyler , annals phys . * 158 * ( 1984 ) 142 ; nucl . b * 250 * ( 1985 ) 465 . s. aoki _ et al . _ [ jlqcd collaboration ] , nucl . phys . proc . suppl . * 106 * , 224 ( 2002 ) . sharpe , phys . d * 56 * ( 1997 ) 7052 . golterman and k.c . leung , phys . d * 57 * ( 1998 ) 5703 . w. detmold _ et al . _ , phys . lett . * 87 * , 172001 ( 2001 ) . j. heitger _ [ alpha collaboration ] , nucl . b * 588 * , 377 ( 2000 ) . m. j. booth , phys . d * 51 * , 2338 ( 1995 ) . s. r. sharpe and y. zhang , phys . d * 53 * , 5125 ( 1996 ) . a. s. kronfeld and s. m. ryan , arxiv : hep - ph/0206058 .

we test the one - loop chiral perturbation theory formula on unquenched lattice data of pseudoscalar meson decay constants . the chiral extrapolation including the effect of the chiral logarithm is attempted and its uncertainty is discussed .

high proper motion sources have been studied for over two centuries now , as one might expect that the fastest moving sources would be the closest to our solar system . since mid xx century , comprehensive searches of high proper motion objects have been performed using photographic plates and later ccd cameras , to find the closest neighbors of the sun ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * recons , among others ) . however , cool objects are intrinsically faint in the optical and emit most of their light in the near infrared ( nir ) . therefore , the searches for unknown nearby sources gradually turn to the nir wavelengths taking advantage of improved camera sensitivities , spatial and temporal resolution @xcite , yielding in the last twenty years the discovery of over two thousand ultra - cool dwarfs . widening the color space of the searches has helped to improve the stellar and sub - stellar density estimates in the solar neighborhood , and the frequency of low - mass companions , among other questions @xcite . the wide - field infrared survey explorer ( _ wise ; _ @xcite ) satellite has revolutionized the field of brown dwarf with color based selections providing the discovery of hundreds of brown dwarfs @xcite . later , the multiepoch nature of the wise mission provided proper motions for over twenty thousand individual sources @xcite , including the discoveries of the third and fourth closest systems to the sun @xcite . these are the closest binary bd , and the coldest bd known . the wise mission also lead to the discovery of the first y dwarfs @xcite . thousands of other interesting objects were found , including young very low mass objects ( m , l and t type ; * ? ? ? * ) , several ultra cool sub - dwarfs @xcite , etc . these discoveries are helping to better understand the role of temperature , metallicity and evolution in very cool atmospheres . finally , a new sample of nearby k and m dwarfs ( d@xmath1100pc ) was created , well suited for exoplanet searches . here we report a follow up study of bright high proper motion objects detected by @xcite and @xcite , concentrating on objects within 50pc , wide co - moving binaries , and possible members of nearby young moving groups . the project summarized here was motivated by the recent discoveries of nearby stellar and sub - stellar objects ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * among others ) , the implications that these findings may have on the stellar census in the solar neighborhood @xcite , new results in the multiplicity of young low - mass stars and brown dwarfs in the field and young moving groups @xcite , and even the dynamic interactions of the solar system @xcite . the paper is organized as follows : section [ sec : data ] describes the catalogs and methods we used to generate a list of objects of interest , and the instruments we used to follow up and characterize them . in section [ sec : spectra ] the methods for classification are presented . section [ sec : dist ] describes the distance measurements and comparison of photometric and spectroscopic results . finally , in section [ sec : discussion ] we discuss individual sources , and present our conclusions in section [ conclusions ] . we started by analyzing bright sources in the new catalogs of high proper motion sources by @xcite and @xcite . we selected the brightest sources with the highest proper motion that were visible in april from the southern skies ( i.e. dec@xmath2 30 ) , with no previous derived spectral types and no data in the eso archive . we also performed a cross check with previously known high proper motion sources from simbad @xcite . if simbad returned a source within 15 showing a similar proper motion , we selected the brighter one between the simbad match and the wise object for spectroscopic follow up . we used a relaxed criterion to select candidate companions , i.e. a source with proper motion @xmath3 200masyr@xmath4 and a position angle of proper motion within @xmath030@xmath5 . these relaxed constraints caused the selection of some spurious pairs , and we discuss this issue in the coming sections . we also created a reduced proper motion diagram in order to find brown dwarf candidates . to derive the spectral types of the selected objects , we obtained spectra in the optical and in the near - infrared for wise j21210032 - 6239194 ( hereafter wise 2121 - 6239 ) . in addition , for the objects located within the area covered by the vista variables in the va lctea survey ( vvv ) we obtained photometry and astrometry . .sample of wise high proper motion objects selected for spectroscopic follow up with the efosc2 at the ntt at la silla observatory , and fire at baade at las campanas observatory . [ cols="^,^,^,^",options="header " , ] the photometric distance error from this work are @xmath620@xmath7 . @xmath8 : we recalculated the proper motion using 2mass and the wise all - sky epoch position with the highest s / n , as we find that the @xcite value for proper motion of lp 721 - 15 is inconsistent with the motion of the sources in the images , this object could be an unresolved binary , and then located furher away , see text for discussion ; @xmath9 : photometric distance ( spectral type derived from t@xmath10 using the relations in @xcite and on - line table maintained by e. mamajek , see text for the link ) ; @xmath11 : parallax distance ; @xmath12 : the paper cites the value for other object of the system ( sips 1910 - 4133a ) and is from photographic plates , our measurement is for sips 1910 - 4133b . the values of proper motion and distances with quoted errors were fitted from vvv data : ( 1 ) @xcite ; ( 2 ) @xcite ; ( 3 ) @xcite ; ( 4 ) @xcite ; ( 5 ) @xcite ; ( 6 ) @xcite ; ( 7 ) @xcite ; ( 8) @xcite ( 9 ) @xcite;(10 ) @xcite we estimated the distances to the objects and their companions from the derived spectral types and t@xmath10 from the 2mass @xmath13 and @xmath14 photometry using the absolute magnitudes from @xcite . for each object we calculated the distances for the two bands separately . the mean difference is 3.1@xmath151.7@xmath7 , with a maximum value of 5.6@xmath7 for object 2massj1403 + 0412 . our final estimate was the average of the two measurements . the sub - type error classification affects the distance estimation by a factor of 15@xmath7 , on average . the photometric uncertainties introduce errors within 1 - 3@xmath7 in the magnitude range we examine . therefore , estimated spectro - photometric distances will have an associated error of @xmath620@xmath7 , of the same order as the differences between our measurements and the values available in the literature ( table [ tab : dists ] ) . a comparison between the values for the distances in the literature and the values we obtain in this study is shown in fig . [ fig : compare_dist ] . the error bars correspond to 15@xmath7 of their distances in each axis . the only exception is 2mass j1546 - 5258 , as this source has three distances estimates in the literature , we plotted the average value of the distance and quote the standard deviation around the mean as the error . four objects in our list lie within the vvv survey footprint : 2mass j14035016 - 5923426 , 2mass j14040025 - 5923551 , 2mass j15464497 - 5254371 , 2mass j15463089 - 5258367 , but the last one is badly saturated in @xmath14 , and reliable positions could not be determined . therefore , we measure the parallaxes only for the first three sources using the procedure developed in the torino observatory parallax program described in @xcite . this has been adapted to the format of the vvv data products ( coordinates and photometry ) delivered by casu . a detailed description of the parallax code can be found in that paper , and we give here only a brief description of the steps involved . for the purpose of the astrometric reduction , we selected reference stars in a circle of radius 1 around the target which , given the high stellar density in these fields , provides an adequate number of reference stars ( above a hundred sources ) . these were selected among the highest s / n objects in the field of view , satisfying the condition that they appear in at least 80% of the frames , and do not exhibit large proper - motions . for example , for 2mass 15464497 - 5254371 the initial number of reference stars was 189 , but in the end only 121 of these were used to build the astrometric reference frame . in total we had 70 vvv images in @xmath16 , at various epochs ( see figure [ fig : par ] ) , spanning more than 3 years . four epochs were excluded from the solution due to their high residuals with respect to the mean solution . despite the relatively small parallax , the long baseline and the parallax factor coverage provide a small final error of 1.80mas . the conversion from relative to absolute parallax , which in any case is quite small ( less than 0.4mas ) , was computed using the galactic model by @xcite , extended to ir - wavelengths . the proper motion and parallaxes for these three sources are listed in table [ tab : dists ] , and fig . [ fig : par ] show the observations of each source and the best fits . the binary system 2mass j14035016 - 5923426 and 2mass j14040025 - 5923551 was on the border of two adjacent chips in the observational sequence and both objects are very bright often saturating in good seeing . this led to high centroiding errors and a sparse reference star set of only 41 and 55 objects for the two fields respectively . the two parallax solutions were therefore completely independent with different reference fields and with observations from different vircam chips . each observation has a quote positional error from the vista pipeline but there is a significant fraction of the total that is systematic in nature from transforming the observations to a common system . for this reason when calculating the errors of the derived parallax we do not use the individual observation errors . the final quote errors on the target parameters are obtained from the covariance matrix of the solution scaled by the error of unit weight . the observations used in the sequence are selected using the standard outlier rejection criteria developed in the torino program following these two criteria : 1 ) the average per coordinate error of the reference stars in a frame must be less than the mean error of all frames plus three standard deviations about the mean ; 2 ) the combined observed - minus - computed coordinate residual of each observation must be less than three times the sigma of the whole solution . the objects 2mass j15464497 - 5254371 , 2mass j14035016 - 5923426 and 2mass j14040025 - 5923551 had 4 , 2 and 2 observations rejected respectively by these criteria . extensive bootstrap - like testing was carried out on the observations to make sure the results were robust . this consisted of iterating through each observation and using as the primary base frame and thus making a solution that that incorporated slightly different sets of reference stars and a different starting point within the sequence . the solution chosen for publication is that one which is closest to the median of all solutions . the majority of the solutions ( @xmath1790@xmath7 ) were all within one sigma of the chosen solution . we determined physical properties and distances for twenty five stars , most of which are within d@xmath0140pc from the sun . only eight of these had previously determined distances . thirteen of the stars are located within d@xmath050pc . given that our selection of targets from the wise high pm list was not systematic , and that the list itself is not a complete census of the objects within d@xmath050pc we did not attempt to do a completeness analysis . the derived spectral types range from early - k to mid - m . the high transverse velocities of 2mass j173453.91 - 620654.6 , 2mass j140336.47 + 041239.5 and 2mass j145749.06 - 390451.1 and the co - moving binary pair 2mass j08291581 - 5850305 ( or l 186 - 122 ) and 2mass j08292286 - 5849209 , make them likely members of the galactic halo . the remaining objects probably belong to the disc population . we adopted binarity criteria requiring : common pms i.e. @xmath18 , where @xmath19 and @xmath20 are the angular separation ( in arcseconds ) and the difference in the magnitude of the proper motion vectors ( in arcseconds per year ) ( see sect . 2.2 . in * ? ? ? * ) , distances in agreement at the 2@xmath21 confidence level , and consistency between the spectral types and the apparent magnitudes . based on these criteria we identify six probable multiple system , five previously reported systems , and discovering one new one . the following stars are most likely real gravitationally bound systems : * 2mass j06571510 - 1446173 ( or lp 721 - 15 ) and 2mass j06571773 - 1446382 ; * 2mass j08291581 - 5850305 ( or l 186 - 122 ) and 2mass j08292286 - 5849209 ; * 2mass j14040025 - 5923551 ( or l 197 - 165 ) and 2massj14035016 - 5923426 ; * 2mass j15480325 - 5811119 ( or lhs 3119 ) and 2mass j15480441 - 5810533 ; * 2mass 19103460 - 4133443 ( or sips1910 - 4133a ) , 2mass 19104599 - 4133407(or sips1910 - 4133b ) and 2mass 19103359 - 4132505 ( or sips1910 - 4132c ) ; * 2mass j20044356 - 7123334 ( or ltt 7914 ) and 2mass j20043661 - 7123532 . their parameters are listed in table [ tab : bins ] . the triple system 2mass 19103460 - 4133443 , 2mass 19104599 - 4133407 b and 2mass 19103359 - 4132505 c was reported by @xcite . the last work discussed the possibility that this is actually a quadruple system because the magnitude of the component b is @xmath00.5 mag brighter than the c component . the spectro - photometric and sed based distances all agree within the 1@xmath21 errors ( see table [ tab : bins ] . when we compare the t@xmath10 obtained via sed fit of components b and c we find a difference of @xmath0200k which would be sufficient to explain the 0.5 mag difference ( that difference can be less given the error bars for temperature estimates ) . but we think that is not necessary to invoke a possible equal mass unresolved binary in component b to explain that difference in magnitude , and it is more likely to be explained as an effect of slight difference ( @xmath0100 - 200k ) in t@xmath10 . 2mass j14040025 - 5923551 and 2massj14035016 - 5923426 are co - moving . our spectra suggest this is a m3+m2.5 nearly equal mass binary . despite the saturation , we were able to measure a parallax from the multiepoch vvv data : 40.35@xmath227.19mas and 49.07@xmath227.06mas ( 24.78@xmath23pc and 20.4@xmath24pc ) respectively . 2mass j20044356 - 7123334 ( or ltt 7914 ) was observed by the radial velocity experiment ( rave ) in its fourth data release @xcite and described there . they derived a t@xmath10=4817 and @xmath25@xmath26=4 and metallicity [ fe / h]=0.05 , based on this we could assume the dwarf nature and apply the relations described before , deriving a distance of 108pc . they also obtained a radial velocity of rv@xmath27=@xmath2850.2@xmath222.2kms@xmath4 our best spectral type for this object is k2 , the reference t@xmath10 for a k2v star is @xmath05000k so our classification might be revised by one sub - type . applying photometric sed fitting we obtained a t@xmath10=4500k@xmath22100 , @xmath25@xmath26=5@xmath221 and metallicity [ fe / h]=0.3@xmath220.5 the photometric spectral type would be k4 , which is 2 spectral types later than the fit from our optical spectroscopy and one later than the type inferred by rave . for a range of photometric distances for k2-k4 types we obtain distances of 100 - 130 pc , implying a tangential velocity of 170 - 225 kms@xmath4 , typical for thick disk or halo objects . interestingly , the available data does not support a low metallicity for this object . for the co - moving star 2mass j20043661 - 7123532 the best photometric fit was t@xmath10=3100k@xmath22100 , @xmath25@xmath26=5.5@xmath221 and metallicity [ fe / h]=-1@xmath220.5 . assuming a spectral type of m4-m5 , the distance would be 80 - 120pc in agreement within the errors to the estimated value for 2mass j20044356 - 7123334 the objects are separated by 38.7on the sky , which corresponds to @xmath04200au for a distance of 110 pc . object 2mass j22275385 - 2337300 ( or lp 876 - 22 ) was observed by mistake , as the real new binary candidate was lp 876 - 1 and 2mass j22274199 - 2337283 . the observed target was classified as m2@xmath221 star , if we compute the distance we obtain 322pc which will put this object in a tangential velocity over 250 km / s and hence probably this object belongs to the halo population , but the distance might be considerably less if we consider that this object might be metal poor , as happened to be with previous sources . we perform the sed fitting to the co - moving pair lp 876 - 1 , 2mass j22274199 - 2337283 and they were classified as a m3.5-m8.5 , that would imply a distance between 40pc-50pc , but further observations are required to settle their true nature . the following objects are not real physical pairs , the argument for rejecting them as binaries are : the total proper motion , position angle of motion , spectral types compared to photometric spectral type and distances estimates for the primary and secondary , do not agree within the expected errors . * 2mass j07523088 - 4709470 and 2mass j07523777 - 4717270 ; * 2mass j09432908 - 0237184 and 2mass j09434389 - 0229570 ; * 2mass j10570299 - 5103351 and 2mass j10573037 - 5102190 ; * 2mass j11163668 - 4407495 ( or lhs 2386 ) and 2mass j11161471 - 4403252 ( see text ) ; * 2mass j13211484 - 3629180 and 2mass j13214404 - 3627316 ; * 2mass j13552455 - 1843080 ( or lp 799 - 1 ) and 2mass j13553933 - 1840586 ; * 2mass j14033647 + 0412395 and 2mass j14040651 + 0418532 ; * 2mass j14233830 + 0138520 and 2mass j14234208 + 0146235 ; * 2mass j14574906 - 3904511 and 2mass j14582414 - 3907504 ; * 2mass j15463089 - 5258367 and 2mass j15464497 - 5254371 . 2mass j07523088 - 4709470 and 2mass j07523777 - 4717270 : the second object moves @xmath02 times faster than the first object ( @xmath175@xmath21 outlier ) and the derived t@xmath10 of the secondary ( the fainter source ) is 200k higher . given the classification of m4.5v for the primary , this would place the secondary at least twice as far . 2mass j11163668 - 4407495 ( or lhs 2386 ) and 2mass j11161471 - 4403252 these were classified and found to be a co - moving pair observed by g.p . kuiper and re - classified in @xcite , he classified lhs 2386 as m3 : we obtain a best fit with m3.5v and m2.5 for 2mass j11161471 - 4403252 . @xcite list these sources as a co - moving binary , because the difference in total proper motion and position angle is small ( 4.5@xmath7 and 1.6@xmath7 respectively ) . also the probability of being a chance alignment is below 10@xmath7 based on the criteria from @xcite . however , the distance we derive for the two stars are only consistent at 3@xmath21 level . we obtain half the distance for lhs 2386 than for 2mass j11161471 - 4403252 . even if 2mass j11161471 - 4403252 is an equal mass binary that would place it around 44 - 50pc @xmath010 - 15pc farther away than we expect for lhs 2386 . in this hypothetical case , the distances will agree within the errors , that would mean that at 5.9of angular separation and distances between 44 and 50pc , the projected physical separation would be @xmath015 - 20 thousand au which would place them as one of the scarce population of very low mass and very wide binaries within 50pc . radial velocities of both stars and a more robust estimation of their distances are needed to disentangle their true nature , but we do not list it as a physical pair with the present evidence . 2mass j14574906 - 3904511 and 2mass j14582414 - 3907504 have pms that agree within 1@xmath7 when comparing the 2mass and wise positions , and the position angle of the motion differ only by one degree , these stars should probably be a real physical pair . on the other hand the inferred spectro - photometric distance for the primary is 328.7pc , while for the secondary using the sed fit we obtain a distance around 40pc for spectral type m7 . in @xcite they found that the primary object is actually a mi.0vi sub - dwarf and the distance derived by @xcite is 215.6pc even assuming this shorter distance the object is not consistent with a real binary and would imply a high tangential velocity of 432 km / s . finally , the angular separation of 445.9means that the projected physical separation at 215.6pc would be 0.466pc . combining these two arguments we argue this is not a real binary . 2mass j15463089 - 5258367 and 2mass j15464497 - 5254371 @xcite derived a distance 44@xmath29pc and estimated a t@xmath10=4669 - 4754k ( according to different fitting functions ) based on hipparcos ( tycho ) data for the first object . other two attempts to measure the distance from photometry are available from @xcite and @xcite they obtained 56pc ( no error bars ) and 77@xmath30pc respectively . our best sed fit yields 4700k , in agreement with @xcite , but our best spectrum fit is between k0v - k2v . if we assume this as the correct spectral type , then the photometric distance we obtain is between 68 - 77pc , for k0-k2 respectively . for the second object we were able to derive a distance based on parallax from vvv , as discussed in the previous section @xmath31= 23.5@xmath221.8 mas ( 42.6@xmath32pc ) . the parallax and proper motion of 2mass j15464497 - 5254371 are shown in figure [ fig : par ] and table [ tab : dists ] . the distance agrees very well with the value derived by @xcite , but is almost 3@xmath21 away from the photometric distance to 2mass j15463089 - 5258367 , in addition to the difference in the proper motion between the primary ( from tycho-2 catalog @xcite ) and our measurements for the candidate companion are too large . the evidence does not support that these two stars form a real binary , the parallax and more accurate proper motions for both sources will be measured by gaia mission , and then the true nature of these objects will be settled . while looking for the available photometry for the object nltt 37178 from the virtual observatory , we found a nearby source , classified as extragalactic ( photometric redshift 0.13 ) with photometry from sdss and galex . in the following years this object will be getting closer until the closest approach to the center , with the closest approach of 0.6 in 48 years . some extended emission in the galaxy is visible on sdss images , and we can expect that the nearby star can act as a lens for the outskirts of this galaxy . deep u and b band pre - lensing observations are needed to characterize the background source . search for microlensing events during the next few years may be promising . the most favorable filters to observe the galaxy will be u , b ( and/or uv filters from space ) . the microlensing event might help to understand the real nature and physical properties of this object , e.g. if it is an unresolved binary or hosts a planet . more robust distance estimations ( parallax ) are necessary , to better constrain the einstein radius for the system . we assumed a mass of 0.2m@xmath33 for the lens and distances of 40pc and 543.9mpc(co - moving radial distance ) for lens and source respectively , and obtain a crude estimate of the einstein radius of 6.4mas . as the lensed source is resolved , we might expect variations on the light curve due to lens magnifying different parts of the galaxy . we performed spectroscopic follow up for over twenty new high proper motion objects found by the wise satellite , and looked for possible new wide binary companions . we found one t2 dwarf probably located within 15pc . we obtained optical spectral types and photometric distances for 24 objects , as well as parallax measurements for 3 of them . we present some additional evidence for six co - moving objects that are likely physical pairs , two of them are new binary candidates . four objects are probable members of the galactic halo given their large tangential velocities . most of the objects analyzed in this study are located within 75pc from the sun , and are bright enough for further follow up and search for planets using state of the art and upcoming nir instruments . the use of relatively loose constraints when selecting possible wide co - moving companions given the in - homogeneity of resources available in the literature prove useful to find new co - moving stars . this causes many false positives , but they can be eliminated a posteriori using multiple arguments , combining proper motion , physical separation and spectral energy distributions in the calculation of photometric distances . it is also important to emphasize the relevance of obtaining distances or spectral types for discriminating chance alignments from real wide binaries ( or co - moving stars ) , even when the proper motion and position probabilities are very low we show two examples in this paper where the hypothesized binaries are most likely not physically related . we also discussed a likely microlensing event due to a star passing in front of a background galaxy . the number of predicted microlensing events of this type will be more frequent as more hpm low mass objects are found in high density environments like the galactic bulge and inner disk , but also with background galaxies . although the lens candidate we present here is not predicted to pass in front of the center of the galaxy the event can be used to study the lens for unresolved companions and planets . in this case this maybe particularly difficult as the lens goes through different parts of the galaxy in short time scales magnifying regions of intrinsically different brightness . these events can also be used to make more detailed structural studies of galaxies at low redshift . c@ c@ c@ c@ c@ c@ c@ c@ c@ star i d & sp . type & t@xmath10 & j@xmath34 & @xmath35 & dist . & @xmath36cos(@xmath37 ) & @xmath38 & remarks + & ( reference ) & [ k ] & [ mag ] & [ ] & [ pc ] & [ mas ] & [ mas ] & + + 2mass j06571510 - 1446173 & m4 & 3300 & 10.678 & 43.6 & 29.1 & 70 & -270 & see * * in table [ tab : dists ] + + 2mass j06571773 - 1446382 & & 3000 & 12.751 & & @xmath0 40 & 61 ( 1 ) & -268 ( 1 ) & + + 2mass j08291581 - 5850305 & k7 & 4200 & 10.206 & 88.49 & 80.3 & 363 ( 4 ) & -81(4 ) & halo wide binary + + 2mass j08292286 - 5849209 & & 2900 & 14.550 & & @xmath0110 & 355 & -73(1 ) & + + + 2mass j14040025 - 5923551 & m2.5 & 3600 & 10.219 & 78.05 & 24.78@xmath23 & 8.27@xmath225.9 & -494.5@xmath225.1 & vvv parallax + + 2mass j14035016 - 5923426 & m3 & 3500 & 10.258 & & 20.4@xmath24 & 11.5@xmath224.6 & -492.2@xmath224.3 & + + + 2mass j15480325 - 5811119 & m1.5(1 ) & 3467(7 ) & 8.379 & 20.7 & 23.5(1 ) & -593 ( 1 ) & -250 ( 1 ) & v@xmath39=30.9 km / s ( 1 ) + & & & & & 33.6(2 ) & & & + + 2mass j15480441 - 5810533 & m3.5 & 3400 & 10.169 & & 29.5 & -503 ( 3 ) & -207 ( 3 ) & + + 2mass 19103460 - 4133443 & & 3500 & 9.851 & 127.78(a - b ) & 35.9 & 72 ( 5 ) & -738(5 ) & triple system + & & & & 54.99(a - c ) & & & & + + 2mass 19104599 - 4133407 & m4 & 3300 & 10.610 & 147.90(b - c ) & 22.8 ( 6 ) & 91 ( 5 ) & -742 ( 5 ) & component b + & & & & & 27.6 & & & + + 2mass 19103359 - 4132505 & & 3100 & 11.147 & & 29.2 & 68 ( 5 ) & -735 ( 5 ) & component c + + \(1 ) @xcite , ( 2 ) @xcite , ( 3 ) @xcite ( 4 ) @xcite , ( 5 ) @xcite , ( 6 ) @xcite , ( 7 ) @xcite . the values without reference were calculated or derived in this work , the photometric distances have a @xmath620@xmath7 error ( spectral type and effective temperature uncertainties are the main source of error ) . the authors thank the referee dr . nigel hambly for his comments and suggestions that helped to the improve the quality of the paper . j.c.b . , d.m . , acknowledges support from : phd fellowship from conicyt , project fondecyt no . d.m and r.a.m acknowledges project support from basal center for astrophysics and associated technologies cata pfb-06 . support for j.c.b , d.m , r.a.m , m.g and r.k is provided by the ministry of economy , development , and tourism s millennium science initiative through grant ic120009 , awarded to the millennium institute of astrophysics , mas . acknowledges financial support from the proyecto fondecyt de iniciacin 11140572 . part of this work was completed at the eso headquarters , garching bei mnich , with support from the eso director gerneral s discretionary fund program . r.a.m . acknowledges eso / chile for hosting him during his sabbatical - leave throughout 2014 . mg acknowledges support from joined committee eso and government of chile 2014 . this research has made use of the simbad database , operated at cds , strasbourg , france this publication makes use of data products from , the two micron all sky survey , which is a joint project of the university of massachusetts and the infrared processing and analysis center / california institute of technology , funded by nasa and nsf . this 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the census of the solar neighborhood is almost complete for stars and becoming more complete in the brown dwarf regime . spectroscopic , photometric and kinematic characterization of nearby objects helps us to understand the local mass function , the binary fraction , and provides new targets for sensitive planet searches . we aim to derive spectral types and spectro - photometric distances of a sample of new high proper motion sources found with the wise satellite , and obtain parallaxes for those objects that fall within the area observed by the vista variables in the va lctea survey ( vvv ) . we used low resolution spectroscopy and template fitting to derive spectral types , multiwavelength photometry to characterize the companion candidates and obtain photometric distances . multi - epoch imaging from the vvv survey was used to measure the parallaxes and proper motions for three sources . we confirm a new t2 brown dwarf within @xmath015pc . we derived optical spectral types for twenty four sources , mostly m dwarfs within 50pc . we addressed the wide binary nature of sixteen objects found by the wise mission and previously known high proper motion sources . six of these are probably members of wide binaries , two of those are new , and present evidence against the physical binary nature of two candidate binary stars found in the literature , and eight that we selected as possible binary systems . we discuss a likely microlensing event produced by a nearby low mass star and a galaxy , that is to occur in the following five years . [ firstpage ] astrometry , parallaxes , binaries , brown dwarfs , techniques : spectroscopic .

the classical scale invariance formalism has been considered as a ( approximate ) symmetry to protect various putative scales within nature from a mutual quadratic destabilization , posing as an economical solution to the hierarchy problem @xcite . in this treatment , we consider a minimal extension of the standard model ( sm ) by adding a complex gauge singlet scalar and three flavors of the right - handed majorana neutrinos , while imposing the @xmath0 as well as the classical scale symmetry . the model is embedded with the renormalizable framework of agravity @xcite , which also respects the imposed symmetries . while the imposed @xmath0 invariance prevents the singlet pseudoscalar from decaying , rendering it a stable dark matter candidate , nonzero masses for the sm neutrinos may be generated using the seesaw mechanism @xcite . a nonzero singlet vacuum expectation value ( vev ) is dynamically generated _ a la _ coleman - weinberg @xcite , which subsequently induces the ( reduced ) planck scale via the scalar non - minimal couplings , as well as the weak scale by means of the higgs portal operators , accommodating the discovered 125 gev state @xcite . it also generates appropriate mass terms for the dark matter , and the right - handed majorana neutrinos . the scale symmetry protects the various scales from a mutual quadratic destabilization . we explore the slow - roll inflationary paradigm within the introduced framework , by identifying the pseudo - nambu - goldstone boson of the ( approximate ) scale symmetry with the inflaton field , and construct the one - loop effective potential for this field . trans - planckian field excursions are accommodated within the agravity framework . we compute the slow - roll parameters and the inflationary observables , and demonstrate the viability of the inflationary paradigm within the introduced model , in accordance with the latest observational data , published by the planck collaboration @xcite . we minimally extend the sm content by introducing one complex gauge singlet scalar . the full scalar potential is conjectured to respect the @xmath0 as well as the scale symmetry , and contains the following operators in its most general form @xmath1 where , @xmath2 and @xmath3 are the sm doublet and singlet , respectively , @xmath4 } \ .\ ] ] the nonzero vevs of the @xmath0-even scalars are generated dynamically ( @xmath5 gev ) ; in particular , once the singlet vev , @xmath6 , is induced via the coleman - weinberg mechanism @xcite , it is transmitted to the sm sector via the higgs portal operators with the coefficients @xmath7 , leading to a mass term for the higgs field , @xmath8 , triggering the spontaneous breaking of the electroweak symmetry @xmath9 interestingly , the @xmath0 symmetry protects the pseudoscalar singlet , @xmath10 , from decaying , rendering it a stable dark matter candidate @xcite . in addition , one may include three flavors of the right - handed majorana neutrinos , @xmath11 , in order to give masses to the sm neutrinos by means of the see - saw mechanism @xcite . assuming a @xmath0- and scale - symmetric singlet sector , the right - handed majorana neutrino masses may be generated via their yukawa interactions with the singlet @xmath12 } -\frac{1}{2}y_{n } { \left ( s + s^ * \right ) } \bar{\mathcal{n}}^{i}\mathcal{n}^{i } \ , \ ] ] where , for simplicity , a flavor - universal yukawa coupling , @xmath13 , is considered . in order to consistently account for the gravitational contributions , pertinent to the inflationary paradigm and the trans - planckian excursion of the inflaton , the described minimal extension of the sm is entirely embedded within the renormalizable , and @xmath0- and scale - symmetric framework of the agravity @xcite . in the jordan frame , the full action is of the form @xmath14 where , in addition to the previously introduced scalar potential , the majorana neutrino and the remaining sm lagrangians , one includes the non - minimal couplings of the scalars to the curvature , @xmath15 and @xmath16 , as well as the pure agravity ( higher - derivative ) operators with the dimensionless couplings @xmath17 . while the @xmath18 operator gives rise to a scalar graviton , the @xmath19 operator ( the weyl term ) produces the usual massless spin-2 graviton together with its `` lee - wick '' ( lw ) partner @xcite , @xmath20 , which is a massive spin-2 ghost with a negative norm . one notes that the einstein - hilbert term , @xmath21 , is absent in due to the classical scale invariance requirement . however , the non - minimal couplings induce such a term , once the nonzero vevs are obtained . therefore , both the ( reduced ) planck and the weak scales are induced dynamically via the vev of the singlet within the current framework . performing the local weyl transformation , one can show that the einstein frame action , with the canonical einstein - hilbert term manifestly exhibited , reads @xmath22}^2 \right\ } } \label{ve},\end{aligned}\ ] ] with @xmath23 the scalar graviton in its `` conformal '' definition , and @xmath24 . the physical vev of the system can be pinpointed by employing the gildner - weinberg procedure @xcite , where one initially minimizes the tree - level potential . this defines the flat direction of the potential at a particular energy scale , @xmath25 . the one - loop corrections then predominantly matter only along this flat direction , where they lift the flatness and specify the true vacuum . performing the tree - level minimization , one obtains @xmath26 where , the ellipses correspond to one - loop contributions due to fixing the cosmological constant to zero at one - loop . since a classical scale - invariant treatment of the cosmological constant problem is outside the scope of this work , such terms should be simply neglected . the relations are examples of the dimensional transmutation phenomenon , and demonstrate the absence of a quadratic sensitivity of various scales to one another . in particular , the weak scale is not quadratically sensitive to the singlet vev scale , since the @xmath27 mixing coupling is not a free parameter . the three @xmath0-even scalars of the framework , @xmath28 , are mixed with one another in the quadratic part of the lagrangian , and can be rotated into the physical mass basis using an orthogonal @xmath29 matrix with the mixing angles @xmath30 , @xmath31 where , @xmath32 corresponds to the physical 125 gev state discovered by the lhc @xcite , @xmath33 is massless at tree - level , serving as the pseudo - nambu - goldstone boson of the ( approximate ) scale symmetry , and @xmath34 denotes the physical scalar graviton . one , subsequently , obtains along the flat direction ( c.f ) @xmath35 once more , signifying the absence of a quadratic destabilization among the separate dynamically - induced scales . the pseudo - nambu - goldstone boson of the ( approximate ) scale symmetry , @xmath33 , can be expressed in terms of the radial combination of the field basis scalars @xmath36 the kinetic term of this scalar has a non - canonical form , @xmath37 } $ ] ( c.f . ) ; however , using the field redefinition ( c.f . and ) @xmath38 its kinetic term can be brought into the canonical form , @xmath39 . note that the kinetic term of the @xmath33 boson becomes canonical at the minimum ; hence , one can set @xmath40 . along the flat direction , the one - loop contributions to the potential can be written according to @xmath41 } \ , \label{v1fin } \\ \mathcal m^{4}\equiv&\ , 5 m_{\theta}^{4}+ m_{\kappa}^{4}+ m_{\chi}^{4 } -6 m_{\mathcal n}^{4}+ m_{h}^{4}+6m_{w}^{4 } + 3m_{z}^{4 } -12m_{t}^{4 } \label{bmodel } \ , \end{aligned}\ ] ] where , the heavy sm degrees of freedom in the loop are also taken into account for completeness . the one - loop corrections is negative at its minimum ; however , the tree - level potential is non - vanishing along the flat direction , due to the gravitational contributions , and can be utilized to fix the minimum of the full one - loop potential ( the cosmological constant at one - loop ) to zero . therefore , one obtains for the full one - loop effective potential along the flat direction @xmath42 } \ .\ ] ] this potential induces a radiative mass term for the @xmath33 scalar @xmath43 and is bounded from below ( and positive definite ) for @xmath44 , implying @xmath45 which , also yields a non - tachyonic mass for the @xmath33 boson . the pseudo - nambu - goldstone boson of the ( approximate ) scale symmetry in its canonical form , @xmath46 , may be identified with the inflaton of the framework , possessing the one - loop effective potential along the flat direction . from this potential the slow - roll parameters can be easily computed @xmath47 where , the field subscripts denote taking the appropriate derivative(s ) of the potential with respect to the argument . additionally , the number of @xmath48-foldings is easily found using the slow - roll equations of motion , and is given by @xmath49 + \frac{\sigma_{c}^2}{8 \bar{m}_\text{p}^2 } \right\ } } \bigg|_{\sigma_{c , e}}^{\sigma_{c , i } } \ , \end{split}\ ] ] with @xmath50 and @xmath51 the values of the @xmath46 inflaton at the beginning and at the end of the inflation , respectively , @xmath52 the exponential integral ( @xmath53 ) , and @xmath54 the logarithmic integral ( @xmath55 ) . the inflation comes to an end , once the condition @xmath56 is reached . the inflaton field value at the horizon crossing point , @xmath57 , can subsequently be obtained from for a given @xmath48-folding number , as a function of the mixing angle @xmath58 . the amplitude of the scalar perturbations , @xmath59 , the scalar spectral index , @xmath60 , the tensor - to - scalar ratio , @xmath61 , and its running , @xmath62 , are defined according to @xmath63 with the subscripted asterisk denoting the field value @xmath57 . given the inflaton potential and the slow - roll parameter definitions , these quantities are easily determined , as functions of @xmath64 , @xmath65 , and the field value @xmath50 at the horizon exit . plane , along with the predictions of the current framework for both the small and large field inflation scenarios ( dashed lines ) . two @xmath48-folding number benchmarks , @xmath66 , are selected and the full range of the mixing angle , @xmath67 , is considered . the diagonal line labeled as `` @xmath68-model '' denotes the chaotic inflation scenario , in the @xmath69 limit . [ fig : nsr ] ] the predictions of the current framework for these observables are displayed within the @xmath70 plane of fig . [ fig : nsr ] , for the small and the large field inflation scenarios . several benchmark values of the mixing angle are exhibited for @xmath66 . in addition , the observational constraints from several planck collaboration published data sets at 68% and 95% c.l . @xcite are incorporated within the plot . in the limit @xmath69 , the inflaton potential reduces to the ordinary chaotic inflation scenario near its minimum @xmath71 with the limiting values @xmath72 therefore , we have @xmath73 for @xmath74 , and @xmath75 for @xmath76 . in contrast , the @xmath77 limit is for the most part compatible with the observational data within the small field inflation scenario . while in the large field inflation scenario the potential becomes increasingly steeper , the small field inflation scenario is characterized by a relatively flat potential with a very small @xmath78 slow - roll parameter . hence the @xmath79 slow - roll parameter dominates the behavior of the spectral index ( c.f . ) , and one obtains for the leading - order behavior in this scenario @xmath80 a minimal extension of the sm has been developed by introducing one complex singlet scalar and three ( mass - degenerate ) right - handed majorana neutrinos . the singlet sector is postulated to respect the @xmath0 as well as the classical scale symmetry , and is embedded within the renormalizable agravity framework . the @xmath0 symmetry prohibits a decay of the singlet pseudoscalar , rendering it a stable dark matter candidate , whereas the sm neutrinos obtain nonzero masses via the seesaw mechanism . the singlet vev , dynamically generated via the coleman - weinberg mechanism , simultaneously induces the weak scale ( via the higgs portal operators ) as well as the ( reduced ) planck scale ( via the scalar non - minimal couplings to the curvature ) . furthermore , it generates mass terms for the right - handed majorana neutrinos ( via yukawa interactions ) and the dark matter . it is shown that the scale symmetry protects the various scales from a mutual quadratic destabilization . identifying the pseudo - nambu - goldstone boson of the ( approximate ) scale symmetry of the model , in its canonical form , with the inflaton field , its one - loop effective potential is constructed , and the viability of the slow - roll inflationary paradigm is validated ; in particular , a trans - planckian field excursion is consistently accommodated utilizing the agravity framework . it is demonstrated that small field inflation scenario can be rendered fully compatible with the latest observational data for the suitable @xmath48-folding numbers , within the minimal classically scale invariant framework . w. a. bardeen , `` on naturalness in the standard model '' , fermilab - conf-95 - 391-t , `` beyond higgs '' , fermilab - conf-08 - 118- t ; h. aoki and s. iso , phys . d * 86 * , 013001 ( 2012 ) [ arxiv:1201.0857 [ hep - ph ] ] . p. minkowski , phys . b * 67 * ( 1977 ) 421 ; t. yanagida , in _ proceedings of the workshop on the unified theory and the baryon number in the universe _ ( o. sawada and a. sugamoto , eds . ) , kek , tsukuba , japan , 1979 , p.95 ; m. gell - mann , p. ramond , and r. slansky , _ supergravity _ ( p. van nieuwenhuizen et al . , eds ) , north holland , amsterdam , 1979 , p.315 ; r. n. mohapatra and g. senjanovi , phys . rev . * 44 * ( 1980 ) 912 ; j. schechter and j. w. f. valle , phys . d * 22 * , 2227 ( 1980 ) .

in this talk , i present the minimal classically scale - invariant and @xmath0-symmetric extension of the standard model , containing one additional complex gauge singlet and three flavors of right - handed majorana neutrinos , incorporated within a renormalizable framework of gravity , consistent with these symmetries ; the agravity . i particularly focus on the slow - roll inflationary paradigm within this framework , by identifying the pseudo - nambu - goldstone boson of the ( approximate ) scale symmetry with the inflaton field , constructing its one - loop effective potential , computing the slow - roll parameters and the inflationary observables , and demonstrating the compatibility of the small field inflation scenario with the latest planck collaboration data sets .

an extensive class of financial time series models is based on two interrelated processes . in particular , many models include an unobservable part that reflects a certain regime or the volatility of the process . a well - known example is given by the garch family . it is typically applied in order to model financial log returns where the unobservable volatility process drives the observable price of an asset . in the following , let @xmath1 denote such a process and @xmath0 its unobservable counterpart . let both @xmath1 and @xmath0 be univariate . a common approach for the analysis of the extremal behavior of such interrelated processes focusses on the joint sequence @xmath3 . more precisely , the process is studied under the condition @xmath4 for @xmath5 and an arbitrary norm @xmath6 on @xmath7 . the connection of this approach to the concept of multivariate regular variation has been discussed extensively in @xcite . we shall follow a more natural point of view where the process @xmath8 is unobservable . that is , we analyze its limiting behavior under the ( observable ) event @xmath9 as @xmath5 . hence , for @xmath10 we focus on the limit distribution of @xmath11 as @xmath12 . we assume @xmath8 to be of a simple markovian structure , i.e.@xmath13 for some measurable mapping @xmath14 and some sequence @xmath15 of i.i.d . innovations on a measurable space @xmath16 . additionally , we will require the sequence of innovations @xmath17 to be independent of @xmath18 for all @xmath19 . based on @xmath8 and the innovations let the observable process be given by @xmath20 for some measurable mapping @xmath21 with @xmath22 . we will always assume that a stationary solution to and exists . now , by @xmath23 as well as by @xmath24 and @xmath25 we have a simple , but flexible model for the dependence between @xmath1 and @xmath0 . however , note that from the recursive definition in we may find a function @xmath26 such that @xmath27 , @xmath28 , with @xmath29 . hence , for ease of notation we may in the following assume that there exists an @xmath30 such that @xmath31 we may interpret @xmath32 and @xmath8 as a generalized hidden markov model which incorporates a large class of models for financial time series , cf . @xcite for the general definition . we shall discuss the garch@xmath2 process ( cf . @xcite ) as a specific example , i.e.@xmath33 and @xmath34 for suitable constants @xmath35 and @xmath36 . here , the sequence @xmath37 is the observable part , e.g. a model for financial log returns , and the series @xmath38 describes the conditional standard deviation ( volatility ) of the process at time @xmath39 . in the basic setup the innovation sequence @xmath15 is assumed to be i.i.d . standard normal . note that the above garch(1,1 ) model satisfies and for @xmath40 we remark that for @xmath41 in the garch@xmath2 setup includes the arch@xmath42 model as a special case , cf . @xcite . for further examples , cf . also remark [ examples ] . it is well - known @xcite that under quite general assumptions about the distribution of @xmath43 , @xmath44 , and about the size of the parameters @xmath45 , @xmath46 and @xmath47 the stationary solutions to and share a common regularly varying ( heavy tailed ) behavior . a heavy tailed behavior of both the volatilities and the log returns is a desirable feature of financial time series as it agrees with commonly accepted stylized facts . accordingly , we will assume regular variation for the stationary solutions to both @xmath48 and @xmath49 , cf . condition 1 below . as it is not clear whether the limit in exists we will discuss those questions in more detail in sections [ firstpart ] and [ uniqueness ] . in section [ specialform ] we will show that under some further assumptions the limiting distribution in has a particularly simple form which can be seen as an extension to similar findings in @xcite . more precisely , outside of the period @xmath50 our results will allow for a representation of the limit process in as a multiplicative random walk , cf . proposition [ mainprop ] . heuristically , if we consider the example given by and , this is the case since a large value of @xmath51 stems most likely from a large value of @xmath52 as the tail of @xmath52 is heavier than the tail of @xmath53 . now , for a large value of @xmath54 , @xmath55 behaves asymptotically like @xmath56 . at the same time , for @xmath57 , the distribution of @xmath53 is influenced by the extremal event of @xmath58 being large while all future @xmath59 are not influenced by this condition . in section [ mrv ] we will analyze connections of our results with multivariate regular variation of the time series @xmath60 . the theoretical results are applied to the garch@xmath2 model in section [ garch ] . they allow for a simple representation of the tail - chain in this case ( cf . proposition [ easysimulation ] ) and are used for monte carlo evaluations of some extremal characteristics in section [ simulation ] . in the following , we will assume that the stationary distribution of @xmath61 , @xmath39 , cf . , is regularly varying with index @xmath62 and that it is tail - balanced , i.e. the following condition holds : @xmath63.\ ] ] we will study the joint extremal behavior of and under the assumption that @xmath64 shares the tail behavior of @xmath65 , i.e. there exists a constant @xmath66 such that @xmath67 analogous to condition 1.a we say that * condition 1.b * holds if the time series @xmath32 satisfies with @xmath64 in place of @xmath65 ( with a possibly different value of @xmath68 ) . furthermore , if both conditions and are satisfied we will say that * condition 1 * holds . [ sequenceistight ] let @xmath8 and @xmath32 be stationary time series given by and and let and be satisfied . then , the family @xmath69 of conditional distributions is tight for all @xmath70 . . then @xmath72 by and the r.h.s . is bounded by @xmath73 for @xmath74 large . therefore , a weak accumulation point of the family of distributions exists . the following lemma shows , however , that it is not necessarily unique . [ nonuniquenesslemma ] there exist time series @xmath8 and @xmath32 of the form and such that condition 1 is satisfied but has more than one weak accumulation point . let @xmath75 i.e. @xmath76 . with @xmath77 we have @xmath78 , so @xmath79 as well . let @xmath80 and @xmath81 for a continuous function @xmath82 to be described below . thus @xmath83 . by independence , any weak accumulation point of @xmath84 equals @xmath85 for some weak accumulation point @xmath86 of @xmath87 , where @xmath88 , @xmath89 denotes the dirac measure in @xmath90 . with @xmath91 we will construct @xmath92 such that @xmath93 has a continuum of weak accumulation points . + let @xmath94 . for the sequence @xmath95 each interval @xmath96 $ ] is mapped onto itself by @xmath92 . on @xmath97 $ ] it interpolates linearly between the values @xmath98 and @xmath99 , and on @xmath100 $ ] between @xmath101 and @xmath98 . the function @xmath92 can be extended on each interval @xmath102 $ ] such that @xmath103 ) \subset [ z_{i},5z_{i } ] $ ] and @xmath104 . the details of the construction of @xmath92 are given in the appendix . we first show that two different weak accumulation points exist . since @xmath105 is nonnegative we drop the absolute value . along the sequence @xmath106 , we have @xmath107 . thus , for @xmath108 it holds that @xmath109 @xmath110 . hence , @xmath111 for all @xmath112 . + now , suppose that @xmath113 . by construction @xmath114 implies @xmath115 , thus @xmath116 . this leads ( at least along a subsequence ) to a different weak limit . for @xmath117 one still has @xmath118 , since @xmath119 implies @xmath120 . + adapting the above argument shows that each sequence @xmath121,@xmath122 , leads to a different weak limit @xmath123 ( at least along a subsequence ) with @xmath124 and @xmath125 for all @xmath126 . in order to study the properties of the limit in in more detail we will make further assumptions about the functional form of @xmath127 and @xmath23 which relate to those given in @xcite . there , the single time series @xmath8 is analyzed and both the existence and the form of the weak limit @xmath128 for all @xmath129 are discussed . under condition 1.a and under an additional assumption ( cf . condition 2.a below ) this so - called tail chain bears resemblance to a multiplicative random walk . the idea behind this condition and the following proposition [ segers ] is that for a stochastic process which behaves roughly like @xmath130 as @xmath131 for a suitable function @xmath132 , the whole process behaves like a multiplicative random walk given an extreme event at time 0 . note that our condition 2.a is a slightly stronger version of ( * ? ? ? * condition 2.2 ) that will allow to simplify some of our proofs in section [ specialform ] . there exists a function @xmath133 such that @xmath134 for all @xmath135 . here , @xmath136 where @xmath137 denotes the indicator function . this condition allows for the case @xmath138 with trivial limit distributions of . [ examples ] if we identify @xmath139 with the volatility process of a financial time series , there exist several examples which satisfy condition 2.a : * `` standard '' garch(1,1 ) models , cf . , with @xmath140@xmath141 , * gjr - garch(1,1 ) models ( cf . @xcite ) which reflect asymmetric behavior of the volatility process with @xmath142 here , @xmath143 , * sr - sarv ( stochastic volatility ) models defined by @xmath144 and volatility sequence @xmath145 or @xmath146 where @xmath147 with @xmath148 is i.i.d . , with a possible dependence between @xmath149 and @xmath43 for a fixed value of @xmath150 ( cf . ( in this case the space @xmath151 of innovations @xmath152 is to be taken as @xmath7 . ) here , @xmath153 or @xmath154 , respectively . if we identify @xmath8 with a volatility sequence , then @xmath155 and the dependence of @xmath132 on @xmath156 is not necessary . for general hidden markov models , however , the extremal behavior of @xmath157 may differ for the cases @xmath158 or @xmath159 . the following proposition puts the aforementioned heuristic @xmath160 for @xmath131 on solid ground . it is taken from @xcite and will be fundamental to our subsequent analysis . here and in the following , `` @xmath161 '' denotes weak convergence of probability measures . [ segers1 ] let @xmath8 ( not necessarily stationary ) be given by and let conditions 1.a and 2.a hold . then for @xmath162 , as @xmath163 , @xmath164 with @xmath165 where @xmath166 , and @xmath167 @xmath168 @xmath169 @xmath170 are independent with * @xmath171 , i.e. @xmath172 * @xmath173 , * @xmath174 are i.i.d . with @xmath175 and @xmath176 note that by embedding the @xmath177 , @xmath178 , the formulation of proposition [ segers1 ] differs slighty from its analog in @xcite . the proof is analogous to the proof of ( * ? ? ? * theorem 2.3 ) and uses the continuous mapping theorem . the joint limit distribution in proposition [ segers1 ] will be an important building block for the derivation of . but in order to derive the limit we also need to specify the behavior of @xmath179 _ before _ the extremal event @xmath180 . going backwards in time , things are not as simple as before . to illustrate this , think of the process @xmath181 with @xmath182 . now , a large value of @xmath183 may either be due to a large value of @xmath184 or due to a large value of @xmath185 . however , if we assume stationarity of @xmath8 in addition to the assumptions of proposition [ segers1 ] , then note the following : for @xmath186 it holds that @xmath187 for all @xmath188 , where @xmath189 . now , for @xmath190 this implies @xmath191 where the r.h.s . converges to @xmath192 since @xmath193 , where @xmath194 denotes the uniform distribution on @xmath195 ( third last equation ) and @xmath196 ( penultimate equation ) . thus , if we assume for the moment that a limit @xmath197 exists , then by a similar reasoning it holds that @xmath198 for all @xmath190 . by rescaling it holds for all @xmath186 . in fact , one can show that for all laws @xmath199 which may evolve in proposition [ segers1 ] from a stationary markov chain @xmath8 there exists a law @xmath200 satisfying and that this law is sufficient to determine the whole backward limit process which is markovian like the forward limit process . details can be found in @xcite , we state the main definitions and results below . a precise form of the limit process for the garch(1,1 ) case is given in section [ garch ] . [ bftc ] a time series @xmath201 is said to be a back - and - forth tail chain with index @xmath202 and forward transition law @xmath86 , denoted by bftc@xmath203 , if * @xmath204 with @xmath205 , * @xmath206 is _ adjoint _ to @xmath86 , i.e. @xmath207 * for all integer @xmath208 and all real @xmath209 , @xmath210 ( cf . proposition [ segers1 ] for the definition of @xmath211 ) where @xmath212 and @xmath213 are independent with @xmath214 * for all integer @xmath208 and all real @xmath215 , @xmath216 where @xmath217 and @xmath218 are independent with @xmath219 [ segers2 ] let @xmath8 be a stationary time series given by and let conditions 1.a and 2.a hold . then , for all @xmath220 , as @xmath163 , @xmath221 with * @xmath222 independent of @xmath201 , * @xmath201 is a bftc@xmath203 where @xmath223 with @xmath224 as in proposition [ segers1 ] . propositions [ segers1 ] and [ segers2 ] show that the assumption about the asymptotic behavior of @xmath127 leads to a very simple form of the tail process for @xmath8 . the key ingredient is the connection between the forward and the backward limit process which is stated in . although this equation is sufficient for a unique determination of @xmath225 from @xmath199 it appears to be cumbersome for specific applications . it is shown in ( * ? ? ? * equation ( 3.4 ) ) that one may derive the law of @xmath226 from that of @xmath224 by noting that @xmath227f(0)\end{aligned}\ ] ] for integrable functions @xmath92 , and @xmath228 such that @xmath229 . in order to discuss similar results for the above case of two connected time series we will introduce an analogous condition for @xmath23 . there exists a function @xmath230 such that @xmath231 for all @xmath232 . this condition allows for the case @xmath233 which is meaningless in practice . if conditions 2.a and 2.b hold we will say that * condition 2 * is satisfied . note that for all examples given in remark [ examples ] , condition 2.b holds due to the multiplicative form of @xmath234 . in the following , we will investigate the uniqueness of the accumulation point of under conditions 1 and 2 . it will turn out that the behavior of the univariate distribution @xmath235 as @xmath5 leads to a sufficient condition . [ onehelpsforall ] let @xmath8 and @xmath32 be stationary time series given by and and let conditions 1 and 2 hold . equivalent are * the weak accumulation point of is unique , and @xmath236 has no mass in zero , * there exists a weak accumulation point @xmath237 of @xmath235 with @xmath238 a.s . consequently , the uniqueness of the limit in may well be derived from _ any _ weak accumulation point . the following lemma will be used in the proof of proposition [ onehelpsforall ] . in addition , it is also of interest in its own right as it provides a criterion for property ( ii ) of proposition [ onehelpsforall ] . [ pointmass ] let the assumptions of proposition [ onehelpsforall ] hold and let @xmath239 be any weak accumulation point of @xmath235 . then @xmath240 with @xmath241 where @xmath242 are defined as in proposition [ segers1 ] , and @xmath243 is given by . for @xmath244 we have @xmath245 with @xmath5 the second term converges to @xmath246 by condition 1 . for the first term we analyze the limit of @xmath247 as @xmath5 . by an application of the continuous mapping theorem ( cf . * theorem 4.27 ) ) in combination with condition 2 and proposition [ segers1 ] this converges to @xmath248 . since @xmath249 is pareto distributed and independent of @xmath250 we may rule out a point mass of @xmath251 in @xmath252 , and conclude that @xmath253 therefore , for a sequence @xmath254 which avoids possible point masses of @xmath255 it follows that @xmath256 the result now follows with @xmath254 if we show that @xmath257 which can be seen as a kind of extension of breiman s theorem ( cf . @xcite ) for the special case of @xmath171 . it follows because @xmath258 @xmath259 , where the first term equals @xmath260 the second term equals @xmath261 , since @xmath262 a.s . thus , @xmath263 @xmath264 @xmath265}(z)+x^\alpha \mathds{1}_{(x , \infty)}(z)dp^{|\chi|}(z)$ ] and follows from monotone convergence . this gives the result . we show that ( ii ) implies ( i ) . let @xmath266 and @xmath267 denote two weak accumulation points . in the following , let @xmath268 , and @xmath269 . we will show that @xmath270 and @xmath271 do not depend on @xmath272 . here , we shall use that ( ii ) implies @xmath273 by lemma [ pointmass ] , which in turn implies that @xmath274 for any weak accumulation point @xmath275 . since the above sets form a generating @xmath276-system , any two weak accumulation points coincide . by tightness ( cf . proposition [ sequenceistight ] ) this implies weak convergence . + consider first @xmath277 and @xmath278 that avoid the at most countably many point masses of the coordinate projections of @xmath266 and @xmath267 . then , @xmath270 is the limit of @xmath279 along a subsequence depending on @xmath280 . for general @xmath74 , insert @xmath281 in . this probability equals @xmath282 by conditions 1 and 2 , and since the variables have point masses at most at zero , this converges to @xmath283 cf . the proof of lemma [ pointmass ] . we have shown that @xmath270 does not depend on @xmath280 . approximation from inside extends this to all @xmath284 and @xmath285 . replacing @xmath286 by @xmath287 the same computation followed by an approximation argument shows the same for @xmath288 . combining these two results for @xmath289 with @xmath290 shows that @xmath291 does not depend on @xmath272 . thus , the same holds for the sets @xmath292 . lemma [ pointmass ] shows that prop . [ onehelpsforall ] ( ii ) , and thus ( i ) , holds if and only if @xmath273 . we give some examples . suppose that @xmath293 and @xmath294 are nonnegative time series and @xmath295 , thus @xmath296 . then , @xmath297 for some @xmath298 implies @xmath299 by breiman s theorem ( cf . @xcite ) and condition 2 holds if @xmath300 is continuous . if @xmath301 for some @xmath302 , then @xmath303 suffices to derive the same result ( cf . e.g. ( * ? ? ? * lemma 2.1 ) ) . for the special case @xmath304 cf . the end of the proof of lemma 3.2 . for further generalizations of breiman s theorem see @xcite . by similar computations it can be shown that under the assumptions of proposition [ onehelpsforall ] uniqueness of the weak limit in is also ensured by @xmath305 , with @xmath306 as in proposition [ segers2 ] . a key step in the argument shows that this condition implies weak convergence of @xmath307 as @xmath163 for all @xmath308 and @xmath309 . we give an example with @xmath310 but @xmath311 , i.e. @xmath310 may ensure uniqueness even if property ( ii ) in proposition [ onehelpsforall ] fails . to this end , let @xmath312 and @xmath313 be nonnegative i.i.d . random variables with @xmath314 for @xmath315 , where @xmath316 solves @xmath317 . with @xmath318 , let @xmath319 for all @xmath320 . then , @xmath321 implies @xmath322par@xmath42 , thus @xmath323 . for @xmath324 let @xmath325 . careful calculations show that @xmath326 ( cf . @xcite for similar arguments ) . but with @xmath327 it holds that @xmath328 , thus @xmath329 by lemma [ pointmass ] . while the existence of a limit in has been analyzed in the preceding section we will now deal with the particular form of the limit . for easy reference we shall introduce the following condition . + there exists a random vector @xmath330 such that @xmath331 we assume that the limit distribution in condition 3 is unique in order to simplify the statement of the proposition below . note , however , remark [ hasnottobeunique ] at the end of this section for a generalization to the case of non - uniqueness . we will use conditions 1 to 3 to derive a result for the form of the limit in which is similar to proposition [ segers2 ] . while conditions 1 and 2 bear a natural resemblance to the assumptions made in @xcite , condition 3 is necessary to ensure that a `` starting point '' for a tail chain exists that covers the time span from @xmath332 to @xmath333 where the @xmath334 and therefore @xmath335 are directly influenced by the event @xmath9 . we will see that outside of this range the behavior of the process @xmath8 corresponds to proposition [ segers2 ] . [ mainprop ] let @xmath8 and @xmath32 be stationary time series given by and and let conditions 1 , 2 and 3 hold . then , for all integers @xmath336 and @xmath309 we have @xmath337 with @xmath338 as in condition 3 , and @xmath339 cf . proposition [ segers1 ] for the definition of @xmath211 . here , @xmath340 are independent , and independent of @xmath341 with @xmath342 further , @xmath343 and @xmath344 are as in definition [ bftc ] . the proof is predecessed by a lemma and a corollary where we only assume that conditions 1 and 2 hold . [ etadeltalemma ] let @xmath345 . for any @xmath346 there is @xmath347 such that for @xmath74 large enough @xmath348 for all @xmath349 . for @xmath350 the statement follows with @xmath351 . so assume that @xmath352 . the l.h.s . equals @xmath353 the second factor converges to @xmath354 by condition 1 . it suffices to show that the first factor becomes small for @xmath355 . to this end , note that by stationarity the first factor equals @xmath356 which by definition of @xmath357 equals @xmath358 we proceed as in the proof of lemma [ pointmass ] . by an application of the continuous mapping theorem with condition 2 and proposition [ segers1 ] this converges to @xmath359 with @xmath360 again , we use that the two limit random variables include @xmath171 as an independent factor which excludes point masses on the positive axis . now , the set @xmath361 is contained in @xmath362 for @xmath363 the probability of this event gets arbitrarily small for @xmath364 small enough . [ makesmainproofeasier ] let @xmath365 and @xmath92 be a bounded uniformly continuous function on @xmath366 with @xmath367 . for any @xmath368 there is @xmath369 such that for @xmath74 large enough @xmath370 for all @xmath371 . since @xmath92 is bounded and uniformly continuous with @xmath367 , there is some @xmath372 such that @xmath373 choose @xmath374 as in lemma [ etadeltalemma ] . for @xmath375 split the expected value in into two by splitting @xmath376 into @xmath377 the first expected value is bounded by @xmath378 by lemma [ etadeltalemma ] , and the second by @xmath379 note that the case @xmath350 and @xmath309 is analogous to the proof of proposition [ segers1 ] ( cf . * theorem 2.3 ) ) . since @xmath380 is independent of @xmath381 the continuous mapping theorem can be applied to derive and leads to the multiplicative structure with independent increments . let now @xmath382 and @xmath383 , and let us assume that proposition [ mainprop ] holds for @xmath384 . let @xmath385 be bounded and uniformly continuous . we will show that @xmath386 with @xmath387 as defined in the statement of the proposition . let us further assume that @xmath388 as soon as @xmath389 . note that an arbitrary function @xmath385 can be split up additively into two functions @xmath390 and @xmath391 with @xmath392 such that the second function satisfies the aforementioned assumption and the first function depends merely on @xmath393 @xmath394 . since the induction hypothesis implies that is satisfied by a function of @xmath395 the assumption about the structure of @xmath92 is no loss of generality . the idea of the proof is to substitute the condition @xmath9 by a corresponding event in @xmath8 . let @xmath396 . then , for @xmath74 large enough @xmath397 for all @xmath371 , where @xmath398 is chosen according to corollary [ makesmainproofeasier ] . we have @xmath399 with the substitution @xmath400 . here , the first factor converges by condition 1 . furthermore , an application of the continuous mapping theorem in connection with propositions [ segers1 ] and [ segers2 ] yields that the whole expression converges to @xmath401 with @xmath167 @xmath174 and @xmath402 as in propositions [ segers1 ] and [ segers2 ] . defining new variables @xmath403 with the same distribution as @xmath404 in the statement of the proposition and independent of @xmath405 @xmath406 @xmath407 , @xmath408 the above expression equals @xmath409 by the definition of @xmath410 . next , note that by the continuous mapping theorem this equals @xmath411 replacing @xmath412 by @xmath413 and again using condition 1 this becomes @xmath414 since both @xmath211 and @xmath92 are uniformly continuous with @xmath415 and @xmath416 this gives @xmath417 for the complementary expression , where @xmath418 with @xmath419 . we may thus conclude from corollary [ makesmainproofeasier ] that @xmath420 tends to 0 as @xmath355 . thus , @xmath421 an application of the continuous mapping theorem in connection with the induction hypothesis yields that the latter expression equals @xmath422 with @xmath404 as in the statement of the proposition . since @xmath423@xmath424 this finishes the proof . [ hasnottobeunique ] if @xmath425 is a random vector such that for a sequence @xmath426 with @xmath427 the relation @xmath428 holds instead of condition 3 then a statement analogous to proposition [ mainprop ] holds true along the sequence @xmath426 . the existence of such sequences is guaranteed by condition 1 , cf . proposition [ sequenceistight ] . in order to simplify notation ( using only @xmath333 instead of @xmath24 and @xmath25 ) , we have assumed that holds instead of . however , under assumption the statement of proposition [ mainprop ] looks very similar , cf . @xcite , theorem 3.5.2 , for details . in this chapter we will show that condition 3 is closely related to the theory of multivariate regular variation . in a time series context this property is well explored in the case of garch@xmath429 processes , cf . @xcite . from the equivalent definitions of multivariate regular variation given in the literature we shall refer to the one used in @xcite . recall that a measurable function @xmath430 is said to be univariate regularly varying with index @xmath431 if @xmath432 for all @xmath433 . we call a random vector @xmath434 multivariate regularly varying if there exists a univariate regularly varying function @xmath435 with index @xmath436 and a non - degenerate , non - zero radon measure @xmath275 on @xmath437^d\setminus\{\mathbf{0}\}$ ] such that @xmath438 where `` @xmath439 '' stands for vague convergence ( cf . @xcite ) in @xmath440 , the space of all nonnegative radon measures on @xmath441 . one can show that the limit measure @xmath275 is necessarily homogeneous , i.e. that @xmath442 holds for all @xmath443 and for all borel sets @xmath444 ( cf . the measure @xmath275 and , consequently , the extremal behavior of @xmath445 are thus completely described by the index @xmath446 of regular variation , a constant @xmath447 and a probability measure @xmath448 on @xmath449 . the latter is the so - called spectral measure . altogether , we have that @xmath450 holds for all @xmath182 ( cf . @xcite ) . it has been shown by @xcite and @xcite ( cf . also @xcite ) that under mild assumptions about the distribution of @xmath451 a stationary garch@xmath429 process is multivariate regularly varying , i.e. for @xmath452 the vector @xmath453 with @xmath454 and @xmath455 as defined in and satisfies . furthermore , one can easily show that the same holds for the vector @xmath456 . now , the fact that a certain vector derived from the processes @xmath8 and @xmath32 is multivariate regularly varying will be useful in the verification of condition 3 as is shown in the following . let us again assume that @xmath457 is stationary and given by and . note that condition 3 is equivalent to @xmath458 for a random vector @xmath459 and for all @xmath460 such that @xmath461 . in the following we will assume multivariate regular variation of @xmath462 on @xmath463 , and show how this concept relates strongly to condition 3 . by continuity from below it suffices to look at such @xmath464 which are bounded away from @xmath465 in order to derive condition 3 from . the assumption of multivariate regular variation of @xmath462 guarantees the existence of a function @xmath435 such that @xmath466 if the denominator is positive ( it is necessarily finite since @xmath467 is bounded away from the origin ) . one easily checks that defines a probability measure for @xmath468 and may be set as the law of the random vector @xmath459 if @xmath469 . because of the aforementioned homogeneity of @xmath275 we note the equivalence @xmath470 thus , @xmath471 implies that the mass of @xmath275 is concentrated on the hyperplane @xmath472 . note that this is not excluded by our definition of regular variation nevertheless , since @xmath275 is non - degenerate and since the process @xmath8 is stationary we find that @xmath473 on the other hand , @xmath471 implies that @xmath474 hence , @xmath471 entails that @xmath475 and @xmath183 are not tail equivalent . this contradicts condition 1 and leads to the following proposition . [ mrvandcond3 ] let @xmath476 be a multivariate regularly varying vector with index @xmath446 and let condition 1 hold . then condition 3 is satisfied . in this section we will apply proposition [ mainprop ] to the special case of a garch@xmath2 process . we suggest a simple way for simulation from the limiting distribution @xmath477 of the tail process of @xmath37 , where @xmath478 are given by and . to this end , we shall make use of proposition [ mainprop ] and initially focus on the tail chain of the volatility sequence @xmath38 conditioned on the event @xmath479 . we will then obtain the desired distribution in by the special structure of @xmath37 . we shall henceforth assume that @xmath35 in order to preclude a degenerate solution to ( [ garchsigma ] ) . let further @xmath480 , @xmath481 , and let @xmath451 i.i.d . standard normal such that there exists a strictly stationary process which satisfies the aforementioned definitions , cf . recall that the case @xmath482 corresponds to the arch(1 ) model . it is well known that the marginal distributions of a stationary garch@xmath429 process with standard normal innovations show a regularly varying behavior . in the case of a garch@xmath2 process there exists a particularly simple characterization of the corresponding index @xmath446 of regular variation of the squared processes @xmath483 and @xmath484 in terms of the unique positive solution to @xmath485^{\alpha}\right)=1\ ] ] ( cf . @xcite , theorem 1 and example 2 ) . in order to apply proposition [ mainprop ] we verify that conditions 1 to 3 are satisfied for a garch@xmath2 process . it follows from the aforementioned regular variation of the marginal distribution of @xmath486 ( and hence also of @xmath487 ) that @xmath488 since @xmath53 is independent of @xmath52 , and all moments of @xmath489 exist we may apply breiman s theorem ( cf . @xcite ) in order to find that @xmath490 and condition 1 is satisfied . by the specifications of @xmath127 and @xmath23 given in and we get that @xmath491 where both @xmath127 and @xmath23 are only defined for @xmath492 . this shows that condition 2 holds where we dropped the second argument in @xmath132 and @xmath300 for simplicity . concerning condition 3 there are several instructive arguments : first , it is a direct consequence of proposition [ mrvandcond3 ] given the multivariate regular variation of the vector @xmath493 . alternatively , it follows from an application of proposition [ onehelpsforall ] in connection with lemma [ pointmass ] using that @xmath494 by . finally , the following lemma states both the existence and the specific distribution of @xmath495 . [ l : czv ] for the stationary processes @xmath37 and @xmath38 given by ( [ garchx ] ) and ( [ garchsigma ] ) there exists a random vector @xmath496 such that @xmath497 @xmath498 where @xmath499 and @xmath500 are independent with @xmath501 and @xmath500 is a symmetric random variable such that @xmath502 is gamma distributed with shape parameter @xmath503 and scale parameter 1 . we show that @xmath504 where @xmath505 is according to the above proposition . the statement then follows from an application of the continuous mapping theorem in connection with condition 2 . now , for @xmath506 and @xmath507 we have @xmath508 where the penultimate equality follows by uniform convergence of regularly varying functions , cf . the last equality holds by substitution and using that @xmath509 . now , the final expression has product form and hence @xmath499 , @xmath510 and @xmath511 are independent random variables with the stated distributions . having checked that all three conditions are met we may now apply proposition [ mainprop ] to the garch@xmath2 setting . [ p : js ] let the stationary time series @xmath37 and @xmath38 be given by ( [ garchx ] ) and ( [ garchsigma ] ) . then , for all @xmath512 , as @xmath12 , @xmath513 with @xmath514 as in lemma [ l : czv ] , and @xmath515 with @xmath516 defined by @xmath517 for an i.i.d . sequence @xmath518 independent of @xmath514 with standard normal distribution , and @xmath519 are i.i.d . random variables on @xmath520 ( i.e. on @xmath521 for @xmath41 ) independent of @xmath522 and @xmath514 with distribution function @xmath523 for @xmath524 . we apply proposition [ mainprop ] . since @xmath525 we omit the variables @xmath526 from proposition [ mainprop ] and restrict ourselves to the distribution of @xmath527 . now , follows from proposition [ segers1 ] . with @xmath528 we are left to show that the distribution defined in equals the distribution of the second component of the adjoint measure @xmath529 corresponding to the bftc(@xmath530 . using ( cf . also ( 4.3 ) in @xcite ) we have that @xmath531+\left(1-e\left[(\alpha_1 \hat{\epsilon}_1 ^ 2+\beta_1)^{\alpha}\right]\right),\ ] ] for a standard normal @xmath532 . this gives since the second summand equals zero by . with @xmath533 or @xmath534 , as above the assertion follows with proposition [ mainprop ] . next , we take proposition [ p : js ] as a starting point for the derivation of . to this end , note that by ( [ garchsigma ] ) we have @xmath535 for a sequence @xmath536 of i.i.d . random variables independent of @xmath537 with @xmath538 now , by an application of the continuous mapping theorem in connection with proposition [ p : js ] and we get @xmath539 with @xmath540 for a sequence @xmath541 with the same distribution as @xmath536 and independent of @xmath542 . note that by the structure of @xmath543 it follows that @xmath544 now , by ( [ eq : cs2 ] ) it holds that @xmath545 , @xmath546 . further , we have that @xmath547 is symmetric where @xmath548 by lemma [ l : czv ] and the definition in ( [ eq : mcl ] ) . for simulation from the r.h.s . of ( [ eq : mcl ] ) it is advantageous to write @xmath549 for @xmath550 , such that replacing for ( [ eq : czv ] ) , ( [ eq : n2 ] ) and ( [ epsilontilde ] ) yields the following proposition . [ easysimulation ] for @xmath220 let * @xmath551 , @xmath552 distributed according to ( [ eq : cs3 ] ) , * @xmath553 , @xmath554 i.i.d . with the distribution of @xmath555 in ( [ eq : pm ] ) , * @xmath556 i.i.d . standard normal * @xmath500 according to lemma [ l : czv ] and * @xmath501 be mutually independent random variables . then , @xmath557 finally , conditioning on @xmath558 as in ( [ eq : ldx ] ) instead of @xmath559 leads to the same limit distribution as in ( [ eq : dwi ] ) but with @xmath560 almost surely . our analysis of two connected time series was originally motivated by an idea to extend the approach considered in @xcite for arch@xmath42 processes to a simulation study for extremal characteristics of the more general garch@xmath2 class . among such extremal measures the extremal index is a well - known example . it characterizes the behavior of extreme events in a time series , i.e. the strength of dependence between subsequent high - level exceedances . more precisely , let @xmath561 for @xmath162 where @xmath562 is a stationary univariate process with marginal distribution function let further @xmath564 be the associated i.i.d . sequence with the same marginal distribution @xmath563 , and let accordingly @xmath565 for @xmath162 . assume that there exists a nonnegative number @xmath566 such that for every @xmath433 there is a sequence @xmath567 such that @xmath568 then , @xmath566 is called the extremal index of the process @xmath562 , and @xmath569 $ ] . note also that under an additional mild mixing condition the extremal index corresponds to the inverse of the mean cluster size of extreme values in the series , cf . @xcite for reference and for further details about the extremal index . now , focussing on the garch@xmath2 model as defined in section [ s : intro ] we find that @xmath570 where @xmath571 cf . @xcite and ( * ? ? ? * section 5.2 ) . in addition to the extremal index we shall in the following also consider two alternative extremal characteristics that may be evaluated by the same simulation approach . the so - called extremal coefficient function discussed in @xcite is given by @xmath572 for @xmath573 . following the notion of usual autocovariances the extremal coefficient function gives the conditional probability of two extreme events separated by a lag @xmath574 . for two reasons we will also briefly describe a modification of this concept , i.e. a probability for threshold exceedances at a lag @xmath575 given that @xmath576 is not only extreme as in , but given that @xmath576 is also at the beginning of an extremal cluster in a time series , cf . * chapter 5 ) for a discussion . more precisely , let @xmath577 where @xmath578 the first reason to touch on this characteristic in our study is its potential to serve as a complement to the extremal coefficient function regarding questions of cluster structures in risk management and related applications that focus on the development of extremal events . the second reason is related to the numerical simulation of ( [ thetam ] ) to ( [ gammam ] ) that will be based on the tail chain concept discussed in section [ garch ] . while the evaluation of @xmath579 and the extremal coefficient function @xmath250 requires a series of runs of either the forward or the backward tail chain it is evident from ( [ gammam ] ) that the simulation of @xmath580 must be based on simultaneous runs of the forward and the backward tail chain at the same time . note at this point that we are not aware of any general closed form solutions for ( [ thetam ] ) to ( [ gammam ] ) that would include the garch@xmath429 model parameters , not even for @xmath581 . our simulation setup generalizes similar methods proposed by @xcite and @xcite . the algorithm used in @xcite is restricted to single time series which satisfy the assumptions of @xcite that do , however , not hold for the log returns in a garch@xmath2 setting . as a generalization of @xcite our algorithm is in principle not restricted to models with symmetric innovations . furthermore , to our knowledge , there have been no approaches to simulate from the backward tail chain so far . taking the limit @xmath5 in to we note that as indicated above all three characteristics can be expressed via the tail chain distribution . by simulation from this distribution ( cf . proposition [ easysimulation ] ) we may therefore evaluate these quantities by monte carlo estimation . in table [ t : es ] we report the results of such a simulation study for @xmath582 and @xmath583 , ( which we use as approximations for @xmath584 and @xmath585 ) and @xmath586 for @xmath587 . the evaluation of probabilities is based on @xmath588 draws . we fix @xmath589 in the table in order to reflect the stylized fact that @xmath590 is close to one in many applications . the last row of table [ t : es ] is motivated by the following example . cccccccccc @xmath46 & @xmath47 & @xmath591 & @xmath592 & @xmath593 & @xmath594 & @xmath595 & @xmath596 & @xmath597 & @xmath598 ' '' '' ' '' '' + 0.99 & 0 & 1.014 & 0.570 & 0.213 & 0.139 & 0.104 & 0.251 & 0.167 & 0.125 + 0.15 & 0.84 & 1.478 & 0.207 & 0.061 & 0.063 & 0.065 & 0.153 & 0.144 & 0.139 + 0.11 & 0.88 & 1.838 & 0.245 & 0.052 & 0.042 & 0.038 & 0.110 & 0.104 & 0.104 + 0.09 & 0.90 & 2.203 & 0.304 & 0.045 & 0.035 & 0.034 & 0.089 & 0.085 & 0.081 + 0.07 & 0.92 & 2.885 & 0.397 & 0.022 & 0.020 & 0.020 & 0.055 & 0.050 & 0.053 + 0.04 & 0.95 & 5.991 & 0.854 & 0.005 & 0.004 & 0.003 & 0.007 & 0.007 & 0.006 + 0.072 & 0.920 & 2.476 & 0.317 & 0.021 & 0.020 & 0.027 & 0.063 & 0.064 & 0.066 + [ ex : garch ] we fit the garch(1,1 ) model given by ( [ garchx ] ) to a data set of log returns of the s&p 500 index from 01.04.80 to 30.03.10 ( 7569 records ) . the estimated parameters @xcite are @xmath599 where the ml standard errors are given in brackets . we include an evaluation of the corresponding extremal measures by the above tail chain approach in the last row of table [ t : es ] . in order to discuss the adequacy of a garch@xmath2 model with regard to the extremal behavior we compare the result of table [ t : es ] with the so - called blocks estimator of the extremal index for the given data @xcite . for a block length of @xmath600 and a threshold corresponding to the empirical 0.95 quantile the estimator yields @xmath601 . here , the brackets represent the simulated 95% confidence interval which is based on @xmath602 independent garch(1,1 ) processes of length 7569 according to . as to the choice of the block length note that extremal events occuring in two distinct blocks are assumed to be independent . here , six trading months correspond to 126 days and appear to be a reasonable order of magnitude . given that our block length is a valid choice the fact that the result falls within the simulated confidence interval indicates a satisfactory agreement of the data set and a garch@xmath2 model with regard to their extremal behavior . [ [ details - on - the - construction - of - f - in - the - proof - of - lemma - nonuniquenesslemma ] ] details on the construction of @xmath92 in the proof of lemma [ nonuniquenesslemma ] + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + let @xmath603 and @xmath604 , i_{2 } = [ 2\frac{1}{4 } z , 3z],i_{3}=[3z,4z],i_{4}=[4z,5z]$ ] . the restrictions @xmath605 will look as follows : @xmath606 is strictly increasing from value @xmath607 to @xmath608 , @xmath609 is symmetric on @xmath610 , on the left half of @xmath610 it increases from @xmath608 to @xmath611 . finally , @xmath612 interpolates linearly between the values @xmath608 and @xmath607 , and @xmath613 between @xmath607 and @xmath611 . + given the definitions of @xmath612 and @xmath613 , we show that @xmath606 , @xmath609 can be defined implicitely . + for the definition of @xmath606 via @xmath614 consider @xmath615 $ ] . the function @xmath614 has to satisfy @xmath616 thus @xmath617 . observe that @xmath618 . thus @xmath619 is strictly increasing and @xmath620 , @xmath621 by the above formula . this shows that @xmath606 is well - defined as the inverse of @xmath614 . + for the existence of @xmath609 on @xmath610 as described it suffices to show that @xmath622 $ ] , is strictly decreasing ( note that @xmath623 ) which follows from @xmath624 . andersen , t. g. : stochastic autoregressive volatility : a framework for volatility modelling . finance * 4 * , 75102 ( 1994 ) . basrak , b. , davis , r.a . and mikosch , t. : regular variation of garch processes . stochastic processes and their applications * 99 * , 95115 ( 2002 ) basrak , b. and segers , j. : regularly varying multivariate time series . stochastic processes and their applications * 119 * , 1055 - 1080 ( 2009 ) beirlant , j. , goegebeur , y. , segers , j. and teugels , j. : statistics of extremes . wiley , chichester ( 2004 ) bingham , n. h. , goldie , c. m. and teugels , j. l. : regular variation . cambridge university press , cambridge ( 1987 ) bollerslev , t. : generalized autoregressive conditional heteroskedasticity . j. econometrics * 31 * , 307327 ( 1986 ) boman , j. and lindskog , f. : support theorems for the radon transform and cramr - wold theorems . technical report , kth stockholm ( 2002 ) breiman , l. : on some limit theorems similar to the arc - sin law . . appl . * 10 * , 323331 ( 1965 ) carrasco , m. and chen , x. : mixing and moment properties of various garch and stochastic volatility models . econometric theory . * 18 * , 1739 ( 2002 ) davis , r. a. and mikosch , t. : extreme value theory for garch processes , in : andersen , t.g . , davis , r.a . , krei , j .- mikosch , t. ( editors ) : handbook of financial time series . springer , new york , 187200 ( 2009 ) de haan , l. , resnick , s.i . , rootzn , h. and de vries , c. g. : extremal behaviour of solutions to a stochastic difference equation with applications to arch processes . stochastic processes and their applications * 32 * , 213224 ( 1989 ) denisov , d. and zwart , b. : on a theorem of breiman and a class of random difference equations . * 44 * , 10311046 ( 2007 ) ehlert , a. : characteristics for dependence in time series of extreme values thesis , university of gttingen ( 2010 ) embrechts , p. , klppelberg , c. and mikosch , t. : modelling extremal events . springer , berlin ( 1997 ) engle , r. : autoregressive conditional heteroscedastic models with estimates of the variance of united kingdom inflation . econometrica * 50 * , 9871007 ( 1982 ) fasen , v. , klppelberg , c. and schlather , m. : high - level dependence in time series models . extremes * 13 * , 133 ( 2010 ) glosten , r. , jagannathan , r. and runkle , d. : on the relation between expected value and the volatility of the nominal excess return on stocks . journal of finance * 48 * , 17791801 ( 1993 ) gomes , m. i. , de haan , l. and pestana , d. : joint exceedances of the arch process . . prob . * 41 * , 919926 . 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( 2006 ) ) janen , a. : on some connections between light tails , regular variation and extremes . thesis , university of gttingen ( 2010 ) kallenberg , o. : foundations of modern probability . springer , new york ( 2002 ) laurini , f. and tawn , j. a. : the extremal index for garch(1,1 ) processes . extremes , published online first , doi : 10.1007/s10687 - 012 - 0148-z ( 2012 ) nelson , d.b . : stationarity and persistence in the garch(1,1 ) model . econometric theory , * 6 * , 318334 ( 1990 ) resnick , s. i. : heavy - tail phenomena . springer , new york ( 2007 ) segers , j. : multivariate regular variation of heavy - tailed markov chains , institut de statistique dp0703 , available on arxiv.org as math.pr/0701411 ( 2007 ) smith , r. l. : the extremal index for a markov chain . * 29 * , 3745 ( 1992 ) taylor , s. : modelling financial time series . wiley , chichester ( 1986 ) trapletti , a. and hornik , k. : tseries : time series analysis and computational finance , url ` http://cran.r-project.org/package=tseries ` , r package version 0.10 - 18 ( 2009 ) wuertz , d. , et . see the source file : fextremes : rmetrics - extreme financial market data , url ` http://cran.r-project.org/package=fextremes ` , r package version 2100.77 ( 2009 )

we study the behavior of a real - valued and unobservable process @xmath0 under an extreme event of a related process @xmath1 that is observable . our analysis is motivated by the well - known garch model which represents two such sequences , i.e. the observable log returns of an asset as well as the hidden volatility process . our results complement the findings of segers [ j. segers , multivariate regular variation of heavy - tailed markov chains , arxiv : math/0701411 ( 2007 ) . available online : http://arxiv.org/abs/math/0701411 ] and smith [ r. l. smith , the extremal index for a markov chain . j. appl . prob . ( 1992 ) ] for a single time series . we show that under suitable assumptions their concept of a tail chain as a limiting process is also applicable to our setting . furthermore , we discuss existence and uniqueness of a limiting process under some weaker assumptions . finally , we apply our results to the garch@xmath2 case .

the optical properties of an excitonic molecule originate from the resonant interaction of its constituent excitons ( xs ) with the light field . for semiconductor ( gaas ) nanostructures we analyze in this paper , the above interaction refers to quasi - two - dimensional ( quasi-2d ) qw excitons and is different in single , mc - free quantum wells and in microcavities . in the first case , the breaking of translational invariance along the growth direction ( @xmath11-direction ) leads to the coupling of qw excitons to a continuum of bulk photon modes . this results in an irreversible radiative decay of low - energy qw excitons into the bulk photon modes and to interface , or qw , polaritons for the qw exciton states lying outside the photon cone @xcite . an interface polariton is the in - plane propagating eigenwave guided by a single qw , and the light field associated with interface polaritons is evanescent , i.e. , it decays exponentially in the @xmath11-direction . in contrast , the mc polariton optics deals with the quasi - stationary mixed states of quasi-2d mc photons and qw excitons @xcite , i.e. , one realizes a nearly pure 2d exciton - photon system with resonant coupling between two eigenmodes ( for a review of the mc polariton optics see , e.g. , refs . [ ] ) . in this case the radiative lifetime of mc polaritons originates from a finite transmission through the cavity mirrors . the main aim of the present work is to develop coherent optics of quasi-2d excitonic molecules in semiconductor microcavities . the xx - mediated optical response from gaas microcavities has been addressed only recently @xcite . the coulombic attractive interaction of cross - circular polarized ( @xmath12 and @xmath13 ) excitons , which gives rise to the xx bound state , has been invoked and estimated in order to analyze the frequency - degenerate four - wave mixing ( fwm ) experiment @xcite . pump - probe specroscopy was used in ref . [ ] to observe the xx - mediated pump - induced changes in the mc reflectivity spectrum . however , in the above first experiments the microcavity polariton resonance has large broadening so that the spectrally resolved xx transition was not detected . only recently the spectrally resolved `` polariton @xmath14 xx '' photon - assisted transition in gaas - based mcs has been observed by using differential reflection spectroscopy @xcite . in particular , the transition is revealed in a pump probe experiment as an induced absorption from the lower polariton dispersion branch to the xx state , at positive pump - probe time delays @xcite . in the latter work the mc rabi splitting @xmath15 , associated with a heavy hole qw exciton , exceeds the xx binding energy @xmath3 for more than a factor of three . the last experiments on excitonic molecules in gaas microcavities use a high - intensity laser field to investigate the xx - mediated changes in the polariton spectrum @xcite and parametric scattering of mc polaritons @xcite . in this work we are dealing with a _ low - intensity _ limit of the xx optics , aiming to study the radiative corrections to the molecule state in a high - quality gaas single qw embedded in a co - planar @xmath4-cavity . recently , the optical properties of large binding energy excitonic molecules in a mc - embedded znse qw has been studied @xcite . the theoretical model we work out can straightforwardly be adapted to the quasi-2d molecules in ii - vi nanostructures . in the previous theoretical studies @xcite the xx radiative corrections are not included , so that the models deal with the optically unperturbed molecule wavefunction @xmath16 and binding energy @xmath3 . according to ref . [ ] , the xx radiative corrections are rather small , even if @xmath17 . the authors argue qualitatively that a volume of phase space , where the resonant coupling of the constituent excitons with the light field occurs , is rather small to affect the xx state . as we show below , an exactly - solvable bipolariton model @xcite , adapted to excitonic molecules in gaas - based quasi-2d nanostructures , yields @xmath1 and @xmath0 of about @xmath18 for microcavities , and @xmath10 and @xmath19 of about @xmath20 for single qws . the calculated values refer to the weak confinement of qw excitons and qw excitonic molecules we deal with in our study . in the weak confinement limit , the qw thickness @xmath21 is comparable with the in - plane radius of the above electron - hole bound complexes , which are still constructed in terms of well - defined transversly - quantized quasi-2d electronic states . in contrast , in the strong confinement limit , @xmath21 is much less than the in - plane radius of qw excitons ( excitonic molecules ) . the radiative corrections to the xx state can not be neglected , because the exciton - photon coupling ( polariton effect ) changes the dispersion of excitons not only in a very close vicinity of the resonant crossover between the relevant exciton and photon energies , but in a rather broad band @xmath22 . here wavevector @xmath23 is given by the resonant condition @xmath24 between the bulk photon and exciton dispersions ( @xmath25 is the background dielectric constant ) . for gaas structures @xmath26 . the dimensionless parameter @xmath27 , which scales the xx radiative corrections , is @xmath28 , where @xmath29 is the molecule radius and @xmath30 is the dimensionality of a semiconductor structure . remarkably , as we demonstrate below , @xmath31 does not depend upon the mc detuning between the @xmath4-cavity mode and @xmath32 , i.e. , is the same for microcavities and single qws . for our high - quality gaas qws with weak confinement of excitons one estimates @xmath33 , so that @xmath34 . the latter value clearly shows that the exciton - photon coupling does change considerably the quasi-2d xx states . even for the x wavevectors far away from the resonant crossover point @xmath23 , the polariton effect can still have a considerable impact on the dispersion of optically - dressed excitons in bulk semiconductors and qws . to illustrate this , note that for bulk gaas , e.g. , the effective mass associated with the upper polariton dispersion branch at @xmath35 is given by @xmath36 , i.e. , by factor four is less than the translational mass @xmath37 of optically undressed excitons . in a similar way , the dispersion of qw excitons dressed by mc photons , which gives rise to @xmath38 and @xmath1 , refers to the in - plane wavevector domain @xmath39 rather than to a close vicinity of the crossover point @xmath40 . an excitonic molecule can be described in terms of two quasi - bound polaritons ( bipolariton ) , if the coupling of the molecule with the light field is much stronger than the incoherent scattering processes . in this case the sequence `` two incoming polaritons ( or bulk photons ) @xmath5 quasi - bound xx state @xmath5 two outgoing polaritons '' is a completely coherent process of the resonant polariton - polariton scattering and can be described in terms of the bipolariton wavefunction @xmath41 . the latter includes an inherent contribution from the outgoing ( incoming ) polaritons and should be found from the bipolariton wave equation . the solution also yields the radiative corrections to the xx energy , i.e. , @xmath42 , where @xmath43 is the `` input '' xx binding energy of an optically inactive molecule . for some particular model potentials of @xmath12-exciton @xmath13-exciton interaction , e.g. , for the deuteron and gaussian potentials , the bipolariton wave equation can be solved exactly @xcite . the bipolariton concept was verified in high - precision experiments with low - temperature bulk cucl @xcite and cds @xcite , and was also applied successfully to explain the xx - mediated optical response from gaas / algaas multiple qws @xcite . the latter experiment dealt with quasi-2d xxs in the limit of strong qw confinement . in this case the bipolariton model shows that the main channel of the optical decay of qw excitonic molecules in mc - free structures is the resonant photon - assisted dissociation of the molecule into two outgoing interface ( qw ) polaritons . note that the coulombic interaction between two constituent excitons of the molecule couples the radiative modes and the interface polariton states , so that an `` umklapp '' process between the modes can intrinsically be realized . the above picture refers to the following scenario of the coherent optical generation and dissociation of qw molecules : `` @xmath12 bulk photon + @xmath13 bulk photon @xmath5 @xmath12 virtual qw exciton + @xmath13 virtual qw exciton @xmath5 qw molecule @xmath5 @xmath12 interface polariton + @xmath13 interface polariton '' . the experiments we report on deal with weakly confined qw excitonic molecules , i.e. , the qw thickness @xmath44 is comparable with the radius of excitons in bulk gaas . the quasi-2d weak confinement allows us to neglect inhomogeneous broadening in the detected x- and xx - mediated signals . the mc - free single qw is used as a reference structure : all the @xmath4-microcavities , which we study , are embedded with a single qw nearly identical to the reference one . by analyzing the coherent dynamics of the xx - mediated signal in spectrally - resolved transient fwm , we infer the xx radiative width in the microcavities and in the reference single qw , @xmath1 and @xmath10 , respectively . the measurements yield @xmath1 larger than @xmath10 by nearly factor two . furthermore , by using pump - probe spectroscopy we also estimate the xx binding energies @xmath45 and @xmath46 . our measurements deal with the mc detuning band @xmath47 . similarly to quasi-2d xxs in high - quality single qws , the main mechanism of the optical decay of mc molecules is their in - plane resonant dissociation into mc polaritons . thus the coherent optical path of the xx - mediated signal in our experiments is given by `` @xmath12 ( pump ) bulk photon + @xmath13 ( pump ) bulk photon @xmath5 mc molecule @xmath5 @xmath12 mc polariton + @xmath13 mc polariton @xmath5 @xmath12 ( signal ) bulk photon + @xmath13 ( signal ) bulk photon '' . the latter escape of the mc polaritons into the bulk photon modes is due to a finite radiative lifetime of mc photons . in order to explain the experimental data , the bipolariton model is adapted to weakly confined quasi-2d molecules in ( gaas ) microcavities and mc - free ( gaas ) single qws . one of the most important features of the optics of excitonic molecules in microcavities is a large contribution to the bipolariton state @xmath41 from @xmath48-mode mc polaritons . the relevant @xmath48-mode polariton states refer to the in - plane wavevectors @xmath49 , i.e. , are short - wavelength in comparison with the @xmath50-mode polariton states activated in standard optical experiments . an `` invisible '' decay channel of the mc molecule into two outgoing @xmath48-mode polaritons in combination with the directly observable dissociation path `` xx @xmath5 @xmath50-mode mc polariton + @xmath50-mode mc polariton '' explain qualitatively the factor two difference between @xmath1 and @xmath10 . the use of the microcavities embedded with a single qw allows us to apply the bipolariton model without complications due to the dark x states in multiple qws @xcite . the bipolariton model quantitatively reproduces our experimental data and predicts new spectral features , like @xmath51 critical points in the detuning dependent @xmath52 and @xmath53 . thus the main results of our study on weakly confined quasi-2d molecules in gaas microcavities are ( i ) rigorous justification of the bipolariton model , ( ii ) importance of the xx radiative corrections , and ( iii ) existence of the efficient `` hidden '' xx decay channel , associated with 0@xmath4-mode mc polaritons . in sec.ii , we apply the bipolariton model in order to analyze the xx radiative corrections , the xx lamb shift @xmath54 and xx radiative width @xmath55 , relevant to our microcavities and reference qw . after a brief discussion of interface and mc polaritons , we demonstrated that in gaas - based quasi-2d structures the xx radiative corrections can be as large as @xmath2 of the ( input ) xx binding energy @xmath56 . it is shown that independently of the mc detuning @xmath57 the xx radiative corrections in microcavities and ( reference ) qws are scaled by the same dimensionless parameter @xmath58 , and that the main xx optical decay channels in microcavities are `` xx @xmath5 1@xmath4-mode mc polariton + 1@xmath4-mode mc polariton '' and `` xx @xmath5 0@xmath4-mode mc polariton + 0@xmath4-mode mc polariton '' against the main decay path in single qws , `` xx @xmath5 interface polariton + interface polariton '' . we also find and classify two _ critical points _ , @xmath59 and @xmath60 , in the spectrum of the xx radiative corrections in microcavities , @xmath61 and/or @xmath62 , and propose to use the critical points for high - precision measurements of the mc rabi splitting and the xx binding energy . in sec.iii , the investigated gaas - based mc sample and the reference gaas single qw are characterized . we describe the fwm measurements at @xmath63k , which allow us to estimate the xx dephasing width for the mc detuning band @xmath64 , @xmath65mev , and the pump - probe experiments at @xmath66k , which yield the bipolariton ( xx ) binding energy in our microcavities , @xmath67mev . in sec.iv , by analyzing a temperature - dependent contribution to the dephasing widths @xmath68 and @xmath69 , which is associated with xx la - phonon scattering , we estimate the corresponding xx radiative widths in the microcavities and reference qw ( @xmath70mev and @xmath8mev ) , and show that the bipolariton model does reproduce _ quantitatively and self - consistently _ both @xmath1 and @xmath10 . it is shown that the @xmath9 limit , which is crucial for the validity of the bipolariton model , starts to hold for excitonic molecules at cryostat temperatures below 10k . we also discuss the underlaying physical picture responsible for the large xx radiative corrections in high - quality quasi-2d ( gaas ) nanostructures . a short summary of the results is given in sec.v . in this section we briefly discuss interface ( quantum well ) and microcavity polaritons , and apply the bipolariton model @xcite in order to calculate the xx radiative corrections and to describe the optical decay channels of excitonic molecules in high - quality gaas - based microcavities and single qws . for a single qw , the resonant coupling of excitons with the light field can be interpreted in terms of the radiative in - plane modes @xmath71 , which ensure communication of low - energy qw excitons with incoming and outgoing bulk photons ( the only photons used in standard pump - probe optical experiments with qws ) , and interface polaritons , which refer to the states outside the photon cone , @xmath72 . the latter in - plane propagating polariton eigenmodes are trapped and waveguided by the x resonance ; they are accompanied by the evanescent , interface light field , i.e. , are invisible at macroscopic distances from the qw . for an ideal qw microcavity the mc photons with in - plane wavevector @xmath73 can be classified in terms of @xmath74- transverse eigenmodes ( @xmath75 ) . the mc polariton eigenstates arise when some of the mc photon eigenmodes resonate with the qw exciton state . as we show below , only @xmath76- and @xmath77- polariton eigenmodes are relevant to the optics of qw excitonic molecules in our mc structures . with increasing mc thickness towards infinity the microcavity polariton eigenstates evolve into the radiative and interface polariton eigenmodes associated with a mc - free single qw @xcite . \(i ) _ the light field resonantly interacting with quasi-2d excitons in a single ( gaas ) qw . _ the interaction of a qw exciton with in - plane momentum @xmath78 with the transverse light field of frequency @xmath79 is characterized by the dispersion equation @xcite : @xmath80 where @xmath37 is the in - plane translational x mass , @xmath81=0 ) is the x energy , @xmath82 is the rate of incoherent scattering of qw excitons , and @xmath83 is the dimensional oscillator strength of exciton - photon interaction per qw unit area . equation ( [ pol ] ) refers to a single ( gaas ) qw confined by two identical ( algaas ) bulk barriers . for @xmath84 , i.e. , for the momentum - frequency domain outside the photon cone , eq . ( [ pol ] ) describes the in - plane polarized transverse interface polaritons ( @xmath85-mode polaritons ) . the evanescent light field associated with the interface polaritons is given by @xmath86 , where @xmath87 . the exciton and photon components of a qw polariton with in - plane wavevector @xmath73 are @xmath88 ^ 2 } \ , \nonumber \\ & & v^2_{\rm ip}(p_{\| } ) = 1 - u^2_{\rm ip}(p_{\| } ) \ , \label{comp}\end{aligned}\ ] ] respectively . here @xmath89 is the polariton dispersion determined by eq.([pol ] ) . note that the @xmath11-polarized transverse interface polaritons ( @xmath90-mode qw polaritons ) associated with the ground - state heavy - hole excitons are not allowed in gaas qws @xcite . the low - energy qw excitons from the radiative zone @xmath91 couple with bulk photons , i.e. , can radiatively decay into the bulk photon modes . in this case eq.([pol ] ) yields the x radiative decay rate into bulk in - plane ( @xmath85- ) polarized transverse photons : @xmath92 one can also re - write eq.([rad ] ) as @xmath93=0 ) @xmath94 , where @xmath95=0 ) = @xmath96 is the radiative width of a qw exciton with in - plane momentum @xmath97 . in high - quality gaas qws at low temperatures , the condition @xmath98 can be achieved , so that the x dispersion within the photon cone is approximated by @xmath99@xmath100@xmath101 . the oscillator strength @xmath102 associated with qw excitons is given by @xmath103 where @xmath104 is the x wavefunction of relative electron - hole motion , and @xmath105 is the dipole matrix element of the interband optical transition . in the limits of strong and weak qw confinement eq.([qwstr ] ) yields @xmath106 respectively , where @xmath107 is the bohr radius of bulk excitons and @xmath108 is the longitudinal - transverse splitting associated with bulk excitons ( in bulk gaas one has @xmath109 and @xmath110 , respectively @xcite ) . thus we estimate the upper limit of the oscillator strength in narrow gaas qws as @xmath111@xmath5@xmath112 . for our gaas qws with weak confinement of excitons one evaluates from eq.([strength ] ) that @xmath113@xmath114@xmath115 . \(ii ) _ the mc polariton dispersion relevant to excitonic molecules in ( gaas - based ) microcavities . _ the dispersion equation for mc polaritons , which contribute to the xx - mediated optics of a @xmath4-cavity we study in our experiments , is given by @xmath116 \ , , \nonumber \\ \label{mc}\end{aligned}\ ] ] where the photon frequencies , associated with the 1@xmath4- and 0@xmath4- microcavity eigenmodes , are @xmath117 and @xmath118 , respectively . here @xmath119 is the cavity eigenfrequency , @xmath120 is the mc thickness , and @xmath121 is the inverse radiative lifetime of mc photons , due to their escape from the microcavity into external bulk photon modes . the mc rabi frequency @xmath122 refers to 1@xmath4-eigenmode of the light field , @xmath123 $ ] ( we assume that the qw is located at @xmath124 so that @xmath125 ) , and is determined by @xmath126 where @xmath127 \sin[(\pi d_z)/l_z ] \simeq 1 - ( \pi/6)(d_z / l_z)^2 $ ] . in turn , the rabi frequency @xmath128 is associated with 0@xmath4-eigenmode of the mc light field , @xmath129 , and @xmath130 from eqs.([qwstr ] ) , ( [ mcstr1 ] ) , and ( [ mcstr0 ] ) one gets @xmath131 because the factor @xmath132 ( for our microcavities @xmath133 and @xmath134 , so that @xmath135 ) , we conclude that @xmath136 . the factor two difference between @xmath137 and @xmath138 originates from the difference of the intensities of the light fields associated with microcavity 1@xmath4- and 0@xmath4-eigenmodes at the qw position , @xmath124 , i.e. , is due to @xmath139=@xmath140=@xmath141 . ) for zero - detuning . the parameters are adapted to the gaas microcavities used in our experiments : @xmath142mev , @xmath143mev , @xmath144 , @xmath145 , and @xmath146=@xmath147ev . the dashed lines show the 1@xmath4-mode mc photon and exciton dispersions ( the 0@xmath4-mode photon dispersion is not plotted ) . , width=302 ] thus , the dispersion eq.([mc ] ) deals with a _ three - branch _ mc polariton model . in fig.1 we plot the polariton dispersion branches , designated by 1@xmath4-ub ( upper branch ) , 1@xmath4-lb ( middle branch ) , and 0@xmath4-lb ( lower branch ) , respectively , and calculated by eq.([mc ] ) for a zero - detuning gaas - based microcavity with @xmath148mev and @xmath149mev . the ratio between the rabi frequencies satisfies eq.([strr ] ) , and the used value of @xmath122 corresponds to that observed in our experiments . for small in - plane wavevectors @xmath150 ( see fig.1 ) the 1@xmath4-ub and 1@xmath4-lb dispersion curves are identical to the upper and lower mc polariton branches calculated within the standard 1@xmath4-eigenmode resonant approximation @xcite . in this case the 1@xmath4-ub and 1@xmath4-lb polaritons are purely 1@xmath4-eigenwaves ; the 0@xmath4-lb dispersion is well - separated from the x resonance so that in eq.([mc ] ) one can put @xmath151 in order to describe the 1@xmath4-ub and 1@xmath4-lb dispersions in the wavevector domain @xmath152 . the anti - crossing between the x dispersion @xmath153 and the mc 0@xmath4-mode photon frequency @xmath154 , which occurs at @xmath155 , gives rise to the mc 0@xmath4-eigenmode dispersion associated with the 1@xmath4-lb and 0@xmath4-lb short - wavelength polaritons with @xmath156 ( see fig.1 ) . this picture is akin to the two - branch polariton dispersion in bulk semiconductors ; for @xmath157 the 1@xmath4-lb and 0@xmath4-lb polariton dispersion can accurately be approximated by eq.([mc ] ) with @xmath158 . in this case eq.([mc ] ) becomes identical to the dispersion equation for bulk polaritons , if in the latter the bulk rabi splitting @xmath159 ( @xmath160mev in gaas ) is replaced by @xmath161 and the bulk photon wavevector @xmath162 is replaced by @xmath163 . note that for the mc 0@xmath4-eigenmode the light field is homogeneous in the @xmath11-direction within the microcavity , i.e. , for @xmath125 . with increasing detuning from the x resonance the 1@xmath4-lb and 0@xmath4-lb polariton dispersions approach the photon frequencies @xmath164 and @xmath165 , respectively , where the low - frequency dielectric constant is given by @xmath166 $ ] . the interconnection between two mc polariton domains occurs via the 1@xmath4-lb polariton dispersion : with increasing @xmath163 from @xmath167 towards @xmath168 the structure of the photon component of 1@xmath4-lb polaritons smoothly changes , as a superposition of two modes , from purely 1@xmath4-mode to purely 0@xmath4-mode . because @xmath169 , the non - zero exciton component of all three mc polariton dispersion branches contributes to the molecule state and , therefore , to the xx - mediated optics of microcavities . the x component , associated with the 0@xmath4-lb , 1@xmath4-lb , and 1@xmath4-ub dispersions , is given by @xmath170 ^ 2 } \nonumber \\ & + & { \omega^4_i ( \omega_{0 \lambda}^{\rm mc})^2 \over \omega_t \omega^{\gamma}_{0 \lambda } [ \omega^2_i - ( \omega^{\gamma}_{0 \lambda})^2]^2 } \bigg]^{-1 } \ , , \label{mccomp}\end{aligned}\ ] ] where @xmath171 are the polariton dispersion branches calculated with eq.([mc ] ) . for a given @xmath163 the x components satisfy the sum rule , @xmath172 . the exciton components , which correspond to the 0@xmath4-lb , 1@xmath4-lb , and 1@xmath4-ub dispersions shown in fig.1 , are plotted in fig.2 . the above polariton branches have non - zero x component when the frequencies @xmath173 resonate with the x state , i.e. , at @xmath174 for the 1@xmath4-ub , at @xmath175 for the 1@xmath4-lb , and at @xmath168 for the 0@xmath4-lb , respectively ( see fig.2 ) . in our microcavities the x component of the @xmath176- , @xmath177- etc . eigenmode mc polaritons is negligible . -lb ( dotted line ) , 1@xmath4-lb ( solid line ) , and 1@xmath4-ub ( dashed line ) polaritons in a zero - detuning gaas microcavity . , width=302 ] a non - ideal optical confinement of the mc photon modes by distributed bragg reflectors ( dbrs ) leads to the leakage of mc photons and gives rise to the radiative rate @xmath121 in eq.([mc ] ) . thus the radiative width of mc polaritons , due to their optical escape through the dbrs , is @xmath178 , where the photon component of the polaritons is given by @xmath179 . note that for our gaas - based macrocavities at low temperatures , @xmath121 is much larger than @xmath82 @xcite . the quasi-2d excitonic molecules in single qws without co - planar optical confinement of the light field can either resonantly dissociate into interface polaritons or decay radiatively into the bulk photon modes . in our optical experiments , which deal with pump and signal bulk photons only , the first route of the xx optical decay can not be visualized directly . thus this channel refers to the `` hidden '' optics associated with the evanescent light field resonantly guided by qw excitons . \(i ) _ resonant dissociation of qw excitonic molecules into outgoing interface polaritons . _ the bipolariton model allows us to calculate the xx radiative width @xmath180 , associated with the resonant dissociation of the molecule with in - plane translational momentum @xmath181 , by solving the wave equation @xcite : @xmath182 \tilde{\psi}_{\rm xx}({\bf p}_{\|},{\bf k}_{\| } ) \nonumber \\ & & + \ f_{\rm ip}({\bf p}_{\|},{\bf k}_{\| } ) \sum_{\bf p'_{\| } } w_{\sigma^+\sigma^-}({\bf p}_{\| } - { \bf p'}_{\| } ) \tilde{\psi}_{\rm xx}({\bf p'}_{\|},{\bf k}_{\| } ) \nonumber \\ & & \ \ \ \ \ \ \ \ \ = \tilde{e}_{\rm xx}^{\rm qw}({\bf k}_{\| } ) \tilde{\psi}_{\rm xx}({\bf p}_{\|},{\bf k}_{\| } ) \ , . \label{bpint}\end{aligned}\ ] ] here @xmath183 and @xmath184 are the bipolariton ( xx ) wavefunction and energy , respectively , @xmath185 is the qw polariton energy determined by the dispersion eq.([pol ] ) , @xmath186 , where @xmath187 is given by eq.([comp ] ) , @xmath78 is the in - plane momentum of the relative motion of the optically - dressed constituent excitons , and @xmath188 is the attractive coulombic potential between @xmath12- and @xmath13- polarized qw excitons . the complex bipolariton energy can also be rewritten as @xmath189 , where @xmath43 is the xx binding energy with no renormalization by the coupling with the vacuum light field . for the non - local deuteron model potential @xmath190 , which yields within the standard schrdinger two - particle ( two - x ) equation the wavefunction @xmath191^{3/2}$ ] for an optically inactive molecule , the bipolariton wave eq.([bpint ] ) is exactly - solvable @xcite . the input parameters of the model are the binding energy @xmath43 and the oscillator strength @xmath102 . thus the exactly - solvable bipolariton model simplifies the exciton - exciton interaction , but treats rigorously the ( interface ) polariton effect . \(ii ) _ resonant decay of qw excitonic molecules into the bulk photon modes . _ the decay occurs when at least one of the constituent excitons of a qw molecule moves within the radiative zone , i.e. , when @xmath192 and/or @xmath193 . note that the exciton - exciton resonant coherent coulombic scattering within the molecule state intrinsically couples the x radiative and qw polariton modes . thus the xx width , associated with the optical decay into the bulk photon modes , is given by @xmath194 \ , , \label{rada}\end{aligned}\ ] ] where @xmath195 and @xmath196 is given by eq.([rad ] ) . in the above integral over the qw radiative zone we approximate @xmath197 by the deuteron wavefunction . for @xmath198 eq.([rada ] ) yields @xmath199 = @xmath200=0 ) . however , for our reference gaas qw with weak confinement of the electronic states one has @xmath201 so that the above simple approximation of eq.([rada ] ) can not be used . . the xx radiative widths associated with the decay into interface polaritons , @xmath202 , and into bulk photon modes , @xmath203 , are plotted separately . the input xx binding energy @xmath204mev . the two circle symbols show @xmath205 and @xmath10 inferred from the experimental data . , width=302 ] in fig.3 we plot the radiative widths @xmath206=0 ) , @xmath207=0 ) , @xmath208=0 ) , and @xmath207=0 ) + @xmath208=0 ) against the oscillator strength of qw excitons @xmath102 . the widths are calculated with eqs.([bpint ] ) and ( [ rada ] ) for the input xx binding energy @xmath209mev . as we discuss in section iii , the oscillator strength @xmath102 of the high - quality reference qw used in our experiments is given by @xmath210=@xmath211 . the above value , which is inferred from the experimental data , refers to the gaas qw sandwiched between semi - infinite bulk algaas barriers and is consistent with that estimated in the previous subsection by using eq.([strength ] ) . a cap layer on top of the reference single qw modifies the evanescent field associated with interface polaritons and reduces the oscillator strength to @xmath212=@xmath213 ( for the details see section iv ) . as shown in fig.3 , for @xmath214 eqs.([bpint ] ) and ( [ rada ] ) yield @xmath207=0 ) @xmath215 and @xmath208=0 ) @xmath216 , so that the total xx radiative width is given by @xmath217=0 ) = @xmath218mev . for @xmath219 one calculates @xmath207=0 ) @xmath220 , @xmath208=0 ) @xmath221 , and @xmath217=0 ) = @xmath222mev . the latter value is indeed very close to the xx radiative width @xmath223mev inferred from our opical experiments with the reference qw ( see section iii ) . the photon - assisted resonant dissociation of qw molecules into outgoing interface polaritons is more efficient than the xx optical decay into the bulk photon modes by factor 4.5 for @xmath224 and by factor 3.8 for @xmath219 , respectively . this conclusion is consistent with that of ref . [ ] , where for the limit of strong qw confinement ( @xmath225 ) the relative efficiency of the two optical decay channels was estimated to be @xmath226 . the latter ratio refers to the idealized case of an extremely narrow gaas qw surrounded by infinitely thick algaas barriers . the resonant optical dissociation of the qw molecules into interface polaritons is much stronger than the radiative decay into the bulk photon modes , because the constituent excitons in their relative motion move mainly outside the radiative zone , with the in - plane momenta @xmath227 . in this case the excitons are optically dressed by the evanescent light field , i.e. , they exist as qw polaritons and , therefore , decay mainly into the confined , qw - guided interface modes . the picture can also be justified by analyzing the joint density of states relevant to the two optical decay channels . note that in both main equations , eq.([bpint ] ) and eq.([rada ] ) , @xmath228 does represent the dimensionless smallness parameter of the ( bipolariton ) model . the bipolariton model for excitonic molecules in @xmath4-microcavities requires to construct the xx state in terms of quasi - bound 0@xmath4-lb , 1@xmath4-lb , and 1@xmath4-ub polaritons . in this case the radiative corrections to the xx state with @xmath229 are given by @xmath230 where @xmath231 \psi_{\rm xx}^{(0)}(p_{\| } ) \ , , \nonumber \\ b&= & { 27 \over 16 } { 1 \over \sqrt{2 \pi } } \ \epsilon_{\rm xx}^{(0 ) } \int_0^{+ \infty } \ ! \ ! \ ! p_{\| } dp_{\| } \ , { \tilde g}(p_{\| } ) \psi_{\rm xx}^{(0)}(p_{\| } ) \ , . \label{mcab}\end{aligned}\ ] ] in eq.([mcab ] ) the bipolariton green function @xmath232 is @xmath233 ^ 2 \over e_{\rm xx}^{\rm mc } - \hbar \omega_i^{\rm mc}(p_{\| } ) - \hbar \omega_j^{\rm mc}(-p_{\| } ) + i \gamma_0 } \ , , \label{green}\ ] ] where @xmath234=@xmath235 , the mc polariton eigenfrequency @xmath236 and the x component @xmath237 ^ 2 $ ] with @xmath238 0@xmath4-lb , 1@xmath4-lb , and 1@xmath4-ub are given by eq.([mc ] ) and eq.([mccomp ] ) , respectively , and @xmath239 . the xx radiative corrections , i.e. , the lamb shift @xmath38 and the radiative width @xmath240 , depend upon the relative motion of the constituent qw excitons over whole momentum space , i.e. , eqs.([mcrad])-([mcab ] ) include integration over @xmath241 . the change of the input xx binding energy , @xmath242 , occurs because in their relative motion the constituent excitons move along the mc polariton dispersion curves , rather than possess the quadratic dispersion , @xmath243 ( the latter is valid only for optically inactive excitons ) . the solution of the exactly - solvable bipolariton model , given by eqs.([mcrad])-([green ] ) , includes all possible channels of the in - plane dissociation of the microcavity molecule into two outgoing mc polaritons , i.e. , `` xx ( @xmath244=0 ) @xmath5 @xmath245th - branch mc polariton ( @xmath246 ) @xmath247 @xmath248th - branch mc polariton ( @xmath249 ) '' . note that the solution of the bipolariton wave eq.([bpint ] ) for excitonic molecules in a single qw can be obtained from eqs.([mcrad])-([green ] ) by putting @xmath250 = ip and replacing @xmath251 and @xmath236 by @xmath252 and @xmath253 , respectively . and @xmath0 , calculated against the mc detuning @xmath57 with eqs.([mcrad])-([green ] ) for the mc rabi energies @xmath254mev and @xmath255mev . the input xx binding energy @xmath56 is 0.9mev ( dash - dotted line ) , 1.0mev ( solid line ) , and 1.1mev ( dashed line ) . , width=302 ] the radiative width @xmath256 and the lamb shift @xmath62 calculated by eqs.([mcrad])-([green ] ) as a function of the mc detuning @xmath257 between the 1@xmath4 cavity mode and qw exciton are plotted in fig.4 for three values of the input xx binding energy , @xmath258mev , 1.0mev , and 1.1mev . by applying eq.([strr ] ) , we estimate for this plot the rabi frequencies , @xmath259 and @xmath260 , relevant to the used three - branch mc polariton dispersion given by eq.([mc ] ) . namely , for @xmath261 , associated with the reference qw , and @xmath262 , eq.([strr ] ) yields @xmath263mev and @xmath264mev . as a result of non - ideal optical confinement in the @xmath11-direction by dbrs , our gaas - based @xmath4-microcavity ( i ) has a smaller value of @xmath259 , i.e. , @xmath265mev and ( ii ) with increasing @xmath163 loses the strength of optical confinement for mc 1@xmath4-mode photons of frequency @xmath266 . the latter means that the mc photon radiative width @xmath121 is @xmath163-dependent and smoothly increases with increasing @xmath163 . the dbr optical confinement is completely relaxed for @xmath49 so that the dispersion eq.([mc ] ) becomes inadequate , and the microcavity 0@xmath4-lb polariton dispersion evolves towards the interface polariton dispersion , associated with the single qw and given by eq.([pol ] ) . thus , in order to model the experimental data with eqs.([mcrad])-([green ] ) , we use @xmath267mev and @xmath268mev , and replace the 0@xmath4-lb polariton dispersion by the interface , qw polariton dispersion with @xmath269 . for this case the plot of @xmath240 and @xmath0 against the detuning @xmath57 is shown in fig.10 ( for details see section iv ) . there are two sharp spikes in the dependence @xmath53 which are accompanied by the jump - like changes of the xx radiative width @xmath52 ( see figs.4 and 10 ) . the above structure is due to van hove critical points , @xmath59 and @xmath60 , in the joint density of the polariton states ( jdps ) relevant to the optical decay `` mc excitonic molecule @xmath244=0 @xmath5 mc polariton @xmath73 + mc polariton @xmath270 '' ( for the critical points we use the classification and notations proposed in ref . [ ] ) . the first critical point @xmath59 in energy - momentum space @xmath271 refers to a negative mc detuning @xmath272 and deals with the condition @xmath273=0@xmath274=0@xmath275=0 ) . this point is marginal for the optical decay `` xx @xmath5 1@xmath4-lb polariton + 1@xmath4-ub polariton '' : for @xmath276 the above channel is allowed , while it is absent for @xmath277 . the critical point @xmath60 occurs at a positive detuning @xmath278 , which corresponds to the condition @xmath273=0@xmath274=0@xmath279=0 ) , and is the main marginal point in the jdps for the xx optical dissociation into two outgoing 1@xmath4-lb polaritons . namely , for @xmath280 the molecule can decay into two 1@xmath4-lb polaritons , while for @xmath281 the optical decay of mc molecules with zero in - plane wavevector @xmath244 into @xmath50-lb polaritons is completely forbidden . with a very high accuracy of the order of @xmath282 , one finds from eq.([mc ] ) that @xmath283=@xmath284^{1/2}$ ] . thus from the energy - momentum conservation law we estimate the detunings @xmath285 : @xmath286 where @xmath287 is the true , `` measured '' binding energy of the bipolariton state @xmath244=0 , i.e. , of the optically dressed molecule . . the microcavity rabi energies are @xmath288mev and @xmath289mev , the mc detuning is zero . the solutions are shown by the bold points @xmath290 ( xx @xmath5 1@xmath4-lb polariton + 1@xmath4-lb polariton ) , @xmath291 ( xx @xmath5 0@xmath4-lb polariton + 0@xmath4-lb polariton ) , and @xmath292 ( xx @xmath5 0@xmath4-lb polariton + 1@xmath4-lb polariton ) . the efficiency of the last decay channel is negligible in comparison with that of the first two . the xx binding energy @xmath293mev . , width=302 ] 1@xmath4-lb polariton + 1@xmath4-lb polariton '' and `` xx @xmath5 1@xmath4-lb polariton + 1@xmath4-ub polariton '' ( the mc detuning @xmath294mev , the marginal solution @xmath295 at @xmath296 refers to the critical point @xmath59 ) ; ( b ) the decay path `` xx @xmath5 1@xmath4-lb polariton + 1@xmath4-lb polariton '' ( the mc detuning @xmath297 ) ; ( c ) the decay path `` xx @xmath5 1@xmath4-lb polariton + 1@xmath4-lb polariton '' ( the mc detuning @xmath298mev , the marginal solution @xmath290 at @xmath296 refers to the critical point @xmath60 ) ; ( d ) the decay channels `` xx @xmath5 0@xmath4-lb polariton + 0@xmath4-lb polariton '' and `` xx @xmath5 0@xmath4-lb polariton + 1@xmath4-ub polariton '' ( this plot is practically independent of @xmath57 ) . the mc rabi frequencies @xmath259 and @xmath260 , and the xx binding energy @xmath56 are the same as in fig.5 . , width=302 ] in order to visualize the optical decay channels of mc excitonic molecules , in figs.5 and 6 we plot the graphic solution of the energy - momentum conservation law , @xmath299 ( @xmath238 0@xmath4-lb , 1@xmath4-lb , and 1@xmath4-ub ) . the roots of the equation are the poles of the bipolariton green function @xmath300 given by eq.([green ] ) . figure 5 , which refers to the zero - detuning gaas - based microcavity , clearly illustrates that apart from the decay path `` xx @xmath5 1@xmath4-lb polariton + 1@xmath4-lb polariton '' there are also the decay routes which involve the 1@xmath4-lb and 0@xmath4-lb microcavity polaritons with @xmath49 , i.e. , `` xx @xmath5 0@xmath4-lb polariton + 0@xmath4-lb polariton '' and `` xx @xmath5 1@xmath4-lb polariton + 0@xmath4-lb polariton '' . the graphic solution of energy - momentum conservation for the wavevector domain @xmath152 is shown in a magnified scale in figs.6a-6c for @xmath301 , 0 , and @xmath278 , respectively . the touching points at @xmath296 between the 1@xmath4-upper and 1@xmath4-lower ( see fig.6a ) and 1@xmath4-lower and 1@xmath4-lower ( see fig.6c ) dispersion branches correspond to the @xmath59 and @xmath60 critical points , respectively . the graphic solution of the energy - momentum conservation law is shown in fig.6d for the vicinity of @xmath302 . according to eq.([mc ] ) , the 1@xmath4-lb and 0@xmath4-lb polaritons with @xmath49 practically do not depend upon the mc detuning @xmath57 , i.e. , the plot shown in fig.6d is not sensitive to @xmath57 . the value of the @xmath1-jump and @xmath0-spike nearby the critical point @xmath59 , i.e. , at @xmath301 , shows that the contribution of the decay path `` xx @xmath5 1@xmath4-ub polariton + 1@xmath4-lb polariton '' is rather small , about 1 - 2@xmath303 only . this is mainly due to a small value of the jdps in the decay channel . the main contribution to the xx radiative corrections in microcavities is due to the frequency - degenerate decay routes `` xx @xmath5 1@xmath4-lb polariton + 1@xmath4-lb polariton '' and `` xx @xmath5 0@xmath4-lb polariton + 0@xmath4-lb polariton '' ( or `` xx @xmath5 interface polariton + interface polariton '' , as a result of the relaxation of the transverse optical confinement at @xmath304 ) . the jdps associated with the first main channel is given by @xmath305 \theta ( \delta_2 - \delta ) \ , , \label{jdps}\end{aligned}\ ] ] where @xmath306 is the heaviside step function . the above jdps is relevant to the calculations done by the bipolariton eqs.([mcrad])-([green ] ) . the appearance of the dimensionless parameter @xmath58 on the right - hand side ( r.h.s . ) of eq.([jdps ] ) is remarkable . thus the same control parameter @xmath307 determines the optical decay of excitonic molecules in the reference single gaas qw and in the gaas - based microcavities . furthermore , the jdps given by eq.([jdps ] ) depends upon the mc detuning only through the step function @xmath308 . the latter dependence gives rise to the critical point @xmath60 . by comparing the xx radiative corrections for @xmath309 and @xmath281 ( see figs.4 and 10 ) , one concludes that the first main decay channel `` xx @xmath5 1@xmath4-lb polariton + 1@xmath4-lb polariton '' has nearly the same efficiency as the second one , `` xx @xmath5 0@xmath4-lb polariton + 0@xmath4-lb polariton '' ( or `` xx @xmath5 interface polariton + interface polariton '' ) . note that the `` virtual '' decay paths , like `` xx @xmath5 1@xmath4-ub polariton + 1@xmath4-ub polariton '' , still contribute to the xx lamb shift in microcavities , according to eqs.([mcrad])-([green ] ) . the investigated sample consists of an mbe grown gaas / al@xmath310ga@xmath311as single quantum well of the thickness @xmath133 and placed in the center of a @xmath4cavity . an alas / al@xmath312ga@xmath313as dbr of 25 ( 16 ) periods was grown at the bottom ( top ) of the cavity . the spacer layer is wedged , in order to tune the cavity mode along the position on the sample . details on the growth and sample design can be found in ref . [ ] . the optical properties of the reference single qw grown under nominally identical conditions are reported in ref . [ ] : the spectra show the ground - state heavy - hole ( hh ) and light - hole ( lh ) exciton absorption lines separated in energy by about 2.6mev . in the mc sample , the coupling of both hh and lh excitons with the 1@xmath4-mode cavity photons results in the formation of three 1@xmath4-eigenmode mc polariton dispersion branches , 1@xmath4-lb , 1@xmath4-mb , and 1@xmath4-ub @xcite . the 1@xmath4-mode polaritons have a narrow linewidth : the ratio between the hh rabi splitting and the polariton linewidths at zero detuning is about twenty @xcite . for the reference gaas qw at temperature @xmath314k the homogeneous width @xmath315 is dominated by the radiative decay . the absorption linewidth , measured along the @xmath11-direction and extrapolated to zero temperature , yields the hh x radiative width of @xmath316ev . note that this value is affected by optical interference which occurs at the position of the qw , @xmath124 , due to bulk photons emitted by the qw excitons and partly reflected back by the top surface ( @xmath317 nm ) of a cap layer . in this case one has a constructive interference which results in the enhancement of the light field at @xmath124 . by treating the optical interference effect , we estimate @xmath318 for the reference qw sandwiched between semi - infinite bulk algaas barriers . this radiative width yields the intrinsic oscillator strength of quasi-2d hh excitons @xmath319=@xmath211 ( see fig.3 ) . the measured characteristics of excitonic molecules in the reference qw are consistent with those reported in ref . [ ] : the xx binding energy @xmath67mev and the xx radiative width @xmath320mev . the latter value is obtained by extrapolating the measured homogeneous width @xmath321 to @xmath322k . the optical experiments with the mc sample were performed using a ti : sapphire laser source which generates fourier - limited 100fs laser pulses at 76mhz repetition rate . two exciting pulses , 1 and 2 , with variable relative delay time @xmath323 propagate along two different incident directions @xmath324 at small angle ( @xmath325 ) to the surface normal . pulse 1 precedes pulse 2 for @xmath326 . the reflectivity spectra of the probe light and the fwm signal were analyzed with a spectrometer and a charge - coupled device camera of 140@xmath327ev fwhm resolution . the sample was held in a helium bath cryostat at t=5k for all the pump - probe measurements and at t=9k in the fwm experiments . the pulse along @xmath328-direction induces only 1@xmath4-lb polaritons , and its spectrum is shown by the dotted line . , width=302 ] in order to measure the bipolariton dephasing we perform spectrally resolved fwm . the fwm signal was detected at @xmath329 in reflection geometry . the spot size of both exciting beams was @xmath330@xmath327 m . in fig.7 we plot the spectrally resolved fwm signal for different polarization configurations of the laser pulses . the positive detuning between the cavity 1@xmath4-eigenmode and hh exciton is @xmath331mev , and the delay time is @xmath323=1ps . pulse 1 of about 500fs duration was spectrally shaped to _ excite only the 1_@xmath4-_lb polaritons _ , and the fwm was probed with the spectrally broad pulse 2 at all 1@xmath4-mode polariton resonances . for co linear and cross - linear polarization configurations , the 1@xmath4-lb polariton to excitonic molecule transition ( 1@xmath4-lb xx ) is observed in the fwm signal ( see arrow in fig.7 ) at a spectral position consistent with that found in our previous pump - probe experiments @xcite . the xx - mediated fwm signal disappears for co circular polarization , in accordance with the polarization selection rules for the two - photon generation of excitonic molecules in a gaas qw . although the analysis of fwm in microcavities can be rather complicated @xcite , the interpretation of our measurements is simplified by the selective excitation of the 1@xmath4-lb polaritons only . the observed ti fwm is a free polarization decay , due to the dominant homogeneous broadening of the x lines in our high - quality 250@xmath332-wide qws @xcite . at positive delays the fwm signal is created by the following sequence . at first , pulse 1 induces a first order polarization associated with 1@xmath4-lb polaritons . the induced polarization decays with the dephasing time @xmath333 of the 1@xmath4-lb polaritons . the dephasing time @xmath333 is dominated by the lifetime of 1@xmath4-mode mc photons . pulse 2 interacts nonlinearly with the induced polarization , and a third order fwm signal is created with an amplitude that decreases with increasing @xmath323 , due to the decay of the first order polarization associated with the 1@xmath4-lb polaritons . fwm intensities at all probed resonances therefore decay nearly with the time constant @xmath334/2 . at negative @xmath323 the fwm signal stems from the two photon coherence of the crystal ground state to the excitonic molecule transition ( 0xx ) induced by pulse 2 . according to energy in - plane momentum conservation , since pulse 1 is resonant with 1@xmath4-lb polaritons only , the fwm signal , associated with bulk photons , is emitted in the direction @xmath335 with the energy of the 1@xmath4-lb xx transition . thus the ti fwm dynamics at negative time delays allows us to study the polarization decay of the 0xx transition @xcite , i.e. , to find @xmath68 . -lb xx transition , when pulse 1 resonantly induces the 1@xmath4-mode lower - branch polaritons only , and at the 1@xmath4-mb xx transition , when pulse 1 resonantly excites only the 1@xmath4-mode middle - branch polaritons . inset : the xx homogeneous linewidth @xmath68 against the mc detuning @xmath57 , measured at @xmath63k with about 4nj/@xmath336 pump fluence . , width=302 ] the @xmath323-dependence of the ti fwm signals associated with the 1@xmath4-lb xx and 1@xmath4-mb xx transitions is shown in fig.8 . as expected , at negative @xmath323 one finds the same dynamics for both transitions . therefore , independently of the 1@xmath4-eigenmode mc polariton branch selectively excited by pulse 1 , we can infer the polarization decay rate of the 0xx transition . the homogeneous linewidth of the 0xx transition @xmath68 measured at low excitation energies per pulse ( @xmath337nj/@xmath336 ) is potted against the mc detuning @xmath57 in the inset of fig.8 . only a weak detuning dependence of @xmath68 is observed for the detuning band @xmath338 . note that the deduced values @xmath339=9k ) @xmath340mev are by factor @xmath341 larger than @xmath342mev measured from the reference qw at nearly the same bath temperature t=10k @xcite ( see the dotted line in the inset of fig.8 ) . the bipolariton energy @xmath343 was found by analysing the pump - probe experiments . pulse 1 acts as an intense pump while pulse 2 is a weak probe . the spectrum of the pump pulse is shaped and tuned in order to excite resonantly the 1@xmath4-lb polaritons only . the spectrally broad probe pulse has a spot size of @xmath344@xmath327 m . in this case the in - plane spatial gradient of the polariton energy is not significant . in order to achieve a uniform pump density over the probe area , the cross - section of the pump pulse is chosen to be by factor two larger than that of the probe light . in ref . [ ] we show a well - resolved pump - induced absorption at the 1@xmath4-lb xx transition in the investigated mc sample . the 1@xmath4-lb xx absorption was observed in the reflectivity spectra at positive pump - probe delay times and for the cross - circularly ( @xmath12- and @xmath13- ) polarized pump and probe pulses , according to the optical selection rules . in particularly , the induced absorption for three different positive mc detunings was measured . here we extend the pump - probe experiment to study the detuning dependence @xmath345 , including @xmath346 . in fig.9 the probe reflectivity spectra measured at @xmath347ps for the cross - circularly polarized pump and probe pulses is plotted . indicated by the arrows ( see fig.9 ) , a spectrally well - resolved pump - induced absorption resonance is observed . in the upper left - hand side ( l.h.s . ) part of fig.9 the energy position of the 1@xmath4-lb , 1@xmath4-mb , and 1@xmath4-ub polariton resonances and of the induced 1@xmath4-lb xx absorption are plotted against the mc detuning @xmath57 . the fit done with a three - coupled - oscillator scheme ( 1@xmath4-eigenmode mc photon , hh exciton , and lh exciton resonances ) are shown by the solid lines . the energies @xmath348 and @xmath349 of the hh and lh excitons ( @xmath350ev and @xmath351ev ) are inferred from the fit , and the molecule energy @xmath352 is determined as the sum of the measured 1@xmath4-lb and 1@xmath4-lb xx transition energies . the bipolariton binding energy , evaluated as @xmath353 , is plotted against the mc detuning @xmath57 in the lower l.h.s . part of fig.9 . we find that @xmath354mev , i.e. , is similar to the value of @xmath46 in the reference single qw and slightly larger than that previously reported in ref . ps and @xmath66k . the pump fluence is about 0.1@xmath327j/@xmath336 . the arrows indicate the 1@xmath4-lp xx pump - induced absorption . upper left inset : the measured energy position of the 1@xmath4-lp , 1@xmath4-mp , 1@xmath4-up resonances ( filled square points ) and of the induced 1@xmath4-lp xx absorption ( unfilled square points ) versus the mc detuning @xmath57 . the fit of the mc polariton branches , associated with lh and hh qw excitons , is shown by the solid lines . lower left inset : the xx binding energy @xmath355 determined as the difference between twice the bare hh exciton energy and the sum of the 1@xmath4-lp and 1@xmath4-lp xx transition energies . , width=302 ] the optical decay of mc bipolaritons can also occur directly , through escape of the photon component of the constituent @xmath12- and @xmath13-polarized mc polaritons into the bulk photon modes . the xx radiative width associated with this channel is given by @xmath356 \big ] p_{\| } dp_{\| } \ , , \label{mcbulk}\end{aligned}\ ] ] where @xmath357 are determined by eq.([mccomp ] ) and @xmath245 runs over 0@xmath4-lb , 1@xmath4-lb , and 1@xmath4-ub . equation ( [ mcbulk ] ) is akin to eq.([rada ] ) and can be interpreted in terms of optical evaporation of the mc excitonic molecules through the dbr mirrors . using the measured radiative linewidth of 1@xmath4-lb polaritons , @xmath358mev , we estimate @xmath359mev , so that the radiative lifetime of mc photons is given by @xmath360ps . in this case eq.([mcbulk ] ) yields @xmath361ev for @xmath362mev and assuming that @xmath121 is @xmath163-independent . thus @xmath363 is less than @xmath364 , estimated with eq.([rada ] ) for the reference qw ( see fig.3 ) , by more than one order of magnitude . this is because instead of the smallness parameter @xmath228 , which appears on the r.h.s . of eq.([rada ] ) , eq.([mcbulk ] ) is scaled by @xmath365 . the radiative width @xmath363 , associated with the decay of xxs into the bulk photon modes , is by two orders of magnitude less than @xmath366 calculated with eqs.([mcrad])-([green ] ) . thus the resonant in - plane dissociation of molecules into outgoing mc polaritons absolutely dominates in the xx - mediated optics of microcavities , so that the total xx radiative width is given by @xmath367 ( see figs.4 and 10 ) . the extremely small value of @xmath363 allows us to interpret a mc excitonic molecule as a nearly `` optically - dark '' state with respect to its direct decay into the bulk photon modes . however it is the resonant coupling between 1@xmath4-mode cavity polaritons and external bulk photons which is responsible for the optical generation and probe of the xx states in microcavities : our optical experiments deal only with bulk pump , probe , and signal photons . in the meantime the bipolariton wavefunction @xmath41 is constructed in terms of 0@xmath4-lb , 1@xmath4-lb , and 1@xmath4-ub polariton states , and umklapp between the mc polariton branches occurs through the coherent coulombic scattering of two constituent polaritons . while the interpretation of the experimental data ( see section iii ) does require three - branch , 1@xmath4-lb , 1@xmath4-mb , and 1@xmath4-ub , polaritons associated with hh and lh excitons , the contribution to the xx optics from the lh xs is very small . this occurs because ( i ) the energy @xmath349 is well - separated from the xx - mediated resonance at @xmath368 ( the relevant ratio between @xmath369 and @xmath370 is equal to 0.16 , i.e. , is much less than unity ) and ( ii ) because a contribution of the lh exciton to the total xx wavefunction is unfavorable in energy , i.e. , is rather minor . we have checked numerically that by the first argument only the lh - x resonance can not change the xx radiative corrections for more than 3 - 5@xmath303 . thus the bipolariton model we develop to analyze the optical properties of mc excitonic molecules and to explain the experimental data deals only with 0@xmath4-lb , 1@xmath4-lb , and 1@xmath4-ub polaritons associated with the ground - state hh exciton . and @xmath0 , calculated versus the mc detuning @xmath57 with eqs.([mcrad])-([green ] ) for the mc rabi energy @xmath371mev and assuming that the dbr optical confinement is completely relaxed for @xmath372 . in this case the 0@xmath4-lb dispersion is replaced by the interface polariton dispersion with the oscillator strength @xmath373 . the input xx binding energy @xmath56 is 0.9mev ( dash - dotted line ) , 1.0mev ( solid line ) , and 1.1mev ( dashed line ) . , width=302 ] in fig.10 we plot the xx radiative corrections against the mc detuning @xmath57 , calculated with eqs.([mcrad])-([green ] ) by using the mc parameters adapted to our gaas microcavities . namely , the 1@xmath4-mode cavity rabi splitting is given by @xmath374mev , and we assume that the dbr optical confinement follows the step function @xmath375 . for @xmath376 the 0@xmath4-lb is replaced by the interface polariton dispersion given by eq.([pol ] ) . due to the absence of the dbr transverse optical confinement at @xmath376 , the resonant optical decay of the constituent excitons into the bulk photon mode is also included in our calculations by using eq.([rada ] ) with integration over @xmath377 from @xmath378 to @xmath23 . from fig.10 we conclude that for the detuning band @xmath379 the radiative width @xmath1 is about @xmath380mev and indeed weakly depends upon @xmath57 , in accordance with our experimental data . a few @xmath327ev @xmath1-jump , associated with the critical point @xmath59 , is too small to be detected in the current experiments . note that the contribution to @xmath1 from the decay channel `` xx @xmath5 1@xmath4-lb polariton + 1@xmath4-lb polariton '' can easily be estimated within a standard perturbation theory : @xmath381 , where the jdps is given by eq.([jdps ] ) . the above estimate yields @xmath382mev and is consistent with the value of the @xmath1-jump around @xmath383 , i.e. , at the @xmath60 critical point ( see fig.10 ) . an observation of @xmath384mev at @xmath281 , when the mc excitonic molecules become optically dark with respect to the decay into 1@xmath4-mode mc polaritons , would be a direct visualization of the hidden decay path `` xx @xmath5 interface polariton + interface polariton '' . the _ relative _ change of the xx radiative corrections is rather small to be observed in the tested mc detuning band @xmath385mev with the current accuracy of our measurements : eqs.([mcrad])-([green ] ) yield @xmath386ev and @xmath387ev ; the energy structure at @xmath388 , i.e. , nearby the critical point @xmath59 , is also of a few @xmath327ev only ( see fig.10 ) . on the other hand , the gaas - based microcavities we have now do not allow us to test the critical point @xmath60 which is located in the mc detuning band 5mev @xmath389 8mev . in the latter case the relative change of @xmath38 and @xmath240 is large enough , about @xmath390mev , to be detected in our experiments . high - precision modulation spectroscopy is very relevant to observation of the critical points , because the derivatives @xmath391 ( @xmath392 ) and @xmath393 ( @xmath392 ) undergo a sharp change in the spectral vicinity of @xmath51 . the modulation of @xmath57 can be done by applying time - dependent quasi - static electric @xcite , magnetic @xcite or pressure @xcite fields . note that the measurement of the detunings @xmath272 and @xmath278 will allow us to determine with a very high accuracy , by using eqs.([vanhove ] ) , the xx binding energy @xmath355 and the mc rabi frequency @xmath394 . a detailed study of the xx lamb shift @xmath38 versus the mc detuning @xmath57 and , in particular , the detection of the critical points @xmath59 and @xmath60 are the issue of our next experiments . in order to estimate the radiative width @xmath1 from the total homogeneous width @xmath68 measured at @xmath395k in our fwm experiment , we assume that apart from the xx radiative decay the main contribution to @xmath68 is due to temperature - dependent xx la - phonon scattering . note that in the experiment we deal with a low - intensity limit , when @xmath68 is nearly independent of the excitation level . thus @xmath396 , where @xmath397 is due to the scattering of qw molecules by bulk la - phonons . the dbr optical confinement does not influence the xx la phonon scattering , so that the width @xmath398 is the same for xxs in the reference single qw and in the microcavities . @xmath397 is given by @xmath399 where @xmath400 , @xmath401 is the longitudinal sound velocity , @xmath402 is the x deformation potential , @xmath403 is the crystal ( gaas ) density , @xmath404 $ ] , and @xmath405 . the form - factor @xmath406[e^{ix}/(1 - x^2/\pi^2)]$ ] refers to an infinite rectangular qw confinement potential and describes the relaxation of the momentum conservation law in the @xmath11-direction . the dimensionless parameter @xmath407 is given by @xmath408 . associated with scattering of qw excitonic molecules by bulk la - phonons . the calculations are done with eq.([xx - la ] ) for the x deformation potential @xmath409ev ( dashed line ) , 10ev ( solid line ) , and 12ev ( dotted line ) . , width=302 ] the values of the deformation potential @xmath402 , published in literature , disperse in the band @xmath410ev . in fig.11 we plot @xmath398 calculated by eq.([xx - la ] ) for @xmath411 8 , 10 , and 12ev . the deformation potential @xmath409ev , which gives @xmath412mev and is close to @xmath413ev reported for gaas in ref . [ ] , fits the temperature dependence @xmath398 measured for the reference qw . in particular , @xmath414mev is inferred from the total @xmath415mev ( see the inset of fig.8 ) . thus from our fwm measurements of @xmath68 at @xmath63k we conclude that the xx radiative width @xmath416 is about @xmath417mev , i.e. , is consistent with the values calculated within the bipolariton model ( see fig.10 ) . in order to apply the bipolariton model [ see eq.([bpint ] ) ] to excitonic molecules in the reference single qw , one should take into account that the reference qw is sandwiched between a thick substrate and a cap layer of the thickness @xmath418 nm . the evanescent light field associated with the qw polaritons is modified by the cap layer . indeed , for the @xmath419 energy detuning from the x resonance , one estimates that @xmath420 so that @xmath421 is not negligible . the estimate refers to two frequency - degenerate outgoing interface polaritons ( @xmath422 ) created in the photon - assisted resonant dissociation of the qw molecule with @xmath423 . at @xmath424 the initial evanescent field splits into two evanescent fields , `` transmitted '' to air ( or vacuum ) and `` reflected '' back towards the qw . the first light field very effectively decays in the @xmath11-direction , with @xmath425 . the `` reflected '' evanecsent light field makes at @xmath11=0 a destructive superposition with the initial evanecsent field , because the reflection coefficient of the top surface of the cap layer is @xmath426 . the destructive superposition stems from the @xmath427-jump of the phase of the `` reflected '' evanescsent field . thus the effective oscillator strength relevant to the qw bipolariton wave eq.([bpint ] ) is given by @xmath428 ^ 2 $ ] . for our reference structure with @xmath429 we estimate @xmath430 . in this case eqs.([bpint])-([rada ] ) yield the total radiative width @xmath217=0 ) @xmath431 ( see fig.3 ) , the value which is very close to @xmath432 obtained from the experimental data . thus the bipolariton model , which attributes the xx radiative corrections mainly to the in - plane dissociation of molecules into outgoing interface / mc polaritons , reproduce quantitatively the xx radiative widths @xmath1 and @xmath10 estimated from the experimental data . the two main channels for the xx decay in microcavities , `` xx @xmath5 1@xmath4-lb polariton + 1@xmath4-lb polariton '' and `` xx @xmath5 0@xmath4-lb ( or interface ) polariton + 0@xmath4-lb ( or interface ) polariton '' in comparison with the one leading decay route in single qws , `` xx @xmath5 interface polariton + interface polariton '' , explain qualitatively the factor two difference between @xmath1 and @xmath10 . the xx - mediated optics of microcavities does require to include the `` hidden '' 0@xmath4-cavity ( or interface , if the transverse optical confinement is relaxated for large @xmath73 ) polariton mode , which is invisible in standard optical experiments and , therefore , is usually neglected . furthermore , with decreasing temperature @xmath314k @xmath68 and @xmath69 effectively approach @xmath1 and @xmath10 , respectively , so that the dephasing of the two - photon xx polarization in the microcavities and the reference qw occurs mainly through the optical decay of the molecules . thus the @xmath433 limit holds for the xx - mediated optics in our high - quality nanostructures and justifies the bipolariton model . the latter interprets the xx optical response in terms of resonant polariton - polariton scattering and requires nonperturbative treatment of both leading interactions , exciton - exciton coulombic attraction and exciton - photon resonant coupling . note that in our calculations with the exactly solvable bipolariton model only two control parameters of the theory , the input xx binding energy @xmath43 and the mc rabi frequency @xmath434 ( or the x oscillator strength @xmath83 for the reference qw ) , are taken from the experimental data . no fitting parameters are used in the numerical simulations . the relative motion of two optically - dressed constituent excitons of the bipolariton eigenstate ( i.e. , of the excitonic molecule ) is affected by the exciton - photon interaction , according to the polariton dispersion law . the optically - induced change of the x energy occurs not only in the close vicinity of the resonant crossover between the initial photon and exciton dispersions , but in a rather broad band of @xmath435 ( or @xmath73 ) . for example , in bulk semiconductors the effective mass associated with the upper polariton dispersion branch at @xmath35 is given by @xmath436 } \ , . \label{mass3d}\ ] ] for bulk gaas eq.([mass3d ] ) yields @xmath437 nearly by factor four less than the translational mass relevant to the pure excitonic dispersion , @xmath438 . from the microcavity dispersion eq.([mc ] ) one estimates for @xmath439 the effective masses associated with the 1@xmath4-eigenmode polariton dispersion branches : @xmath440 } \ , , \label{massmc}\ ] ] where we assume that @xmath441 . in particular , for a zero - detuning gaas - based microcavity eq.([massmc ] ) yields @xmath442=@xmath443=@xmath444 . in the meantime , at relatively large in - plane momenta @xmath445 the @xmath50-lb polariton energy smoothly approaches the exciton dispersion , i.e. , @xmath446 \big|_{p_{\| } \sim p_{\|}^{(1\lambda ) } } \rightarrow [ \hbar ( \omega^{\rm mc}_{\rm 1\lambda})^2 \omega_t]/[2 ( c^2 p_{\|}^2/\varepsilon_b ) ] \propto 1/p_{\|}^2 $ ] , according to eq.([mc ] ) . while the above difference is rather small in absolute energy units , being compared with the in - plane kinetic energy of the exciton , @xmath447 , it can not be neglected . for example , the difference @xmath448 becomes equal to @xmath449 at @xmath450 . note that for the above value of the in - plane wavevector @xmath163 the photon component , associated with 1@xmath4-lb polaritons , is negligible , i.e. , @xmath451 . because it is a balance between the positive kinetic and negative interaction energies of the constituent excitons that gives rise to an excitonic molecule , the described optically - induced changes of the x effective mass at @xmath296 and the nonparabolicity of the x dispersion at large @xmath163 are responsible for the large xx radiative corrections in quasi-2d gaas nanostructures . in this paper we have studied , both theoretically and experimentally , the optical properties of qw excitonic molecules in semiconductor ( gaas ) microcavities . we attribute the main channel of the xx optical decay to the resonant dissociation of mc molecules into outgoing mc polaritons , so that the xx - mediated optical signal we detect is due to the resonant radiative escape of the secondary mc polaritons through the dbrs . the bipolariton model has been adapted to construct the xx wavefunction @xmath41 in terms of two ( 1@xmath4-ub , 1@xmath4-lb and 0@xmath4-lb ) mc polaritons quasi - bound via coulombic attraction of their exciton components . the mc bipolariton wave equation gives the radiative corrections to the xx state in microcavities . the following conclusions summarize our results . \(i ) the radiative corrections to the excitonic molecule state in gaas - based microcavities , the xx lamb shift @xmath0 and the xx radiative width @xmath1 , are large ( about @xmath452 of the xx binding energy @xmath43 ) and definitely can not be neglected . \(ii ) while usually the qw exciton mediated optics of semiconductor microcavities is formulated in terms of two 1@xmath4-mode polariton dispersion branches only ( 1@xmath4-ub and 1@xmath4-lb , according to the terminology used in our paper ) , we emphasize the importance of the 0@xmath4-mode lower - branch polariton dispersion : the coulombic interaction of the constituent excitons , which is responsible for the xx state , does couple intrinsically three relevant mc polariton branches , ( 1@xmath4-ub , 1@xmath4-lb , and 0@xmath4-lb ) . furthermore , the xx decay path `` xx @xmath5 0@xmath4-lb polariton + 0@xmath4-lb polariton '' is comparable in efficiency with the optical decay into 1@xmath4-lb polariton modes , i.e. , `` xx @xmath5 1@xmath4-lb polariton + 1@xmath4-lb polariton '' . due do the relaxation of the dbr optical confinement for in - plane wavevectors @xmath453 , with increasing @xmath163 the 0@xmath4-lb evolves towards the interface polariton dispersion associated with qw excitons . however , the short - wavelength lb polaritons with @xmath304 always contribute to the xx - mediated optics of microcavities . \(iii ) the zero - temperature extrapolation of the experimentally found xx dephasing width @xmath454=9k ) yields @xmath455=0k ) @xmath456mev and is in a quantitative agreement with the result of the exactly solvable bipolariton model , @xmath457mev . from the analysis of the experimental data we conclude that the bipolariton model of mc excitonic molecules , which requires @xmath458 limit , is valid for our high - quality gaas - based nanostructures at @xmath314k . for the reference gaas qw without the dbr transverse optical confinement we find @xmath459=0k ) @xmath460mev . the latter value is also quantitatively consistent with that calculated by solving the qw bipolariton wave equation , @xmath461mev . the nearly factor two difference between @xmath462 and @xmath240 clearly demonstrates the existence of the additional decay channel for a quasi-2d excitonic molecule in microcavities [ `` xx @xmath5 interface polariton + interface polariton '' in mc - free single qws versus `` xx @xmath5 0@xmath4-lb ( or interface ) polariton + 0@xmath4-lb ( or interface ) polariton '' and `` xx @xmath5 1@xmath4-lb polariton + 1@xmath4-lb polariton '' for mc - embedded qw molecules ] . \(iv ) the critical van hove points , @xmath463 and @xmath464 , in the jdps of the resonant optical channel `` xx ( @xmath244=0 ) @xmath14 two 1@xmath4-mode mc polaritons '' can allow us to find accurately the molecule binding energy @xmath355 and the mc rabi frequency @xmath15 . thus , by using time - dependent mc detuning @xmath465 , we propose to develop high - precision modulation spectroscopies in order to detect the rapid changes of the xx radiative corrections at @xmath466 [ spikes in the xx lamb shift @xmath467 and jumps in the xx radiatve width @xmath468 and estimate @xmath355 and @xmath15 . we appreciate valuable discussions with j. r. jensen and j. m. hvam . support of this work by the dfg , epsrc and eu rtn project hprn - ct-2002 - 00298 is gratefully acknowledged .

the optical properties of excitonic molecules ( xxs ) in gaas - based quantum well microcavities ( mcs ) are studied , both theoretically and experimentally . we show that the radiative corrections to the xx state , the lamb shift @xmath0 and radiative width @xmath1 , are large , about @xmath2 of the molecule binding energy @xmath3 , and definitely can not be neglected . the optics of excitonic molecules is dominated by the in - plane resonant dissociation of the molecules into outgoing 1@xmath4-mode and 0@xmath4-mode cavity polaritons . the later decay channel , `` excitonic molecule @xmath5 0@xmath4-mode polariton + 0@xmath4-mode polariton '' , deals with the short - wavelength mc polaritons invisible in standard optical experiments , i.e. , refers to `` hidden '' optics of microcavities . by using transient four - wave mixing and pump - probe spectroscopies , we infer that the radiative width , associated with excitonic molecules of the binding energy @xmath6mev , is @xmath7mev in the microcavities and @xmath8mev in a reference gaas single quantum well ( qw ) . we show that for our high - quality quasi - two - dimensional nanostructures the @xmath9 limit , relevant to the xx states , holds at temperatures below 10k , and that the bipolariton model of excitonic molecules explains quantitatively and self - consistently the measured xx radiative widths . a nearly factor two difference between @xmath1 and @xmath10 is attributed to a larger number of the xx optical decay channels in microcavities in comparison with those in single qws . we also find and characterize two critical points in the dependence of the radiative corrections against the microcavity detuning , and propose to use the critical points for high - precision measurements of the molecule binding energy and microcavity rabi splitting .

supernova 1993j was discovered on 28 march 1993 in the spiral galaxy m81 ( at a distance of 3.63 mpc ; freedman et al . although it is factor of 65 further away than sn 1987a , it is relatively close for a sn and has been the subject of intense observational campaigns in a number of wavelengths regions ( chevalier 1997 , and references therein ) . shortly after the explosion , its spectrum was found to contain hydrogen lines - a type ii supernova - but within a few weeks strong helium lines developed . the spectrum looked then more like a type ib supernova , as though the envelope of hydrogen had lifted to reveal a helium layer ( filippenko et al . this , together with the observed peculiarities observed in the lightcurve , led many to conclude that the dying star , identified as a non - variable red supergiant in images taken before the explosion ( aldering et al . 1994 ) , must have lost a considerable amount of its outer envelope of hydrogen gas to a companion before it exploded ( nomoto et al . 1993 ; podsiadlowski et al . 1993 ; woosley et al . 1994 ) . but no companion had been seen until recently when the brightness of sn 1993j dimmed sufficiently that its spectrum showed the features of a massive star superimposed on the supernova ( maund et al . 2004 ) . given the estimates of the progenitor mass , the remnant is probably a neutron star rather than a black hole ( nomoto et al . 1993 ; podsiadlowski et al . 1993 ; woosley et al . 1994 ) , and the neutron star is generally expected to manifest as a pulsar . the newly born neutron star is characterized by its initial rotation rate and magnetic dipole moment . it will spin down , generating radiation and accelerating charged particles at the expense of its rotational energy . we expect the reprocessing of rotational energy to produce a bright flat - spectrum synchrotron nebula , but this soft emission has not been observed . here we developed a simple shocked wind model for the luminosity of the pulsar nebulae in sn 1993j ( section 2 ) , which is then compared with the current luminosity limits in order to place bounds on the dipole emission of the nascent neutron star and its birth properties ( section 4 ) . we also investigated the impact of the soft photon field radiation of the binary companion on the central pulsar wind ( section 3 ) , along with the types of observation that would help to unambiguously demonstrate whether or not the system is disrupted as a result of the supernova . our conclusions are discussed in section 4 . neutron stars formed in core collapse sn explosions are thought to emit radiation as pulsars , the crab ( rickett & seiradakis 1982 ) being the canonical example . pulsars are generally presumed to lose rotational energy by some combination of winds of relativistic particles and magnetic dipole radiation at the frequency of the neutron star s rotation . in the generic pulsar model , the field is assumed to maintain a steady value , and the luminosity declines as the spin rate slows down ( shapiro & teukolsky 1983 ) . the spindown law is then given by @xmath3 , so that @xmath4 , where @xmath5 is the characteristic time scale for dipolar spindown , @xmath6 is the dipolar field strength at the poles , @xmath7 is the radius of the light cylinder , @xmath8 is the initial rotation period in milliseconds , and @xmath9 . the power output of the pulsar is typically assumed to be @xmath10 , regardless of the detailed process by which the dipole emission is converted to the energy of charged particle acceleration and high - energy electromagnetic radiation . the pulses themselves , even when seen in the gamma - rays , constitute an insignificant fraction of the total energy loss . large fraction of the power output , on the other hand , goes into a bubble of relativistic particles and magnetic field surrounding the pulsar ( chevalier & fransson 1992 ) . this bubble gains energy from the pulsar , and loses it in synchrotron radiation losses and in doing work on the surrounding supernova gas , sweeping it up and accelerating it . the density structure of the supernova gas is determined by the initial stellar structure , as modified by the explosion ( arnett 1988 ) . the density structure has been relatively well determined for models of sn 1987a it shows an inner flat profile outside of which is a steep power - law decrease with radius ( e.g. shigeyama & nomoto 1990 ) . chevalier & soker ( 1989 ) approximated the velocity profile by an inner section @xmath11 and an outer section @xmath12 , where @xmath13 and @xmath14 for sn 1987a . the two segments of the density profile are assumed to be continuous and they intersect at a velocity @xmath15 . for an explosion with total energy @xmath16 and mass @xmath17 , one has @xmath18^{1/2},\ ] ] where @xmath19^{(3-m)/2}{(3-m)m_{\rm t } \over 4\pi}.\ ] ] sn 1987a was an unusual type ii supernova because of its small initial radius . this is in contrast to sn 1993j where the early lightcurve indicated that the progenitor star was more extended , about ten times larger than that of sn 1987a . the lack of a plateau phase in this case is attributed to the low mass of the hydrogen envelope , about 0.1 to 0.6 @xmath20 ( nomoto et al . 1993 ; podsiadlowski et al . 1993 ; woosley et al . 1994 ) , although the initial stellar mass was probably @xmath21 . woosley et al . ( 1994 ) , for example , estimated that the final presupernova star main - sequence star that lost most of its hydrogen envelope to a nearby companion ( @xmath22 , initially 4.5 au ) . ] had a helium and heavy element core of @xmath23 , a low density hydrogen envelope of @xmath24 , and a radius of about @xmath25 cm . the core collapse of the progenitor of 1993j resulted in the deposition of about @xmath26 erg of kinetic energy per solar mass ( woosley et al . the expanded density profile shows an outer steep power - law region with @xmath27 and an inner , relatively flat region ( blinnikov et al . 1998 and fig . 4 therein ) . the bend in the profile occurs at a velocity of @xmath28 . the inner density profile is clearly not a single power law in @xmath29 . however , it is likely that hydrodynamic instabilities , similar to those believed to occur in sn 1987a ( e.g. fryxell et al . 1991 ) , lead to mixing and smooth out any sharp features in the density profile . thus , a reasonable smooth , flat inner profile may be a good approximation . in the outer parts of the density structure , on the other hand , radiative transfer effects are more important for the explosion of an extended star and can lead to the formation of a dense shell in the outer layers . the shell is , however , expected out in the steep power - law region of the density profile , where the pulsar bubble is not likely to reach . in what follows , we assume @xmath13 or @xmath30 and @xmath14 . the first stage of evolution involves the interaction of the pulsar bubble with the inner density section . with the assumptions that the pulsar luminosity is constant during this phase ( i.e. @xmath31 ) and the supernova gas is swept up into a thin shell of mass @xmath32 and velocity @xmath33 , the radius of the pulsar bubble can be written as ( chevalier & fransson 1992 ) @xmath34^{1/(5-m)}t^{(6-m)/(5-m)},\ ] ] where @xmath35 . the shell velocity is @xmath36 @xmath37 $ ] @xmath38 for @xmath39 $ ] , where @xmath40 , @xmath41 erg , @xmath42 erg s@xmath43 , and @xmath44 yr . the density transition velocity is @xmath45e_{51}^{1/2}m_{{\rm t},5}^{-1/2}$ ] @xmath38 for @xmath39 $ ] , so it is likely that the shell remains within the inner part of the density profile . the density of the uniform gas shell is @xmath46 $ ] @xmath47 for @xmath39 $ ] , where @xmath48 . the presence of the companion implies that the supernova gas should be transparent to optical radiation ( i.e. the optical depth to electron scattering should be less than unity ) , which gives @xmath49 for @xmath13 and @xmath50 @xmath51 for @xmath52 . for x - ray energies @xmath53 10 kev , electron scattering also provides the main opacity . however , in this case the bound electrons contribute as well as the free ones , so that the optical depth is always given by electron scattering irrespectively of the degree of ionization . at lower x - ray energies @xmath54 3 kev , photoionization provides additional opacity ( bahcall et al . 1970 ) and may delay the time at which the envelope becomes transparent by an additional factor @xmath55 ( this estimate is for a predominantly neutral envelope of pure hydrogen ) . the delay is somewhat larger if the envelope is rich in heavy elements . the matter that is swept up by the pulsar bubble is subject to rayleigh - taylor instability and may form filaments ( vishniac 1983 ) . the optical depth to electron scattering could then be much less if the envelope is sufficiently irregular that some lines of sight to the pulsar traversed relatively little envelope mass . under such favourable circumstances , the opacity of the envelope is a less serious problem ( except at late times when grain formation may occur in the supernova ejecta and dust absorption could significantly increase the opacity at visual wavelengths ) . radio measurements suggest that any pulsar nebula in the center of sn 1993j is fainter than @xmath56 erg s@xmath43 ( bietenholz et al . 2003 ) . the material immediately surrounding the putative pulsar is , however , expected to be totally opaque until @xmath57 at which time the radio flux ( @xmath58 ghz ) will suddenly appear . here @xmath59k is the temperature of the swept - up material . the lack of detection of a synchrotron nebula at radio wavelengths should not be taken as strong evidence against the presence of a bright pulsar ( bahcall et al . 1970 ) . on the other hand , the lack of an x - ray nebula indicates that , if one exists , its radiative power must be less than @xmath60 erg s@xmath43 ( zimmermann & aschenbach 2003 ) . a compelling model for the optical / x - ray properties of the crab nebula was developed by rees & gunn ( 1974 ) . in this model , the central pulsar generates a highly relativistic , particle dominated wind ( with lorentz factor @xmath61 ) that passes through a shock front and decelerates to match the expansion velocity set by the outer nebula . the emission from the pulsar bubble is thought to provide a larger luminosity source than the radiative shock front itself ( chevalier & fransson 1992 ) . the wind particles acquire a power - law energy spectrum of the form @xmath62 ( for @xmath63 , where @xmath64 is the particle lorentz factor and @xmath65 is its minimum value ) in the shock front and radiative synchrotron emission in the down stream region ( chevalier 2000 ) . under the basic simplification that the emitting region can be treated as a one zone ( i.e. no spatial structure in the nebula ) , and assuming that a balance between injection from the shock front and synchrotron losses is established , the number of radiating particles at a particular @xmath64 is given by @xmath66 ( chevalier 2000 ) , where @xmath67 is the magnetic field in the emitting region , and @xmath68 @xmath69 s@xmath43 . in what follows , it is assumed that the energy density in the emitting region ( which is approximately determined by the shock jump conditions ) is divided between a fraction @xmath70 in particles and a fraction @xmath71 in the magnetic field . these efficiency factors are constrained by @xmath72 . with this assumption , the magnetic field in the emitting region is @xmath73 , where @xmath74 is the shock wave radius . the electron energy is radiated at its critical frequency @xmath75 , where @xmath76 and @xmath77 are the electron charge and mass , respectively . if the electrons and positrons cool rapidly by synchrotron radiation , as thought to be the case for the crab nebula , the luminosity produced from the pulsar power is given by @xmath78 ( chevalier 2000 ) , where @xmath79 and @xmath80 in cgs units . if the particle spectrum is similar to that of the crab nebula ( @xmath81 ) , the x - ray luminosity ( @xmath82 hz ) becomes @xmath83 where cgs units are used . we note that the x - rays from a putative pulsar nebula in sn 1993j with @xmath84 could provide the additional energy input required to reproduce the observed h@xmath85 luminosity at @xmath86350 days ( houck & fransson 1996 ) . the above relation can be used to predict the spin - down power , @xmath87 , of the putative pulsar in wind nebula where a pulsar has not yet been observed ( chevalier 2000 ) . in the case of sn 1993j , the value @xmath74 can be determined by the condition that the envelope should be transparent to optical radiation ( section 2.1 ) . the need for a particle dominated shock suggests that @xmath88 . we set @xmath89 , although there is little dependence to @xmath90 . the value of @xmath61 is difficult to constraint . here we use @xmath91 , the value typically assumed for the crab nebula ( kennel & coroniti 1984 ) . a better estimate of @xmath61 is given in section 3 . when these parameters are substituted into equation ( [ lx ] ) , the _ xmm _ upper limit of @xmath92 ( 0.3 - 10 kev ; zimmermann & aschenbach 2003 ) yields @xmath93 . on the basis of the _ asca _ data , kawai et al . ( 1998 ) found the relation @xmath94 , where @xmath95 is the nebular luminosity in the 1 - 10 kev range , from which we estimate @xmath96 . this estimate is in fair agreement with our predicted value . at the time of the explosion , the progenitor of 1993j has a mass of 5.4 @xmath20 ( with a helium - exhausted core of 5.1 @xmath20 ) , the secondary has a mass of 22 @xmath20 . the orbital period of the system is @xmath86 25 yr , and the companion has an orbital velocity of @xmath86 6 km s@xmath43 ( maund et al . there may have been a large kick imparted by the explosion mechanism , and that would be very interesting to study , but baring that , let us assume the companion star is bound in an eccentric orbit with the newly born pulsar 90 @xmath97 ( brandt & podsiadlowski 1995 ) . ] . the bulk of the pulsar energy @xmath87 would be primarily in the form of a magnetically driven , highly - relativistic wind consisting of @xmath98 , @xmath99 and probably heavy ions with @xmath100 and @xmath101 . under the foregoing conditions , the relativistic wind ( which is likely to be undisturbed by the presence of the binary companion and @xmath102 are the velocity and mass - loss rate of the stellar companion . ] since @xmath103 ) would escape the compact remnant while interacting with the soft photon field of the companion with typical energy @xmath104 ( maund et al . the scattered photons whose energy is boosted by the square of the bulk lorentz factor of the magnetized wind ( i.e. @xmath105 ) propagate in a narrow @xmath106 beam owing to relativistic aberration . the rate of energy loss of a relativistic particle moving in a radiation field with an energy density @xmath107 is about @xmath108 in the thompson limit ( landau & lifshitz 1975 ) . here @xmath109 is the luminosity of the optical star and @xmath110 is the binary separation ( @xmath111 ; maund et al . the total luminosity of scattered hard photons @xmath112 is then equal to the total particle energy losses in the course of motion from the pulsar to infinity @xmath113 , where @xmath114 denotes the efficiency in extracting energy from the relativistic outflow ( chernyakova & illarionov 1999 ) . the resulting radiation pressure on electrons in the ejecta will brake any outflow whose initial lorentz factor exceeds some critical value @xmath115 , converting the excess kinetic energy into a directed beamed of scattered photons . with @xmath116 , equation ( [ lx ] ) yields @xmath117 . as the stellar companion emits a black body spectrum , of effective temperature @xmath118 , the local photon energy density is given by @xmath119 - 1 } , \label{den}\ ] ] where @xmath120 is the soft photon energy in units of @xmath121 . the scattered photons are boosted by the square of the lorentz factor so that the local spectrum has a black body shape enhanced by @xmath122 . as can be seen in fig . [ fig1 ] , the resulting spectrum is the convolution of all the locally emitted spectra ( i.e. @xmath123 $ ] ) and it is not one of a blackbody . note that klein - nishina effects are important for incoming photon energies such that @xmath124 . the maximum energy of the scattered photons in this regime is @xmath125 . we note that the total luminosity emitted by the relativistic outflowing wind through the compton - drag process in the direction of the observer could be highly anisotropic and may change periodically during orbital motion for an eccentric ( or a highly inclined ) orbit . the time dependence , in this case , will be very distinctive and such effects should certainly be looked for . despite presumptions that a neutron star may have been created when the progenitor star of sn 1993j exploded and its core collapsed , no pulsar has yet been seen . if one exists , its radiative power must be less than @xmath126 or lower if the pulsar nebula has a high radiative efficiency . the well - known crab pulsar and its nebula for comparison , put out @xmath127 erg s@xmath43 ; and originally , when it was spinning faster , the luminosity might have been seven times greater still . fig . [ fig2 ] shows the pulsar spin - down luminosity divided by @xmath128 times the square of the distance , the total pulsar energy output at earth . the ten gamma - ray pulsars ( including candidates ) are shown as large circles . the solid curves show the dipole emission of the putative pulsar in 1993j ( at @xmath129 years ) for various assumptions regarding its initial period and magnetic field strength . the pulsar spinning down by magnetic dipole radiation alone , with initial rotation periods of 10 - 30 ms as extrapolated for galactic young pulsars can have a spin - down luminosity below @xmath130 ( see shaded region in fig . [ fig2 ] ) only if either the magnetic field is relatively weak @xmath131 g or if it is so strong ( i.e. @xmath132 g ) that the pulsar luminosity decays rapidly . if weak magnetic moments could be ruled out in the near future by x - ray and infrared observations , then we find that , if undetected , the putative pulsar in sn 1993j could be a magnetar . with a spectrum similar to that of the crab ( see fig . [ fig2 ] ) , the pulsed emission is likely to be below the detection limit of current instruments and may deprive us of the opportunity to witness the emergence of a gamma - ray pulsar in the immediate future . gev radiation with luminosities high enough to be detected with _ argo _ , and the _ veritas _ experiment now under construction , could be produced by the interaction of the pulsar wind with the soft photon field of its companion provided that the system remains bound and @xmath133 ( i.e. similar or larger to than inferred for the crab pulsar ) . its detection will surely offer important clues for identifying the nature of the progenitor and possibly constraining whether or not kicks have played an important role . if there is not a neutron star in sn 1993j , could there be a black hole instead ? had matter fallen back onto the nascent neutron star ( with mass of 1.4 @xmath20 ) on a timescale of 100 seconds to a few hours after the explosion , enough mass may have been accreted to push the object over the minimum thought to be necessary for the creation of a black hole . matter would continue to accrete onto the black hole , but the resulting radiation would be trapped in the outflow so that the escaping luminosity would be small . because the cores of massive stars increase with initial stellar mass , this picture would be more plausible if the initial mass of the sn 1993j progenitor star , @xmath8615 @xmath20 , were close to the mass limit above which stars collapse directly to black holes ( e.g. fryer 1999 ) . even if the neutron star is rotating only slowly or has a weak magnetic field , one would expect surrounding material to fall back onto it , giving rise to a luminosity @xmath134 erg s@xmath43 ( see fig . 1 ) . the remaining possibility of a weak pulsar with little surrounding mass will be difficult to rule out . the thermal emission from a newly formed neutron star gives a luminosity of @xmath135 erg s@xmath43 . the neutron - star emission should have a characteristic soft x - ray emission , but its detection will be arduous . if future observations fail to identify a pulsar in 1993j , then the youngest pulsar that we know will still be psr j0205 + 6449 ( camilo et al . 2002 ) , associated with sn 1181 . this work was stimulated by conversations with m. j. rees and s. smartt . discussions with j. bahcall and r. chevalier are gratefully acknowledged , as is helpful correspondence from the referee ph . this work is supported by the w.m . keck foundation ( ams ) and nasa through a chandra postdoctoral fellowship award pf3 - 40028 ( er - r ) .

the recent report of a binary companion to the progenitor of supernova 1993j may provide important clues for identifying the nature of the nascent compact object . given the estimates of the progenitor mass , the potential power source is probably a pulsar rather than an accreting black hole . if there is a pulsar , one would expect the rotational luminosity to be stored and reprocessed by the supernova remnant , but no pulsar nebula has yet been seen . the lack of detection of an x - ray synchrotron nebula should be taken as strong evidence against the presence of a bright pulsar . this is because absorption by the surrounding supernova gas should be negligible for the light of the companion star to have been detected . a model is developed here for the luminosity of the pulsar nebula in sn 1993j , which is then used to predict the spin - down power of the putative pulsar . if one exists , it can be providing no more than @xmath0 . with an initial rotation period of 10 - 30 ms , as extrapolated from young galactic pulsars , the nascent neutron star can have either a weak magnetic field , @xmath1 g , or one so strong , @xmath2 g , that its spin was rapidly slowed down . the companion star , if bound to the neutron star , should provide ample targets for the pulsar wind to interact and produce high - energy gamma - rays . the expected non - pulsed , gev signal is calculated ; it could be detected by current and future experiments provided that the pulsar wind velocity is similar to that of the crab nebula . [ firstpage ] pulsars : general stars : supernovae supernovae : individual ( sn 1993j ) gamma rays : theory radiation mechanisms : non - thermal

the idea of physical systems characterised by two different temperatures has been proposed a long time ago for the models of spin - glasses and neural networks with partially annealed disorder @xcite . in these models , the two temperatures @xmath3 and @xmath4 are related to two different degrees of freedom , which are evolving at two essentially different time scales . as an example , one may consider a system in which the fast spin variables are connected with the thermal bath kept at the temperature @xmath3 , while the slow spin - spin coupling parameters are connected with another thermal bath maintained at the temperature @xmath4 . it can be easily shown that in the stationary ( non - equilibrium ) state the statistical properties of such systems are described by the usual replica theory of disordered systems with a _ finite _ value of the replica parameter @xmath5 ( see also @xcite ) . unfortunately , generalisation of this idea to the case when dynamics of two types of degrees of freedom is characterised by two comparable ( or equal ) time scales turned out to be rather problematic : it seems that there is no generic explicit expression for the stationary probability distribution function which would generalise the gibbs distribution of the equilibrium case @xmath6 @xcite . however , there is a particular case for which one can find an explicit and a rather non - trivial expression for the stationary distribution function . namely , this is the case when the two degrees of freedom @xmath7 and @xmath8 related to the thermal baths with the temperatures @xmath9 respectively , experience a potential which is a _ quadratic _ function of @xmath7 and @xmath8 @xcite . during last decade theoretical investigations of such type of systems were mostly concentrated on the studies of nonequilibrium fluctuations and energy transfer @xcite . recently this type of model was studied both theoretically @xcite and experimentally @xcite from the point of view of the entropy production and memory effects . in this paper , keeping in mind putative experimental realisation of such a type of systems , we are going to discuss the two - temperatures situation reformulated in terms of the two - dimensional diffusion of a brownian particle in a parabolic potential . the diffusion process is defined in terms of langevin dynamics with two different effective temperatures in the @xmath0 and the @xmath1 directions . in the stationary state this system is described by a non - trivial distribution function @xmath2 , which can be computed explicitly . unlike for the equilibrium case ( @xmath10 ) , this non - equilibrium stationary state is characterised by the presence of nontrivial space dependent particle s flows @xmath11 . moreover , these flows exhibit a `` symmetry breaking '' rotor , @xmath12 ( directed perpendicular to the @xmath13-plain ) , the sign ( or the direction ) of which is determined by the temperature difference @xmath14 . the paper is organised as follows . in section [ model ] we define our model and present the explicit solution for the stationary particle s probability distribution function @xmath2 . in section [ obs ] we compute putative `` observable '' quantities of the system , such as the variances of the particle displacements in the @xmath0 and the @xmath1 directions , the rotor @xmath15 of the particle s flows as well as the average rotation velocity . in section [ sims ] we report the results of the numerical simulations and compare them with our analytical predictions . finally , in section [ conc ] we conclude with a brief recapitulation of our results . we consider stochastic , over - damped langevin dynamics of a particle moving in a two - dimensional space in a presence of an external potential @xmath16 . the particle instantaneous position @xmath17 is defined by projections on the @xmath0 and the @xmath1 axes , @xmath18 and @xmath19 , respectively . the time evolution of @xmath18 and @xmath19 is described by following equations : @xmath20 here @xmath21 is _ anisotropic _ stochastic noise , with zero mean and correlation function @xmath22 where @xmath23 and @xmath24 are two _ different _ `` temperatures '' and @xmath16 has the following parabolic form : @xmath25 the shape of the potential is controlled by the parameter @xmath26 . to keep the particle localised near the origin , we have to impose the constraint @xmath27 . this follows from the requirement that both eigenvalues of the potential , @xmath28 , must be positive ; in the case @xmath29 , there is a direction in the plane @xmath30 at which the potential @xmath16 has a negative curvature which allows the particle to escape to infinity . in the stationary regime , the probability distribution function @xmath2 of the particle position obeys the stationary fokker - planck equation : @xmath31 \ ; + \ ; \frac{\partial}{\partial y}\bigl [ t_{y}\frac{\partial p(x , y)}{\partial y } \ ; + \ ; p(x , y ) \frac{\partial u(x , y)}{\partial y } \bigr ] \ ; = \ ; 0\ ] ] in the trivial isotropic case , @xmath32 , the solution of the above equation is simply the equilibrium gibbs distribution @xmath33 . one can easily show that in the generic _ anisotropic _ case with arbitrary @xmath23 and @xmath24 , the solution of the stationary equation ( [ 5 ] ) reads : @xmath34 where the following shortenings have been used @xmath35 and @xmath36 further on , @xmath37 is the normalisation constant ( the `` partition function '' ) , defined as @xmath38 one immediately observes that @xmath37 exists , so that the system has the stationary solution , only for @xmath27 . using the above probability distribution function we can straightforwardly calculate the variances of the particles position with respect to the @xmath0 and the @xmath1 axes : @xmath39 the characteristic quantity , which can serve as the measure of anisotropy in the system under study , is defined as the ratio of these two quantities : @xmath40 in the trivial decoupled case , @xmath41 , we find @xmath42 , while in the isotropic case , @xmath10 we have @xmath43 for all values of the coupling parameter @xmath26 . note next that in the strongly anisotropic case , e.g. , when @xmath44 , one has @xmath45 in other words , in the strongly anisotropic case the values of both @xmath46 and @xmath47 are defined by the largest @xmath48 , while the value of the ratio @xmath49 becomes a @xmath48-independent constant . in the stationary case the current @xmath50 is defined as follows : @xmath51 using eqs.([4])-([6 ] ) we obtain : @xmath52 \ , p(x , y ) \\ \label{21 } j_{y } & = & \bigl[(1 - t_{y}\gamma_{2 } ) y + u ( 1 - t_{y}\gamma_{3 } ) x \bigr ] \ , p(x , y)\end{aligned}\ ] ] note that in the isotropic case , @xmath32 , we have @xmath53 , so that @xmath54 . in the anisotropic case @xmath9 the above non - trivial pattern of currents can be characterised in terms of the rotor : @xmath55 in general , the rotor @xmath15 is a rather complicated function of two variables @xmath7 and @xmath8 , but it is remarkable that the function @xmath15 has a non - zero ( and very simple ) value at the origin at @xmath56 : @xmath57 note that this quantity changes sign from minus ( `` left rotation '' ) at @xmath58 , to plus ( `` right rotation '' ) at @xmath59 . due to the presence of a non - zero particle s current rotor , one finds that the mean particle s rotation velocity is also non - zero . indeed , for a given value of the particle s linear velocity @xmath60 located in the point @xmath61 on the two - dimensional plane , its angular velocity is @xmath62 where @xmath63 is the vector product directed along the @xmath64-axis . thus , the mean rotation velocity @xmath65 in the limit of an infinite observation time can be defined as follows : @xmath66 changing averaging over time by averaging over ensemble ( which will be justified in what follows by numerical simulations ) we get : @xmath67 here the average current @xmath68 is defined in eqs.([20])-([21 ] ) . according to eq.([6 ] ) , the probability distribution function @xmath69 can be represented as follows @xmath70 where @xmath71 substituting the explicit expressions for the components @xmath72 and @xmath73 of the current , eqs.([20])-([21 ] ) , and using eqs.([7])-([10 ] ) , we get @xmath74 substituting eq.([29 ] ) into eq.([26 ] ) and performing simple integrations we obtain @xmath75 where the parameter @xmath76 is defined in eq.([10 ] ) . one can easily prove that the maximal value of the mean angular velocity is @xmath77 , and it is achieved either in the limits @xmath78 ( which corresponds to @xmath79 for finite @xmath23 ) or in the limit @xmath80 ( which corresponds to @xmath81 for a finite @xmath24 ) , and the value of the coupling parameter @xmath82 . to verify our analytical predictions and the underlying assumption that the time - average can be replaced by the ensemble average , we perform numerical simulations of appropriately discretised langevin equations eqs.([1 ] ) . substituting the potential @xmath83 into eqs.([1 ] ) we first write these equations explicitly : @xmath84 where the variances of the thermal noise components are defined by @xmath85 , @xmath86 and @xmath87 . discretising eq . ( [ eq : delta0 ] ) with a time step @xmath88 , we have : @xmath89 where @xmath90 and @xmath91 are delta - correlated random numbers with gaussian distribution of unit half - width , @xmath92 , @xmath93 , which are the conditions of a smooth motion . in that case for a free motion of a particle ( @xmath94 ) which starts at the origin ( @xmath95 ) , the diffusion coefficients are @xmath96 , @xmath97 and the variances of the displacement are given by @xmath98 and @xmath99 . in the case of the symmetric potential ( @xmath41 ) , one has in the stationary regime @xmath100 and @xmath101 , independently of @xmath102 . for asymmetric potential @xmath103 , we will compute the mean angular velocity @xmath104 given in eq.([25 ] ) and the measure of anisotropy @xmath105 that is described by eq.([14 ] ) . the numerical simulation has been done for the time step @xmath106 . the averaging has been performed over the total time period @xmath107 time units , the numerical inaccuracy has been evaluated by splitting the whole time interval into @xmath108 sub - intervals . in fig . [ fig : u](a),(b ) we plot numerical results for the ratio of variances @xmath109 and for the mean angular velocity @xmath104 , calculated as the time - average of @xmath110 , as functions of @xmath26 for @xmath111 , @xmath112 . for comparison we also show our analytical predictions in eq.([14 ] ) and eq.([25 ] ) , respectively , and find a perfect agreement . this justifies the replacement of the time - average by the ensemble average in our analytical calculations . of variances of particle s displacements along the @xmath0 and the @xmath1-axes vs the parameter @xmath26 . ( symbols and the color - code is as in ( b ) ) ; ( b ) the mean angular velocity @xmath104 as a function of @xmath26 for different @xmath113 . solid lines are our predictions in eqs.([14 ] ) and ( [ 25 ] ) . , title="fig:",scaledwidth=49.0% ] of variances of particle s displacements along the @xmath0 and the @xmath1-axes vs the parameter @xmath26 . ( symbols and the color - code is as in ( b ) ) ; ( b ) the mean angular velocity @xmath104 as a function of @xmath26 for different @xmath113 . solid lines are our predictions in eqs.([14 ] ) and ( [ 25 ] ) . , title="fig:",scaledwidth=49.0% ] further on , in fig . [ fig : t](a),(b ) we plot the same quantities as functions of @xmath114 ( with @xmath111 ) for @xmath115 . we again observe a very good agreement between our numerical and analytical results . and ( b ) the mean angular velocity @xmath104 as functions of @xmath114 for different values of @xmath115 . solid lines define our theoretical predictions in eqs.([14 ] ) and ( [ 25 ] ) , and the symbols denote the results of numerical simulations . , title="fig:",scaledwidth=49.0% ] and ( b ) the mean angular velocity @xmath104 as functions of @xmath114 for different values of @xmath115 . solid lines define our theoretical predictions in eqs.([14 ] ) and ( [ 25 ] ) , and the symbols denote the results of numerical simulations . , in the present work we studied a simple stochastic `` toy model '' with only two degrees of freedom which are connected to two thermostats maintained at two _ different _ temperatures @xmath23 and @xmath24 , respectively . the model describes diffusion of a particle on a two - dimensional plane in a presence of a parabolic potential such that the stochastic noises in the @xmath0 and the @xmath1 directions have different strength ( @xmath23 and @xmath24 , respectively ) . we determine the stationary state probability distribution function for the position of the particle . despite its relatively simple structure , it turns out to be rather non - trivial , revealing interesting qualitative physical phenomena . in particular , in the stationary state one finds a rather sophisticated pattern of particles density currents ( which would be identically equal to zero in the equilibrium case ) characterised by the non - zero rotor . moreover , due to the presence of this flux rotor one observes the phenomenon which could be interpreted as a `` spontaneous symmetry breaking '' , namely one finds non - zero value for the average particle s rotation ( around the origin ) velocity . this value is proportional to @xmath116 , eq.([30 ] ) , being positive ( left rotation ) for @xmath117 and negative ( right rotation ) for @xmath118 . it should be stressed , however , that except for recently proposed two - temperature electric analog system @xcite , for the moment the considered model has no experimental realization . thus , the aim of the present work is somewhat provocative : we would like argue that the systems of such type are sufficiently interesting to stimulate investigations for their `` hardware '' implementations . we also believe that modifications of our toy model towards a system that could be realised in practice and at the same time would not loose its interesting behavior ( rotation ) , is possible . this work was supported in part by the grant irses dcp - physbio n269139 . vd acknowledges the support of prof . dietrich at the mpi stuttgart , where parts of this work were done . 99 a.crisanti , a.puglisi and d.villamaina , phys . rev . e , * 85 * , 061127 ( 2012 ) ; a.puglisi and d.villamaina europhys.lett . * 88 * , 30004 ( 2009 ) ; d.villamaina , a.baldassarri , a.puglisi and a.vulpiani j.stat.mech . p07024 ( 2009 )

we study a planar two - temperature diffusion of a brownian particle in a parabolic potential . the diffusion process is defined in terms of two langevin equations with two different effective temperatures in the @xmath0 and the @xmath1 directions . in the stationary regime the system is described by a non - trivial particle position distribution @xmath2 , which we determine explicitly . we show that this distribution corresponds to a non - equilibrium stationary state , characterised by the presence of space - dependent particle currents which exhibit a non - zero rotor . theoretical results are confirmed by the numerical simulations . _ keywords _ : two - dimensional diffusion , parabolic potential , non - equilibrium stationary state , rotating flows

in massive mimo systems a base station ( bs ) with a large number of antennas ( @xmath9 , a few hundreds ) communicates with several user terminals ( @xmath10 , a few tens ) on the same time - frequency resource @xcite . there has been recent interest in massive mimo systems due to their ability to increase spectral and energy efficiency even with very low - complexity multi - user detection and precoding @xcite . however , physically building cost - effective and energy - efficient large arrays is a challenge . specifically in the downlink , the power amplifiers ( pas ) used in the bs should be highly power - efficient . due to the trade - off between the efficiency and linearity of the pa @xcite , highly efficient but non - linear pas must be used . the efficiency of the pa is related to the amount of backoff necessitated ( to reduce non - linear distortion ) by the peak to average ratio of the input waveform . for minimum backoff and hence maximum efficiency , the input waveform should have a constant or nearly _ constant envelope _ ( ce).[multiblock footnote omitted ] with this motivation , in @xcite we had proposed a ce precoding algorithm for the frequency - flat multi - user mimo broadcast channel , which was then extended to frequency - selective channels in @xcite . with @xmath9 sufficiently larger than @xmath10 and i.i.d . rayleigh fading , numerical studies done in both these papers revealed that in order to achieve a desired per - user ergodic information rate , the proposed ce precoding algorithm needed only about @xmath11 db extra total transmit power compared to that required under the less stringent and commonly used total average transmit power constraint . it was also observed that even under a stringent per - antenna ce constraint , an @xmath12 array gain is achievable , i.e. , with every doubling in the number of bs antennas the total transmit power can be reduced by @xmath13 db while maintaining a fixed information rate to each user ( assuming that the number of users is fixed ) . however , in the ce precoding algorithm proposed in @xcite the phase angle of the complex baseband signal transmitted from each bs antenna is unconstrained ( i.e. , it s principal value lies in @xmath14 $ ] ) , and therefore it is possible that the phase could vary very fast between consecutive channel uses . a phase variation of @xmath15 ( or more ) between consecutive channel uses will result in zero crossings in the baseband signal , which with practical pas could lead to distortion in the transmitted signal . in this paper , we address this problem by proposing a ce precoding algorithm with an additional constraint that the difference of the phase angle transmitted in consecutive channel uses be limited to the interval @xmath0 $ ] for a fixed @xmath1 ( the special case of @xmath16 was considered in @xcite ) . it is shown that the complexity of the proposed ce algorithm is independent of @xmath3 and is the same as the algorithm proposed in @xcite . numerical studies on the i.i.d . rayleigh fading channel suggest that an @xmath12 array gain is achieved even under the additional phase angle variation constraint . to achieve a desired per - user information rate , the extra total transmit power required under the time variation constraint when compared to the special case of no time variation constraint ( i.e. , @xmath2 ) , is _ small _ when @xmath3 is close to @xmath6 and @xmath17 . for example , with @xmath8 , @xmath18 , @xmath19 single - antenna users and a desired per - user rate of @xmath6 bit - per - channel - use ( bpcu ) , the magnitude of the phase variation is limited to @xmath20 and the extra transmit power required is less than @xmath21 db . in the previous works and also in this paper , without loss of generality we assume single - antenna users.[multiblock footnote omitted ] it is assumed that the bs has knowledge of the channel vector to each user.[multiblock footnote omitted ] the complex baseband constant envelope signal transmitted from the @xmath22-th bs antenna at time @xmath23 is of the form @xmath24 & = & \sqrt{\frac{p_t}{n } } \ , e^{j \theta_i[t ] } \,\,\,,\,\,\ , i = 1,2,\cdots , n,\end{aligned}\ ] ] where @xmath25 , @xmath26 is the total power transmitted from the @xmath9 bs antennas and @xmath27 \in [ -\pi \,,\ , \pi)$ ] is the phase of the ce signal transmitted from the @xmath22-th bs antenna at time @xmath23 . the equivalent discrete - time complex baseband channel between the @xmath22-th bs antenna and the @xmath28-th user ( having a single - antenna ) has a finite impulse response of length @xmath29 samples , denoted by @xmath30\,,\ , h_{k , i}[1 ] \,,\ , \cdots \,,\ , h_{k , i}[l-1])$ ] . the signal received at the @xmath28-th user ( @xmath31 ) at time @xmath23 is given by @xmath32 & = & \sqrt{\frac{p_t}{n } } \,\ , \sum_{i=1}^n \ , \sum_{l=0}^{l-1 } \ , h_{k , i}[l ] e^{j \theta_i[t - l ] } \,\,+\,\ , w_k[t ] \,,\,\end{aligned}\ ] ] where @xmath33 \sim { \mathcal c}{\mathcal n}(0,\sigma^2)$ ] is the awgn at the @xmath28-th user at time @xmath23 ( awgn is i.i.d . across time and across the users ) . for the sake of brevity let us denote the vector of phase angles transmitted at time instance @xmath23 by @xmath34 = ( \theta_1[t ] , \cdots , \theta_n[t])$ ] . in the following we briefly summarize the ce precoding algorithm proposed in @xcite . suppose that , at time instances @xmath35 we are interested in communicating the information symbol @xmath36 \in { \mathcal u}_k \subset { \mathbb c}$ ] to the @xmath28-th user . let @xmath37 \vert^2 ] = 1 \,,\ , k=1,\cdots , m$ ] . also , let @xmath38 = ( \sqrt{e_1 } u_1[t ] , \cdots , \sqrt{e_m } u_m[t ] ) \ , \in \ , { \mathcal u}_1 \times \cdots \times { \mathcal u}_m$ ] be the vector of information symbols to be communicated at time @xmath23 . in @xcite , we had proposed an algorithm for finding the transmit phase angles @xmath39 in such a way that the received noise - free signal at each user is almost the same as the information symbol intended for that user , i.e. , @xmath40 e^{j \theta_i[t - l ] } \approx \sqrt{p_t } \sqrt{e_k } u_k[t ] \,\,,\,\ , \forall \ , k=1,2,\ldots , m \,,\ , t=1,2,\ldots , t$ ] . in @xcite we find the transmit phase angles as a solution to the optimization problem in ( [ nls_joint_eqn ] ) , where @xmath41 = ( \theta_1^u[t ] , \cdots , \theta_n^u[t ] ) \,,\ , t=1,\ldots , t$ ] denotes the vectors of transmit phase angles , for the given information symbol vectors @xmath38 \,,\ , t=1,\ldots , t$ ] . the main idea in ( [ nls_joint_eqn ] ) is to choose the transmit phase angles in a way so as to minimize the energy of the difference between the received noise - free signal and the intended information symbol for all users . note that the objective function @xmath42 in ( [ nls_joint_eqn ] ) is a function of @xmath43 variables ( @xmath9 phase angles transmitted at @xmath44 time instances ) . finding an exact solution to the problem in ( [ nls_joint_eqn ] ) is prohibitively complex , and therefore in @xcite we had proposed a low - complexity near - optimal solution to ( [ nls_joint_eqn ] ) . the ce precoding idea is primarily based on our previous work in @xcite ( for frequency - flat channels ) where we had analytically shown that for a broad class of frequency - flat channels ( including i.i.d . fading ) , for a fixed @xmath10 and fixed symbol energy levels ( @xmath45 ) , by having a sufficiently large @xmath17 it is always possible to choose the transmit phase angles in such a way that the received signals at the users are arbitrarily close to the desired information symbols . note that for the ce precoding method , the transmit phase angles can take any value in the interval @xmath46 ( see ( [ nls_joint_eqn ] ) ) . therefore it is possible that between consecutive time instances , the phase angle transmitted from a bs antenna could change by a large magnitude , which will distort the transmit signal at the output of the pa . to address this issue , in this paper we propose a ce precoder where for each bs antenna the difference between the phase angles transmitted in consecutive time instances is constrained to lie in the interval @xmath47 $ ] for a given @xmath1 , i.e. , @xmath48 \ , - \ , \theta_i[t-1 ] \vert \leq \alpha \pi $ ] for all @xmath49 . this constraint ensures that the maximum variation in the transmitted phase angle between consecutive time instances is at most @xmath50 ( e.g. , with @xmath8 the maximum phase angle variation is only @xmath51 ) . in this paper , under the time - variation constraint we propose an optimization problem to find the transmit phase angles for given information symbols @xmath52 \,,\ , k=1,\ldots , m \,,\ , t=1,\ldots , t$ ] , as given by ( [ nls_joint_eqn2 ] ) . exactly solving ( [ nls_joint_eqn2 ] ) has prohibitive complexity , and therefore in the following we propose a low - complexity near - optimal solution to ( [ nls_joint_eqn2 ] ) . the essential idea of this low complexity solution is to iteratively optimize @xmath42 as a function of one variable at a time while fixing the other variables to their previous values . in one iteration of this low - complexity algorithm , we have @xmath43 sub - iterations . in the first sub - iteration we start with @xmath53 $ ] and minimize @xmath42 as a function of @xmath53 $ ] while keeping the other @xmath54 variables fixed to their previous values . we then update @xmath53 $ ] with its optimum value and then move onto the second sub - iteration where we minimize @xmath42 as a function of @xmath55 $ ] while keeping the other variables fixed . in general , in the @xmath56-th sub - iteration we minimize @xmath42 as a function of @xmath57 $ ] ( i.e. , the phase angle transmitted from the @xmath58-th bs antenna in the @xmath59-th time instance ) while keeping the other variables fixed . @xmath60,\cdots,\theta^{u}[t ] ) & = & \arg \hspace{-5 mm } \min_{\substack{\\ ( \theta[1 ] , \theta[2 ] , \cdots , \theta[t ] ) \\ \vert \theta_i[t ] - \theta_i[t-1 ] \vert \ , \leq \ , \alpha \pi \\ i=1,\ldots , n \,,\ , t=1,\ldots , t } } \ , f(\theta_1[1 ] , \cdots , \theta_n[1 ] , \cdots , \theta_1[t ] , \cdots , \theta_n[t]).\end{aligned}\ ] ] @xmath61 & = & \arg \hspace{-10 mm } \min_{\substack { \\ \\ \theta_r[q ] \\ \hspace{3 mm } \vert \theta_r[q ] \ , - \ , \theta_r[q-1 ] \vert \ , \leq \ , \alpha \pi } } \hspace{-16 mm } \sum_{t = q}^{\min(t , ( q + l -1 ) ) } \sum_{k=1}^m { \bigg \vert } s_{r , q}(k , t ) + \frac{h_{k , r}[t - q ] e^{j \theta_r[q]}}{\sqrt{n } } { \bigg \vert}^2 \,\,,\,\ , \mbox{where } \,\ , s_{r , q}(k , t ) \ , { \stackrel { \delta } { = } } \ , { \big ( } \sum \limits_{i=1}^n \hspace{-7 mm } \sum \limits_{\substack{l=0 \,,\ , \\ \hspace{6 mm } ( i , l ) \ne ( r,(t - q))}}^{l-1 } \hspace{-6 mm } \frac{h_{k , i}[l ] e^{j \theta_i[t - l ] } } { \sqrt{n } } { \big ) } - \sqrt{e_k } u_k[t ] \nonumber \\ & = & \arg \min_{\substack{\theta_r[q ] \\ ( \theta_r[q ] \ , - \ , \theta_r[q-1 ] ) \ , \in \ , [ -\alpha \pi \,,\ , \alpha \pi ] } } \ , \sum_{t = q}^{\min(t , ( q + l -1 ) ) } \sum_{k=1}^m { \bigg \vert } s_{r , q}(k , t ) + \frac{h_{k , r}[t - q ] e^{j \theta_r[q-1 ] } \ , e^{j ( \theta_r[q ] - \theta_r[q-1 ] ) } } { \sqrt{n } } { \bigg \vert}^2 \nonumber \\ & = & \theta_r[q - 1 ] \ , + \ , \arg \min_{\substack{\omega \ , \in \ , [ -\alpha \pi \,,\ , \alpha \pi ] } } \ , \sum_{t = q}^{\min(t , ( q + l -1 ) ) } \sum_{k=1}^m { \bigg \vert } s_{r , q}(k , t ) + \frac{h_{k , r}[t - q ] e^{j \theta_r[q-1 ] } \ , e^{j \ , \omega } } { \sqrt{n } } { \bigg \vert}^2 \nonumber \\ & = & \theta_r[q - 1 ] \ , + \ , \arg \max_{\omega \in [ -\alpha \pi \,,\ , \alpha \pi ] } \ , \re { \bigg ( } \ , e^{j \omega } \ , { \big \ { } \ , - \ , \sum_{t = q}^{\min(t , ( q + l -1 ) ) } \sum_{k=1}^m \ , h_{k , r}[t - q ] e^{j \theta_r[q-1 ] } s^{*}_{r , q}(k , t ) { \big \ } } { \bigg ) } \nonumber \\ & = & \theta_r[q - 1 ] \ , + \ , \left \ { \begin{array}{cc } \alpha \pi \ , , & \,\,\ , - \pi \ , \leq \ , c \ , < \ , - \alpha \pi \\ - c \ , , & \,\,\ , - \alpha \pi \ , \leq \ , c \ , < \ , \alpha \pi \\ - \alpha \pi \ , , & \,\,\ , \alpha \pi \ , \leq \ , c \leq \pi \end{array } \right . \,\,,\,\ , \mbox{where } \,\ , c \ , { \stackrel { \delta } { = } } \ , \mbox{arg } { \big ( } - \hspace{-4 mm } \sum_{t = q}^{\min(t , ( q + l -1 ) ) } \sum_{k=1}^m \ , h_{k , r}[t - q ] e^{j \theta_r[q-1 ] } s^{*}_{r , q}(k , t ) { \big ) } .\end{aligned}\ ] ] since the channel is causal and has a memory of @xmath29 time instances , it follows that in the summation on the right hand side of the definition of @xmath42 in ( [ nls_joint_eqn ] ) , only the terms corresponding to @xmath62 depend on @xmath57 $ ] . given this fact , the minimization of @xmath42 only w.r.t . @xmath57 $ ] is given by ( [ new_eqn_1 ] ) . in ( [ new_eqn_1 ] ) , for any complex number @xmath63 , @xmath64 \ , | \ , e^{j \phi } = z/\vert z \vert \}$ ] is the principal value of the phase angle of @xmath63 and @xmath65 denotes the conjugate of @xmath63 . from ( [ new_eqn_1 ] ) it is clear that the new value of @xmath57 $ ] depends on @xmath66 . note that , for every different @xmath67 we need not recalculate @xmath66 explicitly using the sum in the r.h.s . of its definition in ( [ new_eqn_1 ] ) . instead , @xmath66 can be calculated by subtracting the current value of @xmath68 e^{j \theta_r[q]}/\sqrt{n}$ ] ( i.e. , value at the start of the @xmath56-th sub - iteration ) from the current value of @xmath69 e^{j \theta_i[t - l ] } } { \sqrt{n } } - \sqrt{e_k } u_k[t ] $ ] , i.e. @xmath70 e^{j \theta_r[q]}}{\sqrt{n}}.\end{aligned}\ ] ] note that with change in @xmath57 $ ] , we also need to change @xmath71 for all @xmath72 . the modified value of @xmath71 after the @xmath56-th sub - iteration is given by @xmath73 e^{j \theta_i[t - l ] } } { \sqrt{n } } \nonumber \\ & & \ , - \ , \sqrt{e_k } u_k[t ] \ , + \ , \frac{h_{k , r}[t - q ] e^{j \theta^{\prime}_r[q]}}{\sqrt{n } } \nonumber \\ & = & s(k , t ) \ , + \ , \frac{h_{k , r}[t - q]}{\sqrt{n } } { \big ( } e^{j \theta^{\prime}_r[q ] } \ , - \ , e^{j \theta_r[q ] } { \big ) } \end{aligned}\ ] ] where @xmath74 $ ] is the new updated value of the phase angle to be transmitted from the @xmath58-th bs antenna at time instance @xmath59 , and is given by ( [ new_eqn_1 ] ) . after the last sub - iteration of an iteration ( i.e. , where we update @xmath75 $ ] ) , we start with the first sub - iteration ( where we update @xmath53 $ ] ) of the next iteration . it is clear that the value of the objective function @xmath42 reduces monotonically from one sub - iteration to the next . numerically , it has been observed that the value of the objective function converges in a few iterations ( @xmath76 ) and further iterations lead to little reduction in the value of @xmath42 . further , the value that the objective function converges to , is observed to be small when @xmath17 . the complexity of each sub - iteration is @xmath77 and is independent of @xmath3 ( see ( [ new_eqn_1 ] ) ) . since we update @xmath43 phase angles in each iteration , the total complexity of each iteration is @xmath78 . with a fixed number of iterations , the overall complexity of the proposed algorithm is @xmath78 , i.e. , a per - channel - use complexity of @xmath79 , which is the same as that of the algorithm proposed in @xcite to solve ( [ nls_joint_eqn ] ) . for a given set of information symbol vectors @xmath38 \,,\,t=1,\cdots , t$ ] , let @xmath80,{\widehat \theta}^u[2],\cdots,{\widehat \theta}^u[t]$ ] denote the output phase angles of the proposed iterative ce precoding algorithm ( see section [ sec - ce ] ) . let @xmath81 $ ] be the phase angle transmitted from the @xmath22-th antenna at time @xmath23 . the signal received at the @xmath28-th user is then given by @xmath82 & \hspace{-3 mm } = & \hspace{-3 mm } \sqrt{p_t } \sqrt{e_k } \ , u_k[t ] \,+\ , \sqrt{p_t } i_k^u[t ] \ , + \ , w_k[t ] \nonumber \\ i_k^u[t ] & \hspace{-3 mm } { \stackrel { \delta } { = } } & \hspace{-3 mm } { \big ( } \sum_{i=1}^n \ , \sum_{l=0}^{l-1 } \ , \frac{h_{k , i}[l ] } { \sqrt{n } } e^{j { \widehat \theta}^u_i[t - l ] } \ , - \ , \sqrt{e_k } u_k[t ] { \big ) } \end{aligned}\ ] ] note that @xmath83 $ ] behaves like multi - user interference ( mui ) . also , let @xmath84 , \cdots , y_k[t])^t$ ] , @xmath85 , \cdots,\sqrt{e_k } u_k[t])^t$ ] , @xmath86 , \cdots , i_k^u[t])^t$ ] and @xmath87 , \cdots , w_k[t])^t$ ] . let @xmath88 \}$ ] denote the impulse responses of the channels between the @xmath9 bs antennas and the @xmath10 users . for a given @xmath89 , an achievable rate for the @xmath28-th user is given by the mutual information @xmath90 @xcite . for any arbitrary distribution on @xmath91 , it is difficult to compute @xmath92 . a lower bound on @xmath93 is an achievable information rate for the @xmath28-th user . therefore , in the following we derive a lower bound to @xmath93 assuming @xmath52 \,,\ , t=1,\cdots , t$ ] to be i.i.d . @xmath94 i.e. , proper complex gaussian having zero mean and unit variance . @xmath95 where @xmath96 denotes the differential entropy operator , and the inequality in step ( a ) is due to the fact that conditioning reduces entropy @xcite . the inequality in step ( b ) follows from the fact that the proper complex gaussian distribution is the entropy maximizer , i.e. , @xmath97 , where @xmath98 $ ] is the autocorrelation matrix of @xmath99 and @xmath100 denotes its determinant @xcite . from ( [ recvk_eqn_tilde ] ) and the definition of @xmath99 in ( [ inf_rate_bnd ] ) we get @xmath101 . since @xmath102 and @xmath103 are independent , it follows that @xmath104 \ , + \ , ( \sigma^2/p_t ) { \bf i}$ ] , where the expectation is over @xmath105 . substituting this expression for @xmath106 in ( [ inf_rate_bnd ] ) , we get @xmath107 \ , + \ , \frac{\sigma^2}{p_t } { \bf i } { \big \vert } } { t } { \bigg ] } ^+\end{aligned}\ ] ] here @xmath108^+ \ , { \stackrel { \delta } { = } } \ , \max ( 0 , x)$ ] and @xmath109 is the vector of the average information symbol energies of the @xmath10 users . the ergodic information rate lower bound for the @xmath28-th user is then given by @xmath110 $ ] ( expectation is over @xmath89).[multiblock footnote omitted ] we consider a frequency selective channel with a uniform power delay profile , i.e. , @xmath111 \vert^2 ] = 1/l \,,\,l=0,1,\cdots,(l-1)$ ] . the channel gains @xmath112 $ ] are i.i.d . rayleigh faded , i.e. , proper complex gaussian ( mean @xmath113 , variance @xmath114 ) . the ergodic sum rate @xmath115 $ ] can be maximized as a function of @xmath116 . this is however difficult . nevertheless , since the users have identical channel statistics , it is likely that the optimal @xmath117 vector has equal components , i.e. , @xmath118.[multiblock footnote omitted ] using numerical methods , for a given @xmath119 we therefore find the optimal @xmath120 which results in the largest ergodic sum rate . with @xmath118 , we observe that all users have the same ergodic rate , i.e. @xmath121 = \cdots = { \mathbb e } [ r_m({\bf h},{\bf e},p_t/\sigma^2 ) ] $ ] . subsequently , we refer to this rate achieved by each user as the per - user ergodic information rate . in fig . [ fig_3 ] we plot the minimum @xmath119 required by the proposed ce precoder to achieve a per - user information rate of @xmath6 bpcu as a function of increasing @xmath9 with fixed @xmath19 users and @xmath122 . the special case of @xmath2 corresponds to an unconstrained ( time - variation ) ce precoder and therefore has the best performance . we see that for a given @xmath9 , more transmit power is required for a smaller @xmath3 . this is expected since a smaller @xmath3 places a more stringent constraint on the time - variation of the transmitted phase angles , which reduces the information rate . however , even with @xmath8 ( i.e. , limiting the magnitude of the time variation between consecutive time instances to be less than @xmath51 ) , the extra transmit power required when compared to @xmath2 is less than @xmath21 db when @xmath9 is sufficiently larger than @xmath10 ( in this case @xmath123 ) . also , for a fixed @xmath3 the extra transmit power required when compared to @xmath2 , decreases with increasing @xmath9 . from the figure , it is also observed that irrespective of the value of @xmath3 , for sufficiently large @xmath17 the required @xmath119 reduces by roughly @xmath13 db with every doubling in @xmath9 ( i.e. , an @xmath12 array gain with @xmath9 bs antennas ) . for the sake of completeness , we have also considered the sum rate achieved under only an average total transmit power constraint ( tapc ) which is clearly less stringent than the per - antenna ce constraint . under tapc , we have plotted an achievable sum rate ( zf - zero - forcing precoder ) and an upper bound on the sum capacity ( cooperative users ) . it can be observed that even with @xmath124 , the extra total transmit power required by the ce precoder when compared to the sum capacity achieving precoder under tapc , is roughly @xmath13 db when @xmath17 . ce precoding with non - linear pas is beneficial , since this @xmath13 db loss is less than the gain in power efficiency that one can achieve by using a non - linear power - efficient pa instead of using a highly linear inefficient pa @xcite . 1 f. rusek , d. persson , b. k. lau , e. g. larsson , o. edfors , f. tufvesson and t. l. marzetta , `` scaling up mimo : opportunities and challenges with very large arrays , '' _ ieee signal process . mag . _ , vol . 30 , no . 1 , 40 - 46 , jan . 2013 . e. g. larsson , o. edfors , f. tufvesson and t. l. marzetta , `` massive mimo for next generation wireless systems , '' _ ieee commun . _ , vol . 52 , no . 2 , pp . 186 - 195 , feb . 2014 . t. l. marzetta , `` noncooperative cellular wireless with unlimited number of base station antennas , '' _ ieee trans . wireless commun . _ , vol . 9 , no . 11 , pp 3590 - 3600 , nov . 2010 . h. q. ngo , e. g. larsson and t. l. marzetta , `` energy and spectral efficiency of very large multi - user mimo systems , '' _ ieee trans . _ , vol . 61 , no . 4 , april 2013 . s. k. mohammed , `` impact of transceiver power consumption on the energy efficiency of zero - forcing detector in massive mimo systems , '' to appear in _ ieee trans . commun . _ , 2014 . s. c. cripps , _ rf power amplifiers for wireless communications , _ artech publishing house , 1999 . s. k. mohammed and e. g. larsson , `` per - antenna constant envelope precoding for large multi - user mimo systems , '' _ ieee trans . _ , vol . 61 , no . 3 , pp . 1059 - 1071 , march 2013 . s. k. mohammed and e. g. larsson , `` constant - envelope multi - user precoding for frequency - selective massive mimo systems , '' _ ieee wireless communications letters _ , vol . 2 , no . 5 , pp . 547 - 550 , october 2013 . t. m. cover , _ elements of information theory , _ _ john wiley and sons _ , second edition , 2006 . f. d. nesser , j. l. massey , `` proper complex random processes with applications to information theory , '' _ ieee trans . info . theory _ , 39 , no . 4 , july 1993 .

we consider downlink precoding in a frequency - selective multi - user massive mimo system with highly efficient but non - linear power amplifiers at the base station ( bs ) . a low - complexity precoding algorithm is proposed , which generates constant - envelope ( ce ) transmit signals for each bs antenna . to avoid large variations in the phase angle transmitted from each antenna , the difference of the phase angles transmitted in consecutive channel uses is limited to @xmath0 $ ] for a fixed @xmath1 . to achieve a desired per - user information rate , the extra total transmit power required under the time variation constraint when compared to the special case of no time variation constraint ( i.e. , @xmath2 ) , is _ small _ for many practical values of @xmath3 . in a i.i.d . rayleigh fading channel with @xmath4 bs antennas , @xmath5 single - antenna users and a desired per - user information rate of @xmath6 bit - per - channel - use , the extra total transmit power required is less than @xmath7 db when @xmath8 . massive mimo , constant envelope .

the study of the decays of @xmath8 mesons is important and interesting for the determination of the flavor parameters of the standard model ( sm ) , the exploration of @xmath9 violation , the search of new physics beyond sm , etc . in recent years , theoretical studies of @xmath10 mesons have been investigated widely in the literatures . they are tested and supported by the experimental data collected by the detectors at the @xmath11 colliders , such as the cleo , babar , and belle . with the bright hope arising from the startup of the cern large hadron collider ( lhc ) @xcite , the heavier @xmath12 and @xmath6 mesons could be produced abundantly and studied in detail at the hadron colliders . it is estimated that one could expect around @xmath13 @xmath14 @xmath15 `` self - tagging '' @xmath6 events per year at the lhc @xcite due to a relatively large production cross section @xcite plus the huge luminosity @xmath16 @xmath17 @xmath18 @xmath19 @xcite . there seems to exist a real possibility to study not only some @xmath6 rare decays , but also @xmath9 violation and polarization asymmetries . the study of the @xmath6 mesons will highlight the advantages of @xmath8 physics . the @xmath6 mesons are the `` double heavy - flavored '' binding systems and share many features with the heavy quarkonia . the first observation of the @xmath6 mesons at the tevatron @xcite provokes the physicist s particular interest in them . many studies and investigation of the properties of the @xmath6 mesons have been made , and will be further scrutinized by the lhc experiments . because the @xmath6 mesons lie below the @xmath20 threshold ( here we only discuss the lightest @xmath21 ground state pseudoscalar @xmath6 mesons , excluding their excited states ) and carry flavors , they can not annihilate into gluon and/or photon so are stable for the strong and/or electromagnetic interaction . because of the flavor quantum numbers @xmath8 @xmath17 @xmath22 @xmath17 @xmath23 , the @xmath6 mesons can decay through the weak interaction only . the @xmath6 mesons have more decay modes than the @xmath24 mesons due to several reasons . one is that many decay modes , such as @xmath6 @xmath1 @xmath24 @xmath25 @xmath26 , are only accessible by @xmath6 mesons because of their sufficiently large masses . another is that the @xmath6 mesons carry open flavors , so either @xmath27 or @xmath28 quarks can decay individually . the potential decays of the @xmath6 mesons permit us to over - constrain quantities determined by the @xmath24 meson decays . the decays of @xmath6 mesons can be divided into three classes : ( 1 ) the @xmath27-quark decay ( i.e. @xmath27 @xmath1 @xmath29 , where the up - type quark @xmath29 @xmath17 @xmath30 , @xmath28 ) accompanied with the spectator @xmath28-quark , ( 2 ) the @xmath28-quark decay ( i.e. @xmath28 @xmath1 @xmath31 , where the down - type quark @xmath31 @xmath17 @xmath32 , @xmath33 ) accompanied with the spectator @xmath27-quark , and ( 3 ) the annihilation channel ( i.e. @xmath0 @xmath1 @xmath34 , @xmath35 , where the lepton @xmath36 @xmath17 @xmath37 , @xmath38 , @xmath39 ) . among the multitudinous @xmath6 decays , the weak annihilation channels are expected to take @xmath4 @xmath40 shares according to the estimates in @xcite for which a major part comes from the tree weak annihilation process @xmath0 @xmath1 @xmath41 which is not helicity - suppressed because of the large charm quark mass and produces a large weak annihilation branching ratio with charm in the final state , while the charmless pure weak annihilation decay @xmath6 @xmath1 @xmath7 is helicity - suppressed like the @xmath42 @xmath1 @xmath7 decay and would have a very small branching ratio . it is highly expected that the lhc experiments might shed light on a better understanding of weak annihilation processes for @xmath6 mesons . in recent years , several attractive methods have been proposed to study the nonleptonic @xmath8 decays , such as the qcd factorization ( qcdf ) @xcite , perturbative qcd method ( pqcd ) @xcite , soft and collinear effective theory @xcite , etc . here , we would like to investigate the charmless pure weak annihilation @xmath6 @xmath1 @xmath7 decay with the pqcd approach due to several reasons . ( 1 ) one reason is that fits of nonleptonic charmless decays @xmath10 @xmath1 @xmath43 , @xmath44 without taking into account weak annihilation contributions are generally of poor quality @xcite ( here @xmath45 and @xmath46 denote the lightest ground pseudoscalar and vector mesons , respectively ) . our present understanding of the weak annihilation contributions remains limited and unclear . so the pure weak annihilation processes , such as @xmath6 @xmath1 @xmath7 decays , are interesting and worthy of study , which will certainly help us to improve our understanding of the weak annihilation contributions . ( 2 ) another is that due to both kinematic improvement from the large phase spaces and dynamic enhancement of the ckm factor @xmath47 , the @xmath6 @xmath1 @xmath7 decay is expected to have a large branching ratio among two - body nonleptonic charmless @xmath48-annihilation @xmath6 @xmath1 @xmath43 processes . in addition to the absence of penguin operators for the tree annihilation process @xmath6 @xmath1 @xmath7 , final state interactions arising from soft gluon exchanges are expected to be extremely small because of the large momenta of the final @xmath49 mesons . therefore a relatively accurate estimation of annihilation contributions could be obtained effectively from the charmless @xmath6 @xmath1 @xmath7 decay . still another is that ref.@xcite obtains a very large @xmath6 @xmath1 @xmath7 branching ratio , about @xmath50 , at @xmath51 orders of magnitude bigger than the estimate @xmath52 of ref.@xcite , but this estimate is not valid because ref.@xcite in their calculation incorrectly uses the measured penguin - dominated @xmath53 branching ratio while the decay @xmath6 @xmath1 @xmath7 is a pure tree weak annihilation and should be related to @xmath42 @xmath1 @xmath7 . in addition , the branching ratio of the charmless decay @xmath6 @xmath1 @xmath7 is estimated to be @xmath54 with the qcdf approach @xcite . recently , this charmless decay is also studied with the pqcd approach and its branching ratio is @xmath5 with the off - mass - shell final states @xcite , which is the same order of magnitude as ours obtained in this paper with the on - shell final states . this paper is organized as follows : in section [ sec2 ] , we will discuss the theoretical framework and give the decay amplitudes for @xmath6 @xmath1 @xmath7 with the perturbative qcd approach . in our calculation , we shall ignore the final state interactions because the final states have very large momenta and move far away before soft gluon exchange . section [ sec3 ] is devoted to the numerical result of the branching ratio . finally , we summarize in section [ sec4 ] . for convenience , the kinematics variables are described in the terms of the light cone coordinate . the momenta of the valence quarks and hadrons in the rest frame of the @xmath6 meson are defined by @xmath65 where @xmath66 @xmath17 @xmath67 . the null vectors @xmath68 and @xmath69 are the plus and minus directions , respectively . @xmath70 is the momentum of @xmath28 quark in the @xmath6 meson . @xmath71 and @xmath72 are the momenta of the light non - strange quark in the @xmath73 and @xmath74 mesons , respectively . @xmath75 denotes the transverse momentum . @xmath76 denotes the longitudinal momentum fraction of the valence quark . the calculation of the hadronic matrix elements is difficult due to the nonperturbative effects arising from the strong interactions . phenomenologically , using the brodsky - lepage approach @xcite , a modified perturbative qcd formalism has been proposed recently under the @xmath77 factorization framework @xcite . taking into account the transverse momentum of the valence quarks in the hadrons , the sudakov factors are introduced to modify the endpoint behavior of the hadronic matrix elements . the amplitudes are factorized into three convolution parts : the `` harder '' functions , the heavy quark decay subamplitudes , and the nonperturbative meson wave functions , which are characterized by the @xmath78 boson mass @xmath79 , the typical scale @xmath80 of the decay processes , and the hadronic scale @xmath81 , respectively . the pqcd approach has been extensively applied to study semileptonic and nonleptonic @xmath8 decays with phenomenological results . more information about pqcd approach can be found in @xcite . the final decay amplitudes can be expressed as @xmath82 where the wilson coefficient @xmath83 is calculated in perturbative theory at the scale of @xmath79 and evolved down to the typical scale @xmath80 using the rg equations . @xmath84 denotes the convolution over parton kinematic variables . @xmath85 is the hard - scattering subamplitude which is dominated by hard gluon exchange and can be factorized . the universal wave functions @xmath86 absorb nonperturbative long - distance dynamics , which can be extracted from experiments or constrained by lattice calculation and qcd sum rules . @xmath27 is the conjugate variable of the transverse momentum of the valence quark of the meson . according to the arguments in @xcite , the amplitude of eq.([eq : am01 ] ) is free from the renormalization scale dependence . within the pqcd framework , the long - distance hadronic information is contained by the the so - called light - cone distribution amplitudes ( lcdas ) which are defined from hadron - to - vacuum matrix elements of nonlocal bilinear operators . although lcdas are not calculable in qcd perturbation theory , some of their properties are well understood for both light and heavy mesons . for example , the lcdas for the @xmath49 meson including higher - twist contributions are systematically presented in @xcite . in our calculation , we only consider two - particle ( valence quarks ) twist-2 and twist-3 lcdas for @xmath49 mesons , and neglect contributions from higher fock states . the lcdas for @xmath49 mesons are written as @xmath87\big\}_{{\beta}{\alpha}}\ ] ] where @xmath88 is the color number . the parameter @xmath89 is the chiral factor @xmath89 @xmath17 @xmath90 . the null vector @xmath91 and @xmath92 are parallel to @xmath93 and @xmath94 , respectively . the expressions of the twist-2 lcdas @xmath95 and the twist-3 lcdas @xmath96 , @xmath97 are collected in appendix [ app01 ] . unlike the @xmath98 and @xmath49 mesons , our knowledge of the lcdas for @xmath6 mesons has been relatively poor until recently ( for a recent view , see @xcite ) , but we know that the @xmath6 mesons are composed of heavy valence quark both @xmath27 and @xmath28 . given @xmath99 @xmath100 @xmath101 @xmath25 @xmath102 , the @xmath6 mesons can be described approximately by nonrelativistic dynamics . in this paper , we will take @xmath103_{{\beta}{\alpha } } , \label{eq : wf - bc-01}\ ] ] where @xmath104 is the decay constant of the @xmath6 meson . as the arguments in @xcite , this simplest form , @xmath105 @xmath17 @xmath106 , is the two - particle nonrelativistic lcdas at the tree level where both heavy valence quarks just share the total momentum of the @xmath6 mesons according to their masses . for a rough estimation of the branching ratio for @xmath6 @xmath1 @xmath7 decay , we will take the simplest form as an approximation , and neglect the relativistic corrections and contributions from higher fock states . the branching ratio in the @xmath6 meson rest frame can be written as : @xmath123 where @xmath93 is the center - of - mass momentum of @xmath49 mesons . the lifetime and mass of the @xmath6 meson are @xmath99 @xmath17 @xmath124 @xmath125 @xmath126 gev and @xmath127 @xmath17 @xmath128 ps @xcite , respectively . other input parameters are @xmath129 if not specified explicitly , we shall take their central values as the default input . the numerical result of the branching ratio is @xmath130{\times } [ 1{\pm}0.3\%(\hbox{ckm}){\pm}1.6\%(f_{b_{c}}){\pm}3.7\%(f_{k } ) ] { \times}10^{-7},\ ] ] where the errors come from the uncertainties of quark masses @xmath101 and @xmath102 , the ckm factor @xmath57 , and the decay constants @xmath104 and @xmath131 . the largest error arises from the parameter of @xmath101 , which can reach @xmath132 . the errors arising from both the ckm factor and the decay constants are relatively small . of course , there are some other uncertainties not considered here , such as the radiative corrections to the lcdas of @xmath6 mesons , the final states interactions , etc . so the results might just be an estimation of the pqcd approach . our estimation of the branching ratio @xmath3 is slightly different with the result in @xcite , although they are calculated with the same pqcd approach resulting in the same order of magnitude @xmath5 . besides the input parameters , the reasons may be ( 1 ) whether the final states are on - mass - shell or not , and ( 2 ) whether the contributions of factorizable topologies are zero or not . with appropriate input parameters , the results in @xcite and ours are in agreement with each other within an error range . as the arguments in @xcite , the inconsistencies among various estimations of the branching ratio @xmath3 , such as @xmath52 based on @xmath42 annihilation by using the relations among the charmless weak annihilation @xmath6 decay channels relying on the @xmath63 flavor symmetry @xcite , @xmath5 ( or @xmath54 @xcite ) based on perturbative one - gluon exchange with the pqcd ( or qcdf ) approach , arise from conceptually different methods . anyway , for weak annihilation to light quarks in the final state , the tree annihilation @xmath0 @xmath1 @xmath133 process is helicity suppressed because of small light quark masses , so that gluon emission either from the initial or final state must occur in this annihilation and the decay amplitude is then @xmath134 as given in pqcd . both the estimations in @xcite and our result are in accordance with an intuitive expectation for nonleptonic charmless @xmath48-annihilation of heavy meson decays which are usually suppressed . there are some additional factors for the tiny estimation of @xmath3 . one the is that although the @xmath135 decay is a tree weak annihilation process , its amplitude is color suppressed and associated with @xmath136 . another is that there is a large destructive interference between the nonfactorizable topologies due to the near equal final state particle masses . this can be clearly found in eq.([eq : amplitude ] ) . the numerical results also confirm the cancellation between the nonfactorizable topologies , and give the strong phases @xmath4 @xmath137 and @xmath4 @xmath138 for fig.[fig1 ] ( a ) and ( b ) , respectively . if the pqcd prediction is right , then there should be some @xmath139 events for @xmath135 decay per year at the lhc . considering the detection efficiency and selection efficiency , there would be just a few events per year . the signal of the pure weak annihilation @xmath135 decay would be very tiny at the lhc . as @xmath8 nonleptonic charmless decays , the charmless pure weak annihilation is expected to be small in @xmath6 nonleptonic decays , so the lhc measurement could confirm our understanding of the annihilation terms in weak decays based on perturbative qcd . in this paper , we study the @xmath6 @xmath1 @xmath7 decay with the pqcd approach , which would call for another reassessment of the weak annihilation processes and might provide some valuable hints of our understanding on perturbative qcd and long - distance contributions . it is found that the contributions of factorizable annihilation topologies are zero , and that there is a large cancellation between the nonfactorizable topologies , which result in the branching ratio @xmath3 @xmath4 @xmath5 . the branching ratio with the pqcd approach is so tiny that the @xmath6 @xmath1 @xmath7 decay might not be measured at the lhc experiments . this work is supported by both national natural science foundation of china ( under grant no . 10805014 ) and the program for science & technology innovation talents in universities of henan province , china ( under grant no . 2010hastit001 ) . we would like to thank the referees for their helpful comments . the expression of the lcdas of the @xmath49 meson incluing higher - twist contributions can be found in @xcite . in our calculation , the twist-2 distribution amplitude @xmath95 and the twist-3 distribution amplitude @xmath96 and @xmath97 are @xcite @xmath140 where the decay constant @xmath131 @xmath17 @xmath141 mev . @xmath80 @xmath17 @xmath142 @xmath143 @xmath144 @xmath17 @xmath145 @xmath143 @xmath67 . the gegenbauer polynomials are @xmath146 the expression of the sudakov factors @xmath121 is @xmath147 where @xmath148 the anomalous dimension of the quark is @xmath149 @xmath17 @xmath150 . the explicit expression of @xmath151 can be found in @xcite . 99 k. aamodt , _ et al . _ ( the alice collaboration ) , eur . j. * c65 * , 111 ( 2009 ) . n. brambilla , _ et al . _ ( quarkonium working group ) , cern-2005 - 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in the framework of the perturbative qcd approach , we study the charmless pure weak annihilation @xmath0 @xmath1 @xmath2 decay and find that the branching ratio @xmath3 @xmath4 @xmath5 . this prediction is so tiny that the @xmath6 @xmath1 @xmath7 decay might be unmeasurable at the large hadron collider .

for @xmath0 let @xmath1 denote the linear space of entire ( real valued ) eigenfunctions @xmath2 of the laplacian @xmath3 whose eigenvalue is @xmath4 @xmath5 the zero set of @xmath2 is the set @xmath6 the zero set decomposes into a collection of connected components which we denote by @xmath7 . our interest is in the topology of @xmath8 and of the members of @xmath7 . let @xmath9 denote the ( countable and discrete ) set of diffeomorphism classes of compact connected smooth @xmath10-dimensional manifolds that can be embedded in @xmath11 . the compact components @xmath12 in @xmath7 give rise to elements @xmath13 in @xmath9 ( here we are assuming that @xmath2 is generic with respect to a gaussian measure so that @xmath8 is smooth , see section [ monochromatic ] ) . the connected components of @xmath14 are the nodal domains of @xmath2 and our interest is in their nesting properties , again for generic @xmath2 . to each compact @xmath15 we associate a finite connected rooted tree as follows . by the jordan - brouwer separation theorem @xcite each component @xmath15 has an exterior and interior . we choose the interior to be the compact end . the nodal domains of @xmath2 , which are in the interior of @xmath12 , are taken to be the vertices of a graph . two vertices share an edge if the respective nodal domains have a common boundary component ( unique if there is one ) . this gives a finite connected rooted tree denoted @xmath16 ; the root being the domain adjacent to @xmath12 ( see figure 2 ) . let @xmath17 be the collection ( countable and discrete ) of finite connected rooted trees . our main results are that any topological type and any rooted tree can be realized by elements of @xmath1 . [ main theorem r a ] given @xmath18 there exists @xmath19 and @xmath20 for which @xmath21 . [ main theorem r b ] given @xmath22 there exists @xmath19 and @xmath15 for which @xmath23 . theorems [ main theorem r a ] and [ main theorem r b ] are of basic interest in the understanding of the possible shapes of nodal sets and domains of eigenfunctions in @xmath11 ( it applies equally well to any eigenfunction with eigenvalue @xmath24 instead of @xmath4 ) . our main purpose however is to apply it to derive a basic property of the universal monochromatic measures @xmath25 and @xmath26 whose existence was proved in @xcite . we proceed to introduce these measures . let @xmath27 be the @xmath28sphere endowed with a smooth , riemannian metric @xmath29 . our results apply equally well with @xmath30 replaced by any compact smooth manifold @xmath31 ; we restrict to @xmath30 as it allows for a very clean formulation . consider an orthonormal basis @xmath32 for @xmath33 consisting of real - valued eigenfunctions , @xmath34 . a monochromatic random wave on @xmath27 is the gaussian random field @xmath35 @xmath36 } a_j \phi_j,\label{e : rwdef}\ ] ] where the @xmath37 s are real valued i.i.d standard gaussians , @xmath38 , @xmath39 is a non - negative function satisfying @xmath40 as @xmath41 , and @xmath42\}$ ] . when choosing @xmath43 the @xmath44 s we consider in forming the @xmath45 s are the square roots of the laplace eigenvalues . to a monochromatic random wave we associate its ( compact ) nodal set @xmath8 and a corresponding finite set of nodal domains . the connected components of @xmath8 are denoted by @xmath7 and each @xmath15 yields a @xmath46 . each @xmath20 also gives a tree end @xmath16 in @xmath47 which is chosen to be the smaller of the two rooted trees determined by the inside and outside of @xmath48 . the topology of @xmath8 is described completely by the probability measure @xmath49 on @xmath9 given by @xmath50 where @xmath51 is a point mass at @xmath52 . similarly , the distribution of nested ends of nodal domains of @xmath2 is described by the measure @xmath53 on @xmath47 given by @xmath54 with @xmath55 is the point mass at @xmath56 . the main theorem in @xcite asserts that there exist probability measures @xmath57 and @xmath26 on @xmath9 and @xmath17 respectively to which @xmath49 and @xmath53 approach as @xmath41 , for almost all @xmath35 , provided one has that for every @xmath58 @xmath59\big . \right|=o(1),\ ] ] as @xmath60 . here , @xmath61 , @xmath62 is the localized wave on @xmath63 defined as @xmath64 , and @xmath65 is the gaussian random field on @xmath63 characterized by the covariance kernel @xmath66 ( see section [ monochromatic ] ) . the probability measures @xmath57 and @xmath26 are universal in that they only depend on the dimension @xmath67 of @xmath31 . monochromatic random waves on the @xmath67-sphere equipped with the round metric are known as random spherical harmonics whenever @xmath68 . it is a consequence of the mehler - heine @xcite asymptotics that they satisfy condition for all @xmath58 . also , on any @xmath27 the fields @xmath45 with @xmath69 satisfy condition for all @xmath58 . finally , monochromatic random waves @xmath45 on @xmath27 with @xmath70 , for some @xmath71 , satisfy condition for every @xmath58 satisfying that the set of geodesic loops that close at @xmath72 has measure @xmath73 ( see @xcite ) . on general manifolds one can define monochromatic random waves just as in @xmath27 . monochromatic random waves with @xmath68 on the @xmath67-torus are known as arithmetic random waves . they satisfy condition for all @xmath74 if @xmath75 , and on @xmath76 with @xmath77 provided we work with a density one subsequence of @xmath44 s @xcite . on general @xmath78 monochromatic random waves with @xmath70 , for some @xmath71 , satisfy condition for every @xmath79 satisfying that the set of geodesic loops that close at @xmath72 has measure @xmath73 ( see @xcite ) . examples of such manifolds are surfaces without conjugate points , or manifolds whose sectional curvature is negative everywhere . our main application of theorems [ main theorem r a ] and [ main theorem r b ] is the following result . [ main theorem]let @xmath27 be the @xmath67-sphere equipped with a smooth riemannian metric . let @xmath57 and @xmath26 be the limit measures ( introduced in @xcite ) arising from monochromatic random waves on @xmath27 for which condition is satisfied for every @xmath58 . 1 . the support of @xmath57 is @xmath9 . that is , every atom of @xmath9 is positively charged by @xmath57 . the support of @xmath26 is all of @xmath17 . that is , every atom of @xmath17 is positively charged by @xmath26 . theorem [ main theorem ] asserts that every topological type that can occur will do so with a positive probability for the universal distribution of topological types of random monochromatic waves in @xcite . the reduction from theorems [ main theorem r a ] and [ main theorem r b ] to theorem [ main theorem ] is abstract and is based on the ` soft ' techniques in @xcite ( see also section [ monochromatic ] ) . in particular , it offers us no lower bounds for these probabilities . developing such lower bounds is an interesting problem . the same applies to the tree ends . theorem [ main theorem ] holds for monochromatic random waves on general compact , smooth , riemannian manifolds @xmath78 without boundary . part ( i ) actually holds without modification . the reason why we state the result on the round sphere @xmath30 is that , by the jordan - brouwer separation theorem @xcite , on @xmath30 every component of the zero set separates @xmath30 into two distinct components . this gives that the nesting graph for the zero sets is a rooted tree . on general @xmath78 this is not necessarily true , so there is no global way to define a tree that describes the nesting configuration of the zero set in all of @xmath31 , for all @xmath80 . however , according to @xcite almost all @xmath12 s localize to small coordinate patches and hence our arguments apply . we end the introduction with an outline of the paper . theorem [ main theorem r a ] for @xmath81 ( which is the first interesting case ) is proved in @xcite by deformation of the eigenfunction @xmath82 the proof exploits that the space @xmath83 is simply the set of orientable compact surfaces which are determined by their genus . so in engineering a component of a deformation of @xmath2 to have a given genus it is clear what to aim for in terms of how the singularities ( all are conic ) of @xmath84 resolve . for @xmath85 , little is known about the space @xmath9 and we proceed in section [ topology ] quite differently . we apply whitney s approximation theorem to realize @xmath86 as an embedded real analytic submanifold of @xmath11 . then , following some techniques in @xcite we find suitable approximations of @xmath87 and whose zero set contains a diffeomorphic copy of @xmath86 . the construction of @xmath2 hinges on the lax - malgrange theorem and thom s isotopy theorem . as far as theorem [ main theorem r b ] , the case @xmath88 is resolved in @xcite using a deformation of @xmath89 and a combinatorial chess board type argument . in higher dimensions , for example @xmath81 we proceed in section [ nesting ] by deforming @xmath90 this @xmath2 has enough complexity ( as compared to the @xmath91 in ) to produce all elements in @xmath17 after deformation . however , it is much more difficult to study . unlike or @xmath89 , the zero set @xmath92 in has point and @xmath4-dimensional edge singularities . the analysis of its resolution under deformation requires a lot of care , especially as far as engineering elements of @xmath17 . the pay off as we noted is that it is rich enough to prove theorem [ main theorem r b ] . in section [ monochromatic ] we review some of the theory of monochromatic gaussian fields and their representations . section [ topology ] is devoted to the proof of theorem [ main theorem r a ] . section [ nesting ] is devoted to the proof of theorem [ main theorem r b ] . the latter begins with an interpolation theorem of mergelyan type , for elements in @xmath1 . we use that to engineer deformations of which achieve the desired tree end , this being the most delicate aspect of the paper . our interest is in the monochromatic gaussian field on @xmath11 which is a special case of the band limited gaussian fields considered in @xcite , and which is fundamental in the proof of ( * ? ? ? * theoem 1.1 ) . for @xmath93 , define the annulus @xmath94 and let @xmath95 be the haar measure on @xmath96 normalized so that @xmath97 . using that the transformation @xmath98 preserves @xmath96 we choose a real valued orthonormal basis @xmath99 of @xmath100 satisfying @xmath101 the band limited gaussian field @xmath102 is defined to be the random real valued functions @xmath2 on @xmath11 given by @xmath103 where @xmath104 and the @xmath105 s are identically distributed , independent , real valued , standard gaussian variables . we note that the field @xmath106 does not depend on the choice of the orthonormal basis @xmath107 . the distributional identity @xmath108 on @xmath96 together with lead to the explicit expression for the covariance function : @xmath109 from , or directly from , it follows that almost all @xmath2 s in @xmath102 are analytic in @xmath110 @xcite . for the monochromatic case @xmath111 we have @xmath112 where to ease notation we have set @xmath113 in this case there is also a natural choice of a basis for @xmath114 given by spherical harmonics . let @xmath115 be a real valued basis for the space of spherical harmonics @xmath116 of eigenvalue @xmath117 , where @xmath118 . we compute the fourier transforms for the elements of this basis . for every @xmath119 and @xmath120 , we have @xmath121 we give a proof using the theory of point pair invariants @xcite which places such calculations in a general and conceptual setting . the sphere @xmath122 with its round metric is a rank @xmath4 symmetric space and @xmath123 for @xmath124 is a point pair invariant ( here @xmath125 is the standard inner product on @xmath11 restricted to @xmath122 ) . hence , by the theory of these pairs we know that for every function @xmath126 we have @xmath127 where @xmath128 is any spherical harmonic of degree @xmath129 and @xmath130 is the spherical transform . the latter can be computed explicitly using the zonal spherical function of degree @xmath129 . fix any @xmath131 and let @xmath132 be the unique spherical harmonic of degree @xmath129 which is rotationally invariant by motions of @xmath122 fixing @xmath133 and so that @xmath134 . then , @xmath135 the function @xmath136 may be expressed in terms of the gegenbauer polynomials ( * ? ? ? * ( 8.930 ) ) as @xmath137 now , for @xmath138 , @xmath139 where we have set @xmath140 . hence , by we have @xmath141 with @xmath142 the last term in can be computed using ( * ? ? ? * ( 7.321 ) ) . this gives @xmath143 as desired . the monochromatic gaussian ensemble @xmath144 is given by random @xmath2 s of the form @xmath145 where the @xmath146 s are i.i.d standard gaussian variables . the functions @xmath147 , @xmath148 with @xmath149 , and those in for which the series converges rapidly ( eg . for almost all @xmath2 in @xmath144 ) , all satisfy , that is @xmath19 . in addition , consider the subspaces @xmath150 and @xmath151 of @xmath1 defined by @xmath152 @xmath153 [ e : approximation ] let @xmath19 and let @xmath154 be a compact set . then , for any @xmath155 and @xmath156 there are @xmath157 and @xmath158 such that @xmath159 that is , we can approximate @xmath2 on compact subsets in the @xmath160-topology by elements of @xmath150 and @xmath151 respectively . let @xmath161 . since @xmath2 is analytic we can expand it in a rapidly convergent series in the @xmath162 s . that is , @xmath163 moreover , for @xmath164 , @xmath165 in polar coordinates , @xmath166 , the laplace operator in @xmath11 is given by @xmath167 and hence for each @xmath168 we have that @xmath169 where @xmath129 is some positive integer . there are two linearly independent solutions to . one is @xmath170 and the other blows up as @xmath171 . since the left hand side of is finite as @xmath172 , it follows that the @xmath173 s can not pick up any component of the blowing up solution . that is , for @xmath174 @xmath175 for some @xmath176 . hence , @xmath177 furthermore , this series converges absolutely and uniformly on compact subsets , as also do its derivatives . thus , @xmath2 can be approximated by members of @xmath150 as claimed , by simply truncating the series in . to deduce the same for @xmath151 it suffices to approximate each fixed @xmath178 . to this end let @xmath179 be a sequence of points in @xmath122 which become equidistributed with respect to @xmath180 as @xmath181 . then , as @xmath182 , @xmath183 the proof follows since @xmath184 . indeed , the convergence in is uniform over compact subsets in @xmath110 . for @xmath185 open , let @xmath186 denote the eigenfunctions on @xmath187 satisfying @xmath188 for @xmath189 . any function @xmath29 on @xmath187 which is a limit ( uniform over compact subsets of @xmath187 ) of members of @xmath190 must be in @xmath186 . while the converse is not true in general , note that if @xmath191 is a ball in @xmath11 , then the proof of proposition [ e : approximation ] shows that the uniform limits of members of @xmath190 ( or @xmath150 , or @xmath151 ) on compact subsets in @xmath192 is precisely @xmath193 . with these equivalent means of approximating functions by suitable members of @xmath144 , and particularly @xmath1 , we are ready to prove theorems [ main theorem r a ] and [ main theorem r b ] . indeed , as shown in @xcite the extension of condition @xmath194 of ( ? * theorem 1 ) suffices . namely , for @xmath195 it is enough to find an @xmath196 with @xmath197 containing @xmath12 as one of its components for theorem [ main theorem r a ] , and for @xmath22 it suffices to find an @xmath196 such that @xmath23 for some component @xmath12 of @xmath197 . in this section we prove theorem [ main theorem r a ] . by the discussion above it follows that given a representative @xmath12 of a class @xmath46 , it suffices to find @xmath19 for which @xmath7 contains a diffeomorphic copy of @xmath12 . to begin the proof we claim that we may assume that @xmath12 is real analytic . indeed , if we start with @xmath198 smooth , of the desired topological type , we may construct a tubular neighbourhood @xmath199 of @xmath200 and a smooth function @xmath201 note that without loss of generality we may assume that @xmath202 . fix any @xmath203 . we apply thom s isotopy theorem ( * ? ? ? * thm 20.2 ) to obtain the existence of a constant @xmath204 so that for any function @xmath205 with @xmath206 there exists @xmath207 diffeomorphism with @xmath208 to construct a suitable @xmath205 we use whitney s approximation theorem ( * ? ? ? * lemma 6 ) which yields the existence of a real analytic approximation @xmath209 of @xmath210 that satisfies @xmath211 . it follows that @xmath212 is diffeomorphic to @xmath213 and @xmath12 is real analytic as desired . by the jordan - brouwer separation theorem @xcite , the hypersurface @xmath12 separates @xmath11 into two connected components . we write @xmath214 for the corresponding bounded component of @xmath215 . let @xmath216 be the first dirichlet eigenvalue for the domain @xmath214 and let @xmath217 be the corresponding eigenfunction : @xmath218 consider the rescaled function @xmath219 defined on the rescaled domain @xmath220 . since @xmath221 in @xmath222 , and @xmath223 is real analytic , @xmath224 may be extended to some open set @xmath225 with @xmath226 so that @xmath227 where @xmath228 is the rescaled hypersurface @xmath229 . note that since @xmath217 is the first dirichlet eigenfunction , then we know that there exists a tubular neighbourhood @xmath230 of @xmath228 on which @xmath231 ( see lemma 3.1 in @xcite ) . without loss of generality assume that @xmath232 . we apply thom s isotopy theorem ( * ? ? ? * thm 20.2 ) to obtain the existence of a constant @xmath233 so that for any function @xmath2 with @xmath234 there exists @xmath235 diffeomorphism so that @xmath236 since @xmath237 has no compact components , lax - malgrange s theorem @xcite yields the existence of a global solution @xmath238 to the elliptic equation @xmath239 in @xmath11 with @xmath240 we have then constructed a solution to @xmath239 in @xmath11 , i.e. @xmath161 , for which @xmath197 contains a diffeomorphic copy of @xmath12 ( namely , @xmath241 . this concludes the proof of the theorem . we note that the problem of finding a solution to @xmath242 for which @xmath243 contains a diffeomorphic copy of @xmath12 is related to the work @xcite of a. enciso and d. peralta - salas . in @xcite the authors seek to find solutions to the problem @xmath244 in @xmath11 so that @xmath243 contains a diffeomorphic copy of @xmath12 , where @xmath245 is a nonnegative , real analytic , potential and @xmath12 is a ( possibly infinite ) collection of compact or unbounded tentacled " hypersurfaces . the construction of the solution @xmath2 that we presented is shares ideas with @xcite . since our setting and goals are simpler than theirs , the construction of @xmath2 is much shorter and straightforward . the proof of theorem [ main theorem r b ] consists in perturbing the zero set of the eigenfunction @xmath246 so that the zero set of the perturbed function will have the desired nesting . the nodal domains of @xmath247 build a n - dimensional chess board made out of unit cubes . by adding a small perturbation to @xmath247 the changes of topology in @xmath248 can only occur along the singularities of @xmath248 . therefore , we will build an eigenfunction @xmath2 , satisfying @xmath249 , by prescribing it along the singularities @xmath250 of the zero set of @xmath247 . we then construct a new eigenfunction @xmath251 which will have the desired nesting among a subset of its nodal domains . the idea is to prescribe @xmath2 on the singularities of the zero set of @xmath247 in such a way that two adjacent cubes of the same sign will either glue or disconnect along the singularity . the following theorem shows that one can always find a solution @xmath2 to @xmath249 with prescribed values on a set of measure zero ( such as @xmath252 ) . we prove this result following the first step of carleson s proof @xcite of mergelyan s classical theorem about analytic functions . in the statement of the theorem the function @xmath254 can be replaced by @xmath256 , where @xmath185 is any open set with @xmath257 . this is because @xmath258 is dense in @xmath259 in the @xmath260-topology . consider the sets @xmath261 and write @xmath262 for the restrictions of @xmath263 to @xmath264 . both @xmath265 and @xmath266 are subsets of the banach space @xmath267 , and clearly @xmath268 . it follows that the claim in the theorem is equivalent to proving that @xmath269 to prove , note that a distribution @xmath270 in the dual space @xmath271 can be identified with an @xmath272-tuple of measures @xmath273 with @xmath274 for each @xmath275 . that is , for each @xmath276 , @xmath277 since @xmath278 , proving is equivalent to showing that for each @xmath279 satisfying @xmath280 for all @xmath281 one has that @xmath280 for all @xmath282 using that each @xmath283 is supported in @xmath264 , we have reduced our problem to showing that @xmath284 we proceed to prove the claim in . fix @xmath283 satisfying the assumption in . given @xmath285 we need to prove that @xmath286 . consider the fundamental solution @xmath287 where @xmath288 is the volume of the unit ball in @xmath11 . note that there exists @xmath289 so that @xmath290 for all @xmath291 therefore , for @xmath292 fixed , @xmath293 and @xmath294 are locally integrable in @xmath11 . in particular , @xmath295 and @xmath296 are integrable on the product @xmath297 , where the @xmath298 s are as in . also , note that @xmath299 by these observations , and since @xmath264 has measure zero , we may apply fubini to get @xmath300 where @xmath301 the claim that @xmath286 follows from the fact that @xmath302 for @xmath303 . to see this , let @xmath304 be large enough so that @xmath305 . then , for @xmath306 , the map @xmath307 is in @xmath308 . applying proposition [ e : approximation ] we know that there exists a sequence @xmath309 for which @xmath310 hence , by the assumption in , for each @xmath306 @xmath311 now , the integral defining @xmath312 converges absolutely for @xmath313 and defines an analytic function of @xmath110 in this set . since @xmath312 vanishes for @xmath306 , and @xmath314 is connected , it follows that @xmath315 as claimed . we will give a detailed proof theorem [ main theorem r b ] in @xmath316 since in this setting it is easier to visualize how the argument works . in section [ higher dimensions ] we explain the modifications one needs to carry in order for the same argument to hold in @xmath11 . + let @xmath317 be defined as @xmath318 its nodal domains consist of a collection of cubes whose vertices lie on the grid @xmath319 . throughout this note the cubes are considered to be closed sets , so faces and vertices are included . we say that a cube is positive ( resp . negative ) if @xmath247 is positive ( resp . negative ) when restricted to it . we define the collection @xmath320 of all sets @xmath187 that are built as a finite union of cubes with the following two properties : + * engulf operation . * let @xmath324 . we proceed to define the engulf " operation as follows . we define @xmath325 to be the set obtained by adding to @xmath326 all the negative cubes that touch @xmath326 , even if they share only one point with @xmath326 . by construction @xmath327 . if @xmath328 , the set @xmath325 is defined in the same form only that one adds positive cubes to @xmath326 . in this case @xmath329 . + * join operation . * given @xmath330 we distinguish two vertices using the lexicographic order . namely , for any set of vertices @xmath331 , for @xmath332 we set @xmath333 in the same way we define @xmath334 replacing the minimum function above by the maximum one . for @xmath330 , let @xmath335 be the set of vertices of cubes in @xmath326 . we then set @xmath336 given the vertex @xmath337 we define the edge @xmath338 to be the edge in @xmath339 that has vertex @xmath337 and is parallel to the @xmath110-axis . the edge @xmath340 is defined in the same way . we may now define the join " operation . given @xmath341 and @xmath342 we define @xmath343 as follows . let @xmath344 be the translated copy of @xmath345 for which @xmath346 coincides with @xmath347 . we join " @xmath348 and @xmath345 as @xmath349 + _ definition of the rough nested domains . _ let @xmath353 . a rooted tree is characterized as a finite set of nodes @xmath354 satisfying that @xmath355 to shorten notation , if @xmath356 is a node with @xmath357 children , we denote the children by @xmath358 . given a tree @xmath47 we associate to each node @xmath359 a structure @xmath360 defined as follows . if the node @xmath356 is a leaf , then @xmath361 is a cube of the adequate sign . for the rest of the nodes we set @xmath362 where @xmath357 is the number of children of the node @xmath363 . it is convenient to identify the original structures @xmath364 with the translated ones @xmath365 that are used to build @xmath361 . after this identification , @xmath366 * we let @xmath370 be the set of edges in @xmath371 through which the structures @xmath372 are joined . we will take these edges to be open . that is , the edges in @xmath370 do not include their vertices . + * we let @xmath373 be the set of edges in @xmath374 that are not in @xmath370 . here @xmath375 is the surface @xmath376 if @xmath363 is a leaf , we set @xmath377 . all the edges in @xmath373 are taken to be closed ( so they include the vertices ) . + * we let @xmath378 be the set of edges that connect @xmath379 with @xmath380 for some @xmath381 . if @xmath363 is a leaf , then we set @xmath382 . note that if @xmath356 , and @xmath383 , then @xmath384 is the set of positive cubes that are in the bounded component of @xmath379 and touch @xmath385 . also , if a negative cube in @xmath386 is touching @xmath369 , then it does so through an edge in @xmath387 . + given a node @xmath363 with children @xmath388 , let @xmath389 be the set of edges in @xmath390 . it is clear that for each @xmath391 the set @xmath389 is connected . also , @xmath392 . since the edges in @xmath370 are open , the structures @xmath373 are connected . we proceed to define a perturbation @xmath393 , where @xmath394 we note that by construction @xmath264 is formed by all the edges in @xmath395 . also , it is important to note that if two adjacent cubes have the same sign , then they share an edge in @xmath264 . the function @xmath224 is defined by the rules @xmath396 , @xmath192 and @xmath326 below . + * _ perturbation on @xmath373 . _ let @xmath356 and assume @xmath397 . we define @xmath224 on every edge of @xmath373 to be @xmath4 . if @xmath398 , we define @xmath224 on every edge of @xmath373 to be @xmath399 . + rule @xmath396 is meant to separate @xmath369 from all the exterior cubes of the same sign that surround it . note that for all @xmath356 we have @xmath400 , where @xmath401 is any of the children of @xmath363 , so rule a is well defined . + * _ perturbation on @xmath402 . _ let @xmath403 be an edge in @xmath378 . then , we already know that @xmath224 is @xmath4 on one vertex and @xmath399 on the other vertex . we extend @xmath224 smoothly to the entire edge @xmath403 so that it has a unique zero at the midpoint of @xmath403 , and so that the absolute value of the derivative of @xmath224 is @xmath404 . we also ask for the derivative of @xmath224 to be @xmath73 at the vertices . for example , if the edge is @xmath405\}$ ] where @xmath406 , we could take @xmath407 . + rule @xmath192 is enforced to ensure that no holes are added between edges that join a structure @xmath369 with any of its children structures @xmath408 . + next , assume @xmath409 . note that for any edge @xmath403 in @xmath370 we have that the function @xmath224 takes the value @xmath4 at their vertices , since those vertices belong to edges in @xmath410 and the function @xmath224 is defined to be @xmath4 on @xmath410 . we have the same picture if @xmath411 , only that @xmath224 takes the value @xmath399 on the vertices of all the joining edges . we therefore extend @xmath224 to be defined on @xmath403 as follows . + * _ perturbation on @xmath412 . _ let @xmath356 and assume @xmath397 . given an edge in @xmath370 we already know that @xmath224 takes the value @xmath4 at the vertices of the edge . we extend @xmath224 smoothly to the entire edge so that it takes the value @xmath399 at the midpoint of the edge , and so that it only has two roots at which the absolute value of the derivative of @xmath224 is @xmath404 . we further ask @xmath224 to have zero derivative at the endpoints of the edge . for example , if the edge is @xmath413\}$ ] where @xmath406 , we could take @xmath414 . in the case in which @xmath411 we need @xmath224 to take the value @xmath415 at the midpoint of the edge . + [ extension ] by construction the function @xmath224 is smooth in the interior of each edge . furthermore , since we ask the derivative of @xmath224 to vanish at the vertices in @xmath264 , the function @xmath224 can be extended to a function @xmath417 where @xmath418 is an open neighborhood of @xmath264 . [ perturbation ] given a tree @xmath47 , let @xmath417 be defined following rules a , b and c and remark [ extension ] , where @xmath418 is an open neighborhood of @xmath264 . since @xmath264 is compact and @xmath419 is connected , theorem [ t : perturbation ] gives the existence of @xmath420 that satisfies @xmath421 for @xmath156 small set @xmath422 we will show in lemma [ limit ] that the perturbation was built so that the nodal domain of @xmath423 corresponding to @xmath356 is constituted by the deformed cubes in @xmath424 after the perturbation is performed . we illustrate how rules a , b , and c work in the following examples . in what follows we shall use repeatedly that the singularities of the zero set of @xmath247 are on the edges and vertices of the cubes . therefore , the changes of topology in the zero set can only occur after perturbing the function @xmath247 along the edges and vertices of the cubes . + * example 1 . * as an example of how rules a and b work , we explain how to create a domain that contains another nodal domain inside of it . the tree corresponding to this picture is given by two nodes , @xmath4 and @xmath425 , that are joined by an edge . we start with a positive cube @xmath426 and work with its engulfment @xmath427 . all the edges of @xmath428 belong to @xmath429 . therefore , the function @xmath423 takes the value @xmath430 on @xmath429 . also , all the positive cubes that touch @xmath428 do so through an edge in @xmath429 . it follows that all the positive cubes surrounding @xmath428 are disconnected from @xmath428 after the perturbation is performed . the cube @xmath428 then becomes a positive nodal domain @xmath431 of @xmath423 that is contractible to a point . next , note that all the negative cubes that touch @xmath428 ( i.e. , cubes in @xmath432 ) do so through a face whose edges are in @xmath429 , or through a vertex that also belongs to one of the edges in @xmath429 . therefore , all the negative cubes are glued together after the perturbation is performed , and belong to a nodal domain @xmath433 that contains the connected set @xmath429 . so far we have seen that @xmath433 contains the perturbation of the cubes in @xmath432 . we claim that no other cubes are added to @xmath433 . indeed , all the negative cubes that touch the boundary of @xmath434 do so through edges in @xmath435 . then , since @xmath423 takes the value @xmath436 on @xmath435 , all the surrounding negative cubes are disconnected from @xmath437 after we apply the perturbation . since along the edges connecting @xmath438 with @xmath439 the function @xmath423 has only one sign change ( it goes from @xmath430 to @xmath436 ) it is clear that @xmath433 can be retracted to @xmath440 . + * example 2 . * here we explain how rule c works . suppose we want to create a nodal domain that contains two disjoint nodal domains inside of it . the tree corresponding to this picture is given by three nodes , @xmath4 , @xmath425 , and @xmath441 . the node @xmath4 is joined by an edge to @xmath425 and by another edge to @xmath441 . assume that @xmath428 and @xmath442 belong to @xmath320 . then , @xmath443 . when each of the structures @xmath444 or @xmath444 are perturbed , we get a copy of the negative nodal domain in example 1 . since in @xmath348 the structures @xmath444 and @xmath444 are joined by an edge , the two copies of @xmath433 will also be glued . the reason for this is that the function @xmath423 takes the value @xmath430 in the middle point of the edge joining @xmath444 and @xmath444 . therefore , a small negative tube connects both structures . in this section we explain what our perturbation does to the zero set of @xmath247 at a local level . given a tree @xmath47 , and @xmath156 , let @xmath445 be defined as in definition [ perturbation ] . using that @xmath2 is a continuous function , and that we are working on a compact region of @xmath11 ( we call it @xmath270 ) , it is easy to see that there exists a @xmath446 , so that if @xmath447 is the @xmath448-tubular neighborhood of @xmath264 , then @xmath423 has no zeros in @xmath449 as long as @xmath450 and @xmath451 where @xmath452 is some positive constant that depends only on @xmath453 . this follows after noticing that @xmath454 takes the value @xmath4 at the center of each cube and decreases radially until it takes the value @xmath73 on the boundary of the cube . the construction of the tubular neighborhood @xmath447 yields that in order to understand the behavior of the zero set of @xmath423 we may restrict ourselves to study it inside @xmath447 for @xmath450 . we proceed to study the zero set of @xmath423 in a @xmath448-tubular neighborhood of each edge in @xmath264 . assume , without loss of generality , that the edge is the set of points @xmath455\}$ ] . + * vertices . * at the vertex @xmath456 the function @xmath224 takes the value @xmath4 or @xmath399 . assume @xmath457 ( the study when the value is @xmath4 is identical ) . in this case , we claim that the zero set of @xmath458 near the vertex is diffeomorphic to that of the function @xmath459 provided @xmath448 ( and hence @xmath460 ) is small enough . to see this , for @xmath461 set @xmath462 to be one of the connected components of @xmath463 intersected with @xmath447 . we apply the version of thom s isotopy theorem given in [ ep , theorem 3.1 ] which asserts that for every smooth function @xmath129 satisfying @xmath464 there exists a diffeomorphism @xmath465 making @xmath466 we observe that the statement of [ ep , theorem 3.1 ] gives the existence of an @xmath467 so that the diffeomorphism can be built provided @xmath468 . however , it can be tracked from the proof that @xmath469 can be chosen to be as in the rhs of . applying [ ep , theorem 3.1 ] to the function @xmath470 we obtain what we claim provided we can verify . first , note that @xmath471 . it is then easy to check that @xmath472 for some @xmath473 depending only on @xmath474 . next , we find a lower bound for the gradient of @xmath423 when restricted to the zero set @xmath475 . note that for @xmath476 we have @xmath477 on the other hand , since @xmath478 for all @xmath479 , we conclude @xmath480 whenever @xmath448 is small enough . this shows that at each vertex where @xmath224 takes the value @xmath399 the negative cubes that touch the vertex are glued together while the positive ones are disconnected . + * edges . * having dealt with the vertices we move to describe the zero set of the perturbation near a point inside the edge . there are three cases . in the first case ( case a ) the perturbation @xmath224 is strictly positive ( approx . @xmath436 ) or strictly negative ( approx @xmath430 ) along the edge . in the second case ( case b ) the perturbation @xmath2 is strictly positive ( approx . @xmath436 ) at one vertex and strictly negative ( approx . @xmath430 ) at the other vertex . in the third case ( case c ) , the edge is joining two adjacent structures so the perturbation @xmath2 takes the same sign at the vertices ( it is approx . @xmath436 ) and the opposite sign ( it is approx . @xmath436 ) at the midpoint of the edge having only two zeros along the edge . + in * case a * the zero set of @xmath458 near the edge is diffeomorphic to the zero set of the map @xmath484 . the proof of this claim is the same as the one given near the vertices , so we omit it . in the picture below the first figure shows the zero set of @xmath247 near the edge while the second figure shows the zero set of @xmath470 . this shows that two cubes of the same sign , say negative , that are connected through an edge are going to be either glued if the perturbation takes the value @xmath399 along the edge , or disconnected if the perturbation takes the value @xmath415 along the edge . + in * case b * , it is clear that the only interesting new behavior will occur near the points on the edge at which the function @xmath2 vanishes . since @xmath485 and @xmath486 , there is only one point at which @xmath2 vanishes ; say the point is @xmath487 . note that @xmath2 was built so that @xmath487 is the only zero of @xmath2 along the edge . we claim that the zero set of @xmath423 near @xmath487 is diffeomorphic to the zero set of the map @xmath488 . the proof of this claim is similar to the one given near the vertices , so we omit it . the only relevant difference is that in order to bound @xmath489 from below , one uses that @xmath490 , and that @xmath491 in a ball of radius @xmath492 centered at @xmath487 while @xmath493 . of course , if one is away from the value @xmath494 , then the analysis is the same as that of case a. the first figure in the picture below shows the zero set of @xmath247 along the edge while the second figure shows the zero set of @xmath470 when @xmath495 . this shows that two consecutive cubes sharing an edge along which the perturbation changes sign will be glued on one half of the edge and disconnected along the other half . + in * case c * , the zero set of @xmath423 is diffeomorphic to that of @xmath496 where @xmath2 satisfies @xmath485 and @xmath497 and @xmath498 . the zero set of @xmath470 when @xmath499 is plotted in the figure below . given a tree @xmath47 and @xmath156 we continue to work with @xmath500 as defined in definition [ perturbation ] . fix @xmath356 , and suppose it has @xmath357 children . assume without loss of generality that @xmath411 . for every @xmath501 the perturbed function @xmath423 takes the value @xmath436 on @xmath502 , and @xmath502 is connected . it follows that for each @xmath503 there exists a positive nodal domain @xmath504 of @xmath423 that contains @xmath502 . we define the set @xmath505 as @xmath506 throughout this section we use the description of the local behavior of @xmath475 that we gave in section [ local ] . in the following lemma we prove that @xmath507 is a nodal domain of @xmath423 . [ nodal d ] let @xmath47 be a tree and for each @xmath156 let @xmath423 be the perturbation defined in . then , for each @xmath156 and @xmath356 , the set @xmath508 defined in is a nodal domain of @xmath423 . let @xmath356 and suppose @xmath363 has @xmath357 children . assume without loss of generality that @xmath397 . by definition , @xmath509 where @xmath504 is the nodal domain of @xmath423 that contains @xmath502 . to prove that @xmath507 is itself a nodal domain , we shall show that @xmath510 for all @xmath511 . the edge @xmath514 is shared by a cube @xmath517 and a cube @xmath518 . note that every cube in @xmath512 has at least one vertex that belongs to an edge in @xmath502 ( same with @xmath513 ) . let @xmath519 be a vertex of @xmath520 that belongs to an edge in @xmath502 . in the same way we choose @xmath521 to be a vertex in @xmath522 that belongs to an edge in @xmath523 . in particular , by rule a we have that @xmath524 and @xmath525 . first , we show that all the cubes in @xmath535 glue to form part of @xmath507 after the perturbation is performed . assume , without loss of generality , that @xmath411 . then , @xmath536 for every child @xmath401 of @xmath363 . all the cubes in @xmath535 have an edge in @xmath502 . since such cubes are positive , and @xmath423 takes the value @xmath436 on @xmath502 , it follows that the cubes become part of the nodal domain that contains @xmath502 . that is , all the cubes in @xmath535 become part of @xmath507 after the perturbation is added to @xmath247 . second , we show that no cubes , other than those in @xmath537 , will glue to form part of @xmath507 . indeed , any other positive cube in @xmath538 that touches @xmath539 does so through an edge in @xmath540 . since the function @xmath423 takes the value @xmath430 on @xmath540 , those cubes will disconnect from @xmath541 after we perturb . on the other hand , any positive cube in @xmath542 is touching @xmath543 through edges in @xmath544 where @xmath545 is the number of children of @xmath401 . since @xmath2 takes the value @xmath430 on @xmath544 , the cubes in @xmath546 will also disconnect from @xmath537 . it is convenient to define the partial collections of nested domains . given a tree @xmath47 , a perturbation @xmath423 , and @xmath356 , we define the collection @xmath547 of all nodal domains that are descendants of @xmath507 as follows . if @xmath363 is a leaf then @xmath548 . if @xmath363 is not a leaf and has @xmath357 children , we set @xmath549 we will use throughout this section that we know how the zero set behaves at a local scale ( as described in section [ local ] ) . let @xmath47 be a tree and for each @xmath156 let @xmath423 be the perturbation defined in . we shall prove that there is a subset of the nodal domains of @xmath423 that are nested as prescribed by @xmath47 . since for every @xmath356 the set @xmath507 is a nodal domain of @xmath423 , the theorem would follow if we had that for all @xmath356 statements ( i ) , ( ii ) and ( iii ) imply that @xmath555 has @xmath556 components . one component is unbounded , and each of the other @xmath357 components is filled by @xmath557 for some @xmath558 . we prove statements ( i ) , ( ii ) and ( iii ) by induction . the statements are obvious for the leaves of the tree . + the proof of claim ( iii ) actually shows that @xmath507 can be retracted to the arc connected set @xmath559 where @xmath560 is the curve introduced in lemma [ nodal d ] connecting @xmath561 with @xmath523 that passes through the midpoint of the edge joining @xmath512 with @xmath513 . assume without loss of generality that @xmath411 . then , for every child @xmath401 , all the faces in @xmath565 belong to cubes in @xmath408 that are negative . also , all the other negative cubes in @xmath566 that touch @xmath565 do so through an edge in @xmath502 . since the function @xmath423 takes the value @xmath436 on @xmath502 , all the negative cubes in @xmath567 are disconnected from those in @xmath566 after the perturbation is performed . while all the negative cubes touching @xmath567 are disconnected , an open positive layer @xmath568 that surrounds @xmath569 is created . the layer @xmath568 contains the grid @xmath502 and so it is contained inside @xmath507 . the result follows from setting @xmath570 . + _ proof of claim ( ii)_. this is a consequence of how we proved the statement ( i ) since both @xmath557 and @xmath571 are surrounded by a positive layer inside @xmath572 . + _ proof of claim ( iii)_. note that @xmath573 and that by the induction assuption @xmath574 has no bounded components . on the other hand , we also have that @xmath575 this shows that , in order to prove that @xmath554 has no bounded components , we should show that the cubes in @xmath576 glue to those in @xmath577 leaving no holes . note that all the cubes in @xmath576 are attached to the mesh @xmath578 through some faces or vertices . assume without loss of generality that @xmath411 . for each @xmath381 the layer @xmath568 is contained in @xmath507 and all the cubes in @xmath384 are glued to the layer thorugh an entire face or vertex . the topology of @xmath507 will depend exclusively on how the cubes in @xmath579 will join or disconnect each other along the edges that start at @xmath502 and end at a distance @xmath4 from @xmath502 . the function @xmath423 takes the value @xmath436 on @xmath502 . also , note that if a pair of positive cubes in the unbounded component of @xmath580 share an edge @xmath403 that starts at @xmath561 and ends at a distance @xmath4 from it , then the end vertex belongs to @xmath373 , and the function @xmath423 takes the value @xmath430 at this point . since the function @xmath423 has only one root on @xmath403 , we have that no holes are added to @xmath507 when applying the perturbation to those two cubes . for cubes in the bounded component that share an edge one argues similarly and uses the value of @xmath423 on @xmath581 where @xmath545 is the number of children of @xmath401 . to finish , we note that two consecutive structures @xmath512 and @xmath513 are joined through an edge separating two cubes as shown in figure [ figure join ] . the function @xmath423 is negative ( approximately equal to @xmath430 ) at the vertices of the edge , and is positive at the middle point ( approximately equal to @xmath582 ) . since along the edge @xmath423 was prescribed to have only two roots , no holes are introduced when joining the structures . the argument in higher dimensions is analogue to the one in dimension 3 . we briefly discuss the modifications that need to be carried in this setting . let @xmath583 we will work with cubes in @xmath11 that we identify with a point @xmath584 . that is , the cube corresponding to @xmath585 is given by @xmath586\}$ ] . as before , we say that a cube is positive ( resp . negative ) if @xmath247 is positive ( resp . negative ) when restricted to it . the collection of faces of the cube @xmath12 is @xmath587\;\ ; \forall k \neq i\}$ ] . the collection of edges is @xmath588 where each edge is described as the set @xmath589 \;\ ; \forall k \neq i , j\}.\ ] ] we note that if two cubes of the same sign are adjacent , then they are connected through an edge or a subset of it . in analogy with the @xmath316 case , we define the collection @xmath320 of all sets @xmath187 that are built as a finite union of cubes with the following two properties : + * engulf operation . * let @xmath324 . we define @xmath325 to be the set obtained by adding to @xmath326 all the negative cubes that touch @xmath326 , even if they share only one point with @xmath326 . by construction @xmath327 . if @xmath328 , the set @xmath325 is defined in the same form only that one adds positive cubes to @xmath326 . in this case @xmath329 . + * join operation . * given @xmath330 we distinguish two vertices using the lexicographic order . for @xmath330 , let @xmath593 be the set of its vertices . we let @xmath337 be the largest vertex in @xmath594 and @xmath595 be the smallest vertex in @xmath594 . given the vertex @xmath337 we define the edge @xmath338 to be the edge in @xmath339 that contains the vertex @xmath337 and is parallel to the hyperplane defined by the @xmath596 coordinates . the edge @xmath340 is defined in the same way . given @xmath341 and @xmath342 we define @xmath343 as follows . let @xmath344 be the translated copy of @xmath345 for which @xmath346 coincides with @xmath347 . we join " @xmath348 and @xmath345 as @xmath597 + * definition of the rough nested domains . * given a tree @xmath47 we associate to each node @xmath359 a structure @xmath599 defined as follows . if the node @xmath356 is a leaf , then @xmath361 is a cube of the adequate sign . for the rest of the nodes we set @xmath600 where @xmath357 is the number of children of the node @xmath363 . we continue to identify the original structures @xmath364 with the translated ones @xmath365 that are used to build @xmath361 . after this identification , @xmath601 + * building the perturbation . * let @xmath356 be a node with @xmath357 children . we define the sets of edges @xmath410 , @xmath378 and @xmath370 in exactly the same way as we did in @xmath316 ( see section [ s : perturbation ] ) . we proceed to define a perturbation @xmath393 , where @xmath394 the function @xmath224 is defined by the rules @xmath396 , @xmath192 and @xmath326 below . + let @xmath602 \to [ -1,1]$ ] be a smooth increasing function satisfying @xmath603 we also demand @xmath604 * _ perturbation on @xmath373 . _ let @xmath356 and assume @xmath397 . we define @xmath224 on every edge of @xmath373 to be @xmath4 . if @xmath398 , we define @xmath224 on every edge of @xmath373 to be @xmath399 . + * _ perturbation on @xmath402 . _ let @xmath605 be an edge that touches both @xmath373 and @xmath606 for some of the child structures @xmath607 of @xmath369 . assume @xmath608 . then we know that we must have @xmath609 and @xmath610 . let @xmath611 be the set of directions in @xmath605 that connect @xmath373 and @xmath606 . we let @xmath612\ ] ] be defined as @xmath613 with this definition , since whenever @xmath614 we have @xmath615 for all @xmath616 , we get @xmath617 . also , whenever @xmath618 we have that there exists a coordinate @xmath619 for which @xmath620 . then , @xmath621 and so @xmath622 . note that @xmath224 vanishes on the sphere @xmath623 and that @xmath624 on @xmath625 because of . if @xmath398 , simply multiply @xmath626 by @xmath399 . + * _ perturbation on @xmath412 . _ let @xmath356 and assume @xmath397 . we set @xmath627 where @xmath628 ranges over the indices @xmath629 . with this definition , whenever @xmath110 is at the center of the edge @xmath630 we have @xmath631 . also , if @xmath632 we have @xmath633 for some @xmath634 , and so @xmath622 . also note that @xmath224 vanishes on a sphere of radius @xmath635 centered at the midpoint of @xmath630 and that the gradient of @xmath224 does not vanish on the sphere because of . if @xmath398 , simply multiply @xmath626 by @xmath399 [ extension2 ] by construction the function @xmath224 is smooth in the interior of each edge . furthermore , since according to we have @xmath636 and @xmath637 , the gradient of @xmath224 vanishes on the boundaries of the edges in @xmath264 . therefore , the function @xmath224 can be extended to a function @xmath417 where @xmath185 is an open neighborhood of @xmath264 . given a tree @xmath47 , let @xmath417 be defined following rules a , b and c and remark [ extension2 ] , where @xmath185 is an open neighborhood of @xmath264 . since @xmath264 is compact and @xmath253 is connected , theorem [ t : perturbation ] gives the existence of @xmath638 that satisfies @xmath421 for @xmath156 small set @xmath422 the definitions in rules a , b and c are the analogues to those in dimension @xmath639 . for example , when working in dimension @xmath639 on the edge @xmath640\}$ ] , we could have set @xmath641 and @xmath642 if two adjacent cubes are connected through a subset of @xmath643 , then the cubes will be either glued or separated along that subset . this is because the function @xmath2 is built to be strictly positive ( approx . @xmath436 ) or strictly negative ( approx . @xmath430 ) along the entire edge . if two adjacent cubes share an edge through which two structures are being joined , then they will be glued to each other near the midpoint of the edge . this is because @xmath2 is built so that it has the same sign as the cubes in an open neighborhood of the midpoint of the joining edge . if two adjacent cubes in @xmath369 of the same sign share a subset of an edge in @xmath644 , then with the same notation as in rule b , there exists a subset of directions @xmath645 so that the set @xmath646 \ ; \forall t=1 , \dots , s \}$ ] is shared by the cubes . by construction , the cubes will be glued through the portion @xmath647 of @xmath648 that joins @xmath649 with the point @xmath650 near the midpoint @xmath651 , while being disconnected through the portion @xmath652 of @xmath648 that joins the point @xmath650 with @xmath653 . this is because @xmath2 is prescribed to have the same sign as the cubes along @xmath647 , while taking the opposite sign of the cubes along @xmath652 . let @xmath654 , with @xmath655 . running a similar argument to the one given in @xmath316 one obtains that all the cubes in @xmath656 will glue to form a negative nodal domain @xmath507 of @xmath423 . we sketch the argument in what follows . all the negative cubes in @xmath657 that touch @xmath369 do so through an edge in @xmath373 since they will be at distance @xmath4 from the children structures @xmath658 . since the perturbation @xmath2 takes a strictly positive value ( approx . @xmath582 ) along any edge in @xmath373 , the negative cubes in @xmath657 will be separated from those in in @xmath659 . simultaneously , for each @xmath129 , all the cubes in @xmath660 are glued to each other since they are negative cubes that touch @xmath661 and @xmath661 is a connected set on which the perturbation @xmath2 takes a strictly negative value ( approx . @xmath430 ) . this gives that @xmath661 belongs to a negative nodal domain of @xmath423 , and that the negative cubes in @xmath660 are glued to the nodal domain after the perturbation is performed . furthermore , two consecutive structures @xmath662 and @xmath663 are joined through an edge in @xmath402 . this edge , which joins a negative cube in @xmath662 and a negative cube in @xmath663 has its boundary inside @xmath661 . since @xmath2 is strictly positive ( approx . @xmath582 ) on @xmath661 , we know that the parts of the two cubes that are close to the boundary will be disconnected . however , since the perturbation was built so that @xmath2 is strictly negative ( approx . @xmath430 ) at the midpoint of the edge , both negative cubes are glued to each other . in fact , one can build a curve @xmath664 contained inside the nodal domain that joins @xmath665 with @xmath666 . it then follows that all the cubes in @xmath667 are glued to each other after the perturbation is performed and they will form the nodal domain @xmath507 of @xmath423 containing @xmath668 . one can carry the same stability arguments we presented in section [ local ] to obtain that at a local level there are no unexpected new nodal domains . for this to hold , as in the @xmath316 case , the argument hinges on the fact that in the places where both @xmath247 and @xmath2 vanish , the gradient of @xmath2 is not zero ( as explained at the end of each rule ) . finally , rule b is there to ensure that the topology of each nodal domain is controlled in the sense that when the cubes in @xmath656 glue to each other they do so without creating unexpected handles . indeed , the cubes in @xmath656 can be retracted to the set @xmath669 where @xmath670 and @xmath671 are the children of @xmath672 . the argument we just sketched also shows that the nodal domains @xmath507 with @xmath356 are nested as prescribed by the tree @xmath47 . indeed , claims ( i ) , ( ii ) and ( iii ) in the proof of theorem [ main theorem r b ] are proved in @xmath11 in exactly the same way we carried the argument in @xmath316 .

this paper is dedicated to the study of the topologies and nesting configurations of the components of the zero set of monochromatic random waves . we prove that the probability of observing any diffeomorphism type , and any nesting arrangement , among the zero set components is strictly positive for waves of large enough frequencies . our results are a consequence of building laplace eigenfunctions in euclidean space whose zero sets have a component with prescribed topological type , or an arrangement of components with prescribed nesting configuration .

the effort to investigate low - level nuclear activity in nearby galaxies has proceeded along two separate , largely independent routes . the more direct and traditional approach utilizes optical spectroscopy of magnitude - limited samples , usually without highly restrictive selection by hubble type ( see ho 1996 for a review ) . these surveys arrive at the consensus that mild nuclear activity attributable to the phenomenon of active galactic nuclei ( agns ) is very common in nearby galaxies , particularly those with a prominent bulge component . along similar lines , sensitive radio interferometric observations have revealed intrinsically weak nuclear sources in an increasingly large number of purportedly `` normal '' galaxies ( e.g. , heeschen 1970 ; ekers & ekers 1973 ; crane 1979 ; van der hulst , crane , & keel 1981 ; condon 1982 ) . to date , the most comprehensive surveys for weak , compact nuclear radio emission have concentrated on elliptical and s0 galaxies . ( as a working definition , `` compact '' here denotes angular scales 5 , which , for the relatively nearby galaxies addressed in this paper , pertain to physical dimensions of several hundred parsecs . ) the two main studies are those by sadler ( 1989 ) and wrobel & heeschen ( 1991 , hereafter wh ) , which selected southern and northern sources , respectively ; both employed the very large array ( vla ) at an observing frequency of 5 ghz ( 6 cm ) , which resulted in a synthesized beam with an angular resolution of @xmath05 . both of these studies achieved sensitivities well below 1 mjy and detected a large number of previously unknown , low - power radio cores associated with the galaxy nuclei . some of the newly discovered sources have radio powers as low as 10@xmath1 w hz@xmath2 at 5 ghz , many orders of magnitude less than the cores of classical radio galaxies or quasars . what is the physical nature of these low - power radio cores in nearby early - type galaxies ? are they truly scaled - down versions of distant , more powerful agns , or can they be explained by less exotic processes ? sadler ( 1989 ) favored the agn interpretation on the basis of optical spectroscopic information available for their sample . subsequent radio observations by slee ( 1994 ) of a radio - bright subset at a much higher angular resolution revealed that most of the radio emission originates from extremely small , parsec - scale cores . the high brightness temperatures inferred ( @xmath3 k ) and the preponderance of flat radio spectra support a nonthermal origin for the emission , at least for the brightest sources they selected . the situation for the northern objects , however , is less clear , for , until recently , optical spectra were not widely available for the wh sample . and unlike the southern sample , there are yet no systematic follow - up radio observations with sufficient angular resolution to yield meaningful constraints on the brightness temperature , nor are there , in general , spectral indices to diagnose the emission mechanism . the wh sample contains fewer high - luminosity galaxies than the sample of sadler et al . because the former is selected both by magnitude and by distance . since the physical origin of nuclear activity depends on galaxy luminosity ( e.g. , phillips 1986 ; ho , filippenko , & sargent 1997b ) , the results of the southern survey can not be readily generalized to the northern survey . indeed , wh argued , on the basis of an observed correlation between the radio and the far - infrared emission , that star formation , not agn activity , predominantly powers the nuclei in their sample . the purpose of this paper is to clarify the nature of the radio nuclei in the wh sample of early - type galaxies with the aid of a newly completed optical spectroscopic survey of nearby galaxies . we will argue that the bulk of the radio emission arises from accretion - powered activity , as was proposed by sadler ( 1989 ) for the southern counterpart of the wh survey . the wh sample formally comprises 216 elliptical and s0 galaxies ( @xmath4 1 ) in the center for astrophysics redshift survey ( huchra 1983 ) that satisfy @xmath5 14 mag and @xmath6 3000 ( see wrobel 1991 ) . this sample naturally overlaps heavily with the palomar optical spectroscopic survey of ho , filippenko , & sargent ( 1995 , 1997a ) , which selected galaxies from the revised shapley - ames catalog of bright galaxies ( sandage & tammann 1981 ) having @xmath7 12.5 mag and declinations @xmath80 . the palomar survey contains 145 e and s0 galaxies , and 109 of these ( 38 e and 71 s0 ) are in common with the wh sample . figure 1 shows the fractional radio luminosity function of the 109 objects calculated using the kaplan - meier estimator ( feigelson & nelson 1985 ) to take into account the significant number of upper limits ( 69 ) present . the fractional luminosity functions of ellipticals and s0s are also shown separately to reinforce the well - known fact ( e.g. , sadler 1989 ) that the strength of the radio source increases with the size of the bulge component . the mean 5-ghz radio power is ( 2.1@xmath91.3 ) w hz@xmath2 for the entire sample , ( 5.5@xmath93.7 ) w hz@xmath2 for the ellipticals alone , and ( 0.3@xmath90.1 ) w hz@xmath2 for the s0s alone . ho ( 1997a ) give a variety of spectroscopic parameters measured for the nuclear regions ; the angular scale of these measurements ( 2@xmath104 ) roughly matches that of the radio synthesized beam . for the present purpose , the most pertinent of the parameters are the spectral classifications of the nuclei and their luminosities as measured in the h emission line . the following analysis uses the distances given in ho ( 1997a ) , which are based on @xmath11 = 75 mpc@xmath2 . as described by ho ( 1997a , 1997b ) , the palomar survey distinguishes four classes of emission - line nuclei : nuclei , seyfert nuclei , low - ionization nuclear emission - line regions ( liners ; heckman 1980 ) , and transition objects ( composite liner+ nucleus ) . a minority of nuclei do not have detectable emission lines down to an equivalent - width limit of @xmath00.25 . the primary spectral classification system is based solely on the relative intensities of the narrow optical emission lines . nuclei , whose spectra closely resemble those of regions and are therefore assumed to be photoionized by young , massive stars , have relatively weak lines of 6300 , 6363 , 6548 , 6583 , and 6716 , 6731 ( compared to , say , h ) . the other three groups , which represent variants of agns , are recognized by their exceptionally strong low - ionization lines of , , and . liners differ from seyferts by their level of excitation , as measured by the ratio of 5007 to h , and transition objects have spectra intermediate between those of liners and nuclei . some members within each of the agn classes show evidence for a weak broad ( widths of several thousand ) h emission line ( ho 1997c ) , and these are designated `` type 1 '' objects . table 1 gives a breakdown of the spectral classification of the objects in common with the wh survey . the first point to notice is that a substantial fraction of the galaxies ( 61% ) have detectable emission lines , in agreement with the high detection rate reported by phillips ( 1986 ) for the southern sample . the somewhat higher detection rate of the palomar sample results from its greater sensitivity and the smaller average distance of the objects . second , nuclei with emission lines are at least ten times more likely to be detected in the radio than those without emission lines . the fraction of emission - line nuclei detected in the radio is 57% , compared with 5% for nuclei not detected in emission , a difference significant at a level of @xmath899.99% according to the @xmath12 for the 2@xmath102 contingency table for these results . expressed in another way , of the 40 nuclei detected in the radio , 38 ( 95% ) have optical emission lines , whereas among the 69 nuclei with radio upper limits , only 29 ( 42% ) do . galaxies detected as radio sources , therefore , have a much higher likelihood of being detected in optical emission lines , and vice versa . this is further reflected in the large difference between the strengths of the emission lines : the galaxies detected as radio sources have an average extinction - corrected h luminosity ( properly accounting for upper limits ) , of ( 4.2@xmath91.1 ) , one order of magnitude higher than the radio - undetected galaxies , which have an average @xmath13(h ) = ( 3.9@xmath90.9 ) . most revealing , however , is the fact that the vast majority of the emission - line nuclei ( 59/67 or @xmath090% ) are spectroscopically classified as agns . the active nuclei are extremely faint in comparison to traditionally - studied agns ( fig . 2 ) ; the average @xmath13(h ) is only @xmath02 . the elliptical galaxies exhibit lower levels of optical line emission than the s0s [ average @xmath13(h ) = 0.8 vs. 2.5 ] , probably reflecting a difference in the central gas content between the two morphological types . the typical electron density of the emission - line regions in these objects , as estimated from the 6716 , 6731 doublet , is 200300 , which , when combined with the h luminosities , imply an ionized hydrogen mass of @xmath010@xmath14 . most of the agns belong to the low - ionization category , namely liners and liner/ composites , as is the case in spiral galaxies ( ho 1997b ) . the fraction of agns occupied by liners among elliptical and s0 galaxies , 85% , is somewhat higher than among spirals , which is @xmath070% ( ho 1997b ) . the preponderance of liners in galaxies with the earliest hubble types and the tendency for the most compact radio cores to be found in bulge - dominated systems ( e.g. , sadler 1995 ) no doubt leads to the common suggestion , first made by heckman ( 1980 ) , that there exists a statistical connection between liners and compact radio sources . indeed , disney & cromwell ( 1971 ) had remarked , before heckman coined the term `` liner , '' on the unusually low ionization state of the spectra in their sample of elliptical galaxies with radio - bright nuclei . however , liners certainly exist in more diverse environments than just early - type galaxies , and it should be stressed that a preferred association between compact radio cores and the liner phenomenon has not yet been established unambiguously . hummel ( 1990 ) , who analyzed a galaxy sample with a more representative mixture of hubble types , did _ not _ find a statistically higher incidence of compact radio emission in liners compared to nuclei . in the present sample , the radio detection rate in nuclei ( 50% ) is also very similar to that in agns ( 58% ) ; furthermore , the radio detection rate does not appear to depend on the agn class . the statistics based on our sample alone should be taken as tentative because of the small number of nuclei ( 8) and seyferts ( 9 ) available . when combined with the findings of hummel et al . , however , they strongly suggest that the presence or absence of nuclear radio emission , at least on a scale of several hundred parsecs , is _ not _ a reliable discriminator between agns and star - forming nuclei or between various optical classes of agns . observations at higher angular resolution may be needed to better isolate a central point source , if present , and spectral information may help to identify the emission mechanism . nuclei rarely exist in elliptical galaxies ; none , in fact , are found in the present sample or in the entire palomar survey ( ho 1997b ) . likewise , the nuclei in the vast majority of the s0 galaxies are also optically identified with agns rather than nuclei . the eight s0 galaxies with nuclei in table 1 , which comprise less than 20% of the s0s with emission - line nuclei , stand out as having somewhat lower optical luminosities ( by about 1 mag ) than those classified as agns . thus , insofar as the optical spectral classification of a nucleus gives a reliable indication of its dominant central energy source , our results , in conjunction with those of sadler ( 1989 ) , indicate that nonstellar activity from agns , not star formation , powers the weak radio cores commonly found in nearby early - type galaxies . several older studies have reported an association of compact radio emission with the presence of optical emission lines in the centers of early - type galaxies ( disney & cromwell 1971 ; ekers & ekers 1973 ; oconnell & dressel 1978 ) . the sample analyzed in this paper supports this result with the finding that both the incidence and the strength of optical emission lines are correlated with the presence of radio emission . although there is a large range of radio power ( @xmath15 ) at any fixed h luminosity [ @xmath13(h ) ] , the two quantities appear correlated when the upper limits are properly taken into account ( fig . 3 ) ; the correlation holds for ellipticals and s0s individually and for both types combined . the relation between @xmath15 and @xmath13(h ) for the ellipticals appears offset and possibly steeper than that for the s0s . for a given line luminosity , ellipticals have stronger radio sources than s0s do , although the spread within each morphological type and the overlap between the two are considerable . the error bars in the lower right corner of the diagram give an estimate of the typical uncertainty associated with the data , although three additional factors most likely further contribute to the scatter in the plot . first , the core radio emission is likely to be at least moderately variable . second , as discussed at length in ho ( 1997a ) , some individual h measurements in the palomar survey can be quite uncertain because of slit losses and imperfect photometric calibration . and finally , although the h luminosities have been corrected for galactic and internal extinction , the latter contribution is not always unambiguous in weak emission - line objects because the balmer decrement , on which the correction depends , can be difficult to determine accurately . as emphasized by sadler ( 1989 ) , it is nontrivial to assess the statistical relation between @xmath15 and @xmath13(h ) because each quantity is itself correlated with the distance and with the total optical luminosity of the galaxy . a possible correlation between @xmath15 and @xmath13(h ) in the presence of one of these third variables was evaluated using the test for partial correlation with censored data described by akritas & siebert ( 1996 ) . choosing the total blue absolute magnitude as the test variable , we find that @xmath15 correlates very significantly with @xmath13(h ) . the partial kendall s @xmath16 coefficient is 0.299 and the variance is 0.0432 ( see akritas & siebert ) , which imply that the null hypothesis that @xmath15 and @xmath13(h ) are uncorrelated can be rejected at a significance level of @xmath17 ; the same holds if distance becomes the test variable . in both cases the correlation is stronger for s0s than for ellipticals , but the correlation for the latter is still highly significant . these results conflict with those of sadler ( 1989 ) , who concluded that early - type galaxies with radio cores do not have a higher incidence of optical emission lines than those without radio cores and that radio power is not correlated with optical emission - line luminosity . these authors used a likelihood ratio test based on a maximum - likelihood estimate of the bivariate luminosity function of @xmath15 and @xmath13 ( ) , where @xmath13 ( ) is the luminosity of the 6583 emission line , and found that the two variables are not strongly correlated , even _ without _ accounting for the known partial correlation of each with a third variable ( distance or total galaxy luminosity ) . to try to understand the root of this discrepancy , ( ) is substituted for @xmath13(h ) . we chose to use @xmath13(h ) instead of @xmath13 ( ) because a luminosity based on a recombination line has a more straightforward physical interpretation than a collisionally - excited line . ] the data of sadler et al . were reanalyzed using the same statistical method employed here . table 4 of sadler et al . lists 167 objects having both radio and optical data , with 123 upper limits for @xmath15 and 76 upper limits for @xmath13 ( ) . the conventional generalized kendall s @xmath16 test ( isobe , feigelson , & nelson 1986 ) yields a correlation coefficient of 0.189 and a significance level of 4 . thus , contrary to the conclusion of sadler et al . , @xmath15 and @xmath13 ( ) _ are _ highly correlated . the same conclusion holds even after allowing for dependence on a third variable . taking the total absolute magnitude of the galaxy again as the test variable , the partial kendall s @xmath16 test formally yields a rather small correlation coefficient of 0.0679 ( variance = 0.0223 ) , but it is still statistically significant ( the null hypothesis can be rejected with a probability of 0.2% ) . it is unclear why the statistical method used by sadler et al . gives an inconsistent result . akritas ( 1997 ) notes that the asymptotic distribution that sadler et al . assume for their likelihood ratio statistic ( equation b8 in their appendix b ) is known to be formally valid only for uncensored data ; its behavior in the presence of censoring is unknown . the majority of the radio sources in the wh sample do not yet have published radio spectral indices or meaningful constraints on their brightness temperatures to enable discrimination between a thermal or nonthermal origin for the radio emission . in this regard , it is instructive to consider a constraint that can be derived from the strength of the h emission , which is sampled roughly over the same region as the vla measurements . a purely thermal source with an electron temperature of 10@xmath14 k generates approximately 10@xmath18 @xmath13(h ) w hz@xmath2 of radio power at 5 ghz ( see , e.g. , ulvestad , wilson , & sramek 1981 ) , with @xmath13(h ) expressed in units of w. figure 4 illustrates that the vast majority of the sources fall well above this threshold , which implies that thermal emission contributes negligibly to the radio continuum at this frequency . this is consistent with the hypothesis that the nuclei are agns , although such an interpretation is not required because supernova remnants can generate nonthermal radio emission . the objects that lie near the threshold are all s0 galaxies , most being systems of fairly low luminosity ( @xmath19 19.5 mag ) . several of the s0s with nuclei , notably , have radio powers consistent with the thermal limit , although not all the objects near this limit are classified as nuclei . more generally , there exists a loose correlation between the radio `` excess '' and the galaxy luminosity . the objects with the largest ratio of radio to optical line luminosity tend to be the most luminous , massive galaxies , although there is a considerable spread of this ratio at any given total luminosity . ( as discussed above in connection with figure 3 , the intrinsic scatter is likely to be smaller than indicated because of observational factors . ) the positive correlation between radio power and optical emission - line luminosity in itself , however , does not point to an obvious physical explanation for such a relationship . a correlation between 20-cm radio power and the luminosity of the 5007 forbidden line has been recognized for quite some time in seyfert galaxies ( de bruyn & wilson 1978 ; whittle 1992b ) , most of which have much higher luminosities than the objects considered here , but even in those objects the physical connection between the two variables is not obvious . whittle ( 1992b ) finds that both radio power and emission - line luminosity are strongly coupled to the bulge luminosity , but a residual correlation remains , as is the case here . powerful radio galaxies also exhibit a correlation between radio power and optical emission - line luminosity ( hine & longair 1979 ; rawlings 1989 ; baum & heckman 1989b ; baum , zirbel , & odea 1995 ; zirbel & baum 1995 ) . it is interesting to consider whether the low - luminosity nuclei in our sample extend the radio - power / line - luminosity correlations previously established for the more powerful agns . specifically , one might expect the radio - quiet ellipticals in our sample to track the radio galaxies because they latter are predominantly giant ellipticals ( e.g. , ledlow & owen 1995 ; zirbel 1996 ) ; the sos , on the other hand , might follow the seyfert population , whose host galaxies tend to be bulge - dominated disk systems ( e.g. , whittle 1992a ; ho 1997b ) . the best - fitting relations between radio power and line luminosity found by zirbel & baum ( 1995 ) are plotted on figure 3 . radio galaxies of fanaroff & riley ( 1974 ) `` class i '' and `` class ii '' obey slightly different relations , and both are shown . the line luminosities used by zirbel & baum ( 1995 ) represent the sum of the h and the 6548 , 6583 lines . to translate their results for this comparison , we assume that 6548 , 6583 = 2.3h , the median value observed in elliptical and s0 galaxies by ho ( 1997a ) . the best - fitting lines shown are those for the core radio power , since in the wh sample most of the radio emission comes from a compact core , and they were derived from objects with @xmath20 . note that , in the units used in figure 3 , the objects in the zirbel & baum sample have log @xmath21 2127 and log @xmath13(h ) @xmath22 38.544.5 . we have also include in the plot the sample of seyfert nuclei compiled by whittle ( 1992a ) ; most of these objects are significantly more luminous than those in our sample . again , for comparison with our data , we computed @xmath23 from whittle s 1.4-ghz flux densities assuming @xmath24 , and we calculated @xmath13(h ) from his luminosities by assuming 5007/h = 10 and h / h = 3.1 , values typical of seyfert nuclei ( e.g. , shuder & osterbrock 1981 ) . making allowance for the large scatter in the diagram , the nearby radio - quiet ellipticals broadly follow the faint end of the relations established by the radio galaxies . the sos , on the other hand , appear to join fairly well with the faint end of the seyfert galaxy sequence , which , as was noticed by baum & heckman ( 1989b ) , is distinctly offset from the radio galaxy sequence . a simple interpretation of these results is that the quiescent nuclei in nearby early - type galaxies are the low - luminosity counterparts of the more distant , more powerful agns . this interpretation is consistent with the spectroscopic evidence presented in 2 . the spectra of the nuclei , with only a few exceptions , decidedly do not resemble the spectra of regions ; instead , most of them look like those of liners . if one adopts the viewpoint that most liners represent another , and perhaps the most common , manifestation of the agn phenomenon ( e.g. , halpern & steiner 1983 ; ferland & netzer 1983 ; ho , filippenko & sargent 1993 ; see reviews by filippenko 1996 and ho 1998 ) , one would conclude that the weak , compact radio emission in the centers of nearby early - type galaxies derives from physical processes similar to those operating in more powerful agns . in turn , the finding that many nearby early - type galaxies do , in fact , contain high - brightness temperature , flat - spectrum radio cores ( slee 1994 ) provides strong support for the agn - like nature of liners , at least for those found in elliptical and s0 galaxies . we have used optical spectroscopic information to interpret the nature of the compact radio emission in a sample of nearby early - type ( elliptical and s0 ) galaxies surveyed with the vla by wrobel & heeschen ( 1991 ) . many of these galaxies have weak radio sources ( 1few mjy at 5 ghz ) on scales of several hundred parsecs or less , some with radio powers as low as 10@xmath1 w hz@xmath2 . the radio sources in ellipticals are more luminous than those in s0s because the radio power increases with the optical luminosity of the bulge component . a substantial fraction of the galaxies ( @xmath060% ) show detectable levels of optical nebulosity down to very sensitive limits ( @xmath00.25 equivalent width for the h emission line ) over the same physical scale probed by the radio observations . the measured h luminosities , 10@xmath2510@xmath26 , imply the presence of 10@xmath2710@xmath28 of warm ( 10@xmath14 k ) ionized hydrogen . among s0 galaxies , the quantity of ionized gas increases with galaxy luminosity ( or size ) , but , as a class , ellipticals have less ionized gas than s0s . the amount of thermal gas implied by the h measurements falls short of producing , by large factors in most cases , the observed radio continuum luminosity . most of the radio emission is therefore nonthermal . both the incidence and the strength of optical line emission correlate with the radio power . at a fixed line luminosity , ellipticals have stronger radio cores than s0s . the relation between radio power and line emission in the nearby radio - quiet ellipticals appears to be an extension of a similar relation seen in powerful radio galaxies . the s0s , on the other hand , follow the faint end the correlation established by luminous seyfert galaxies . it is suggested that the weak nuclear sources seen in nearby early - type galaxies are simply the low - luminosity counterparts of more distant , luminous agns . the spectroscopic evidence supports this interpretation . the vast majority of the objects with detectable emission lines emit optical spectra that differ substantially from those of regions . instead , most of them are classified as liners , a few as seyferts . if liner nuclei are predominantly accretion - powered sources , as suggested by some studies , then most of the radio nuclei can be identified as agns . the author is supported by a postdoctoral fellowship from the harvard - smithsonian center for astrophysics and by nasa grants go-06837.01 - 95a and ar-07527.02 - 96a from the space telescope science institute ( operated by aura , inc . , under nasa contract nas5 - 26555 ) . i thank m. akritas and the statistical consulting center for astronomy at penn state university for guidance on some statistical issues . the referee , john stocke , offered constructive criticisms and helpful suggestions . this work has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration .

many nearby early - type ( elliptical and s0 ) galaxies contain weak ( milli - jansky level ) nuclear radio sources on scales a few hundred parsecs or less . the origin of the radio emission , however , has remained unclear , especially in volume - limited samples that select intrinsically less luminous galaxies . both active galactic nuclei and nuclear star formation have been suggested as possible mechanisms for producing the radio emission . this paper utilizes optical spectroscopic information to address this issue . a substantial fraction of the early - type galaxies surveyed with the very large array by wrobel & heeschen ( 1991 ) exhibits detectable optical emission lines in their nuclei down to very sensitive limits . comparison of the observed radio continuum power with that expected from the thermal gas traced by the optical emission lines implies that the bulk of the radio emission is nonthermal . both the incidence and the strength of optical line emission correlate with the radio power . at a fixed line luminosity , ellipticals have stronger radio cores than s0s . the relation between radio power and line emission observed in this sample is consistent with the low - luminosity extension of similar relations seen in classical radio galaxies and luminous seyfert nuclei . a plausible interpretation of this result is that the weak nuclear sources in nearby early - type galaxies are the low - luminosity counterparts of more powerful agns . the spectroscopic evidence supports this picture . most of the emission - line objects are optically classified as seyfert nuclei or low - ionization nuclear emission - line regions ( liners ) , the majority of which are likely to be accretion - powered sources .

as cmb anisotropy experiments have gotten more ambitious , our need for powerful statistical methods has become more urgent . for the cmb data sets that have been obtained up to now , including cobe @xcite ( _ dmr4 _ ) , the 1994 south pole data of @xcite ( _ sp94 _ ) and the 1993 - 95 saskatoon data of @xcite ( _ sk95 _ ) , which we analyze jointly here , it is possible to do a relatively complete bayesian statistical analysis if the primary anisotropies are assumed to be gaussian and the non - gaussian galactic foregrounds are not large . even for these experiments , this is feasible only because of compression , in which the data set is acted upon by linear operators which project it onto subspaces of the full data . in the past the linear combinations of the data were defined by what made intuitive sense ( weighted sums of different frequency channels or weighted averages of pixel separations below the beam scale ) , and could still leave too many pixels to deal with in a complete analysis . here we use the rigorous signal - to - noise eigenmode approach to data compression @xcite to reduce the sets to the manageable @xmath4 important combinations . as we approach the era of megapixel data sets promised by map and cobras / samba , via the era of tens of thousands of pixels promised by long duration balloon experiments , the question of how to come as near to optimal compression as possible given computer limitations becomes of paramount importance . this happy day of too many pixels is now upon us . the goal is to estimate the parameters @xmath5 of a target set of theories with angular power spectra @xmath6 , where @xmath7 is the cmb power spectrum as usually defined and the @xmath8 are the spherical harmonic coefficients of the temperature fluctuations for the theoretical signal . ] by first determining the likelihood function @xmath9 for each theory , and then comparing the likelihoods as a function of the parameters . using bayes theorem , we can write the probability of the theoretical parameters , given the observations and the class of theories being tested , @xmath10 , where the assumed prior probability distribution @xmath11 can reflect _ a priori _ maximal ignorance , or take into account constraints from other information such as large scale structure observations . the proportionality constant is related to the probability that the class of theories is correct given the observations . to give preferred values and errors for a specific cosmological parameter of interest such as the hubble parameter , one often integrates ( marginalizes ) over the other parameters , such as the density fluctuation power amplitude on cluster scales , @xmath12 , and the primordial density fluctuation spectral index @xmath13 . in this paper , we assume the inflationary model for structure formation with gaussian adiabatic ( scalar ) density perturbations and possibly gravitational wave ( tensor ) perturbations . we explore constraints in the parameter space @xmath14 . we assume reheating occurs sufficiently late to have a negligible effect on @xmath15 . the total density parameter , @xmath16 , is expressed in terms of the densities in baryonic cold , hot and vacuum matter , of which @xmath17 can cluster . the age of the universe is @xmath18 and @xmath19 is the hubble parameter in units of @xmath20 ; @xmath21 is a function of @xmath22 , so one parameter is redundant . the scalar and tensor tilts are @xmath23 and @xmath24 . , apart from small corrections . this determines the level of tensor anisotropies compared with scalar . the tensor tilt is related to the deceleration parameter @xmath25 of the universe during inflation by @xmath26 plus small corrections . here , we take one of two cases . ( no - gw case ) : @xmath27 , thus no gravity wave contribution ( for nearly critical acceleration , @xmath28 , as in natural inflation ) . ( gw case ) : @xmath29 if the scalar tilt is negative ( subcritical but nearly uniform acceleration ) and @xmath27 if the scalar tilt is positive . ] with current errors on the data , this space is too large for effective parameter estimation . instead we restrict our attention to various subregions , such as @xmath30 , where either @xmath31 and @xmath32 is a function of @xmath21 or @xmath33 and @xmath34 is a function of @xmath21 . the @xmath18 s we choose are ( 11 , 13 , 15 gyrs ) with @xmath35 , but @xmath31 . for these cases , the `` standard '' nucleosynthesis value @xmath36 was chosen . we constrain @xmath37 , @xmath38 . for @xmath31 models , we have roughly @xmath39 $ ] , where @xmath40 . a recent estimate for globular cluster ages is @xmath41 @xcite . for the case @xmath33 , we have also let @xmath42 vary , over the range @xmath43 , and we have also done a limited exploration of the 13 gyr , @xmath44 , @xmath45 , space with tilt , using @xmath46 from bond and souradeep . the reason for restricting the paths through parameter space is because of the length of time required for a complete statistical treatment of each data set per model @xmath15 . in fig . [ fig : cl13gyr ] , the bandpowers @xcite associated with current experiments are compared with some of the @xmath15 s in the parameter space we are exploring , here the 13 gyr , @xmath47 , tilted sequence with gravity waves included and the 13 gyr @xmath48 sequence with @xmath49 , with amplitude normalized to best - fit the 4-year dmr data . in both cases , @xmath50 . the curves are very similar if we allow for a mix of hot and cold dark matter with the same @xmath51 as these cdm models , and the other parameters fixed . the solid dark curve is the `` standard '' untilted cdm model . the bandpowers for the three experiments analyzed here , _ dmr4 _ , _ sp94 _ , _ sk95 _ , are the darker heavier data points . because of the differing angular scales involved we gain a long lever arm with which we can constrain cosmological parameters more strongly than with any individual experiment . the lower panels in the figures are closeups of the first and second `` doppler peak '' regions . [ fig : clbestfit ] gives the best fit models , described in [ sec : combine ] . = 4.2 in = 4.2 in = 4.2 in = 4.2 in we are given the data in the form of a measured mean @xmath52 of the anisotropy in the @xmath53th pixel , along with the variance about the mean @xmath54 for the measurements . in general , there are pixel pixel correlations in the noise , defining a correlation matrix @xmath55 with off - diagonal components as well as the diagonal @xmath56 . also there is usually more than one frequency channel , with the generalized pixels having frequency as well as spatial designations . the theoretical signal also has a correlation matrix , @xmath57 , which is a linear combination of a product of the @xmath6 times a `` window function matrix '' @xmath58 encoding the possibly frequency - dependent beam , the chopping strategy , sky coverage , for the experiment : @xmath59 $ ] ( see @xcite ) . the `` window function '' usually reported for an experiment is @xmath60 . the likelihood function is @xmath61 here @xmath62 denotes transpose . the noise correlation matrix @xmath63 consists of the pixel errors @xmath64 and the correlation of any unwanted residuals @xmath65 , such as galactic or extragalactic foregrounds . one can think of @xmath65 as increasing the noise for selected correlation patterns in the medium . with a large enough noise in these patterns , they are effectively projected out from the data . constraints such as averages , gradients ( dipoles , quadrupoles ) and known spatial templates , which may be frequency dependent ( iras or dirbe combined with appropriate extrapolations ) can also be modelled in the total @xmath66 , as `` nuisance variables '' to be integrated ( marginalized ) over . denoting each constraint @xmath67 on pixel @xmath53 by @xmath68 , where the template for constraint @xmath67 is @xmath69 and the amplitude is @xmath70 , we need only replace @xmath52 in eq . ( [ eq : likenc ] ) by @xmath71 , then integrate over the amplitudes @xmath70 , assuming some prior probability distribution . this is most easily done if we assume the @xmath70 are distributed as very broad gaussians , reflecting our ignorance of their values ( or , if we know their likely range , incorporating that as prior information in the gaussian spreads ) . the integration over @xmath70 then yields eq.([eq : likenc ] ) with the residual noise matrix given by @xmath72 , where @xmath73 is the assumed prior variance for the constraint amplitudes . as the eigenvalues of @xmath74 become very large , the effect of the constraint matrix is to project onto the data subspace orthogonal to that spanned by @xmath69 . although one can directly use the likelihood equation in this projection limit ( using @xmath75 for the constraint prior ) , it is computationally simpler to use the gaussian prior . ( taking into account constraints with amplitudes that are not linear multipliers times the template is much more complex . ) a suitable @xmath65 can also allow us to focus attention only on a specified band in @xmath76-space for power spectrum estimation . in practice , we do not compute the quantities @xmath77 and @xmath78 $ ] directly ; instead we go to a basis ( linear combination of the data ) in which @xmath79 and @xmath80 are diagonal . first , we whiten the noise matrix using the nonorthogonal transformation provided by its `` hermitian square root , '' @xmath81 ; we apply the same transformation to @xmath79 and diagonalize this in turn with the appropriate matrix of eigenvectors , @xmath82 : @xmath83 , which has units of @xmath84 . we then transform the data into the same basis , @xmath85 , now in units of @xmath86 . the transformed theory matrix @xmath87 still depends on the theoretical amplitude ( @xmath12 , etc . ) as a simple multiplier , which enables the likelihood to be easily calculated as a function of this parameter . in the new basis , the noise and signal have diagonal correlations and @xmath88 , so @xmath89 is useful as a theory - dependent @xmath90 power spectrum which gives a valuable picture of the data and shows how well the target theory fares ( fig . [ fig : snmode])@xcite . the modes are sorted in order of decreasing @xmath90-eigenvalues , @xmath91 , so low @xmath92-modes probe the theory in question best . this expansion is a complete ( unfiltered ) representation of the map . the optimal method for data compression is to use sharp signal - to - noise filtering , keeping only those high @xmath90 modes with @xmath93 and deleting low @xmath90 ones . we also find it extremely useful to look carefully at the power in the low @xmath90 modes to determine whether further residuals need to be added to the generalized noise : a poor model for the noise can give false indications of what the data is saying and misrepresent the signal . filtering using @xmath90-modes has a long history in signal processing where it is called the karhunen - loeve method @xcite , and it is now being widely adopted for analysis of astronomical databases . for an all - sky experiment with uniform , uncorrelated pixel variances , the eigenmodes are the spherical harmonics , and the eigenvalues the expected coefficients @xmath8 . for a more complicated experiment , the high-@xmath90 modes probe the peak of the experiment s window function in @xmath76-space . low-@xmath90 modes are more complex . for experiments with more than one frequency channel , differences between channels should show no cmb signal , and so the eigenvalue should be @xmath94 . nearby pixels , oversampling the beam , should also show very little signal the smooth fall from high @xmath90 to low traces the beam in much the same way that the window function falls as a gaussian @xmath95 $ ] at high @xmath76 . we expect these low @xmath90 modes to be largely independent of the theory used to calculate the appropriate @xmath79 , which enables these modes to be used as a diagnostic of both the analysis procedure and the experiments themselves . we now discuss the anisotropy experiments we use . the six cobe / dmr four - year maps @xcite are first compressed into a ( a+b)(31 + 53 + 90 ghz ) weighted - sum map , with the customized galactic cut advocated by the dmr team , basically at @xmath96 but with extra pixels removed in which contaminating galactic emission is known to be high , and with the dipole and monopole removed . galactic coordinate pixels are used ; slight differences arise with ecliptic coordinate pixels . although one can do full bayesian analysis with the map s @xmath97 pixels , this `` resolution 6 '' pixelization of the quadrilateralized sphere is oversampled relative to the cobe beam size , and there is no effective loss of information if we do further data compression by using `` resolution 5 '' pixels , @xmath98 @xcite . the weighted sum of channels is an exact use of our optimal signal - to - noise compression . the resolution degradation is not optimal but is nearly so ( @xmath99 is nearly constant for separations @xmath100 below the scale of the beam , so adjacent pixel differences have tiny signal but the usual data - noise ) . the galactic cut is also not optimal , but could be made so by using explicit templates for galactic foregrounds to include in @xmath80 , as described above . the combined effect reduces the pixel number from @xmath101 to @xmath102 . further compression by a factor of two or so is possible without much information loss @xcite . a strong indication of the robustness of the _ dmr _ data set is the insensitivity of the band - powers to the degree of signal - to - noise filtering and to which frequencies are probed . for @xmath65 , we include templates for the monopole , dipole and quadrupole , the latter allowing for a galactic foreground contaminant , which we know is there at low @xmath76 in the 31 ghz channel . the dmr data probes @xmath103 well , with useful information out to @xmath104 . the _ sp94 _ experiment @xcite is similar to a classic single - differencing chopping experiment , except that differencing is associated with the oscillation of the beam about the pixel position . it probes @xmath105 . the number of frequency channels and spatial pixels is sufficiently small ( 301 ) that no compression is needed : all 7 frequencies in the ka and q bands at @xmath106 and @xmath107 ghz are simultaneously analyzed . there are 14 constraints , average and gradient removals for each frequency . taking differences in @xmath108 in frequency at the same spatial position is insensitive to the primary signal but has the usual pixel noise for each channel , so @xmath90 filtering would tend to remove those modes and strong compression would result . because the beams do vary somewhat with frequency , however , the compression would remove some information , unlike for cobe . the _ sk95 _ experiment @xcite probes a much larger band in @xmath76-space , from @xmath109 to @xmath110 . even before the data was delivered to us a significant amount of frequency and spatial compression already took place . in this paper , for parameter estimation , we use the `` cap '' data ( 2016 pixels , including rebinned data from _ sk94 _ , with 48 constraints associated with average removals ) . the sk experiment measured the temperature directly by making slightly - curved radial scans from the north celestial pole about 8 degrees in length , which covered the cap as the earth rotated . the data was binned in ra , but , instead of binning in declination , it was projected in software onto what are in effect 3 to 19 beam `` chopping '' configurations . adding the ring data to the cap , involving sweeps in a ring around the ncp at @xmath111 brings the total to 2400 pixels , with 52 constraints . in @xcite , we show that the `` cap '' and `` cap+ring '' parameter estimates agree to much better than `` one sigma '' even though the ring adds substantially more data . one potential concern is that only one hemt band is represented in the data . we have also extensively analyzed the sk94 data set on its own @xcite , with 1344 pixels and 28 constraints , which has only 3 to 9 beam template projections and substantially fewer hours of integration than the 94 + 95 data , but the advantage of having both ka and q band information so the frequency spectrum can be checked . we agree with @xcite that the spectrum of the sk94 3 to 9 templates is consistent with a cmb origin , and inconsistent with likely galactic foregrounds . ( we come to the same conclusion for the _ sp94 _ data , in agreement with @xcite . ) sk95 had data only from the q - band . in analyzing sp94 and sk94 - 95 , it is essential to include errors in the overall calibration of @xmath112 . for sk94 - 95 it is estimated to be a gaussian with standard deviation @xmath113 ; for sp94 , @xmath114 . let @xmath115 denote the calculated likelihood assuming no such errors ; then the likelihood with the uncertainty included is @xmath116{\cal l}_0(\sigma_8^\prime ) $ ] . it is unfortunate that after all of the effort that has gone into these superb experiments , an astronomical issue like the brightness of cas a ( for sk95 ) results in a substantially poorer constraint on @xmath12 than one obtains assuming no such calibration uncertainty . as we mentioned in [ sec : intro ] , we have chosen to order our path through parameter - space using the cosmological age of the models , @xmath18 . to examine the phenomenology of the experiments we shall use a one - parameter sequence of @xmath15 shapes , with the overall bandpower of the experiment ( or the related @xmath12 ) as another parameter . while it was usual in the past to use a power law in @xmath117 , @xmath118 @xcite , it is evident from fig . [ fig : cl13gyr ] that this would be a very bad fit to the _ sk95 _ data , although it is a reasonable representation over the limited @xmath76 range for both the _ dmr4 _ and _ sp94 _ data . the sequence we use is the first panel in fig . [ fig : cl13gyr ] , the tilted cdm sequence for the standard cdm model , @xmath119 , with @xmath13 variable . we use the gw case , @xmath120 if @xmath121 , @xmath122 otherwise . these models have an age of @xmath123 . [ fig : contour ] shows @xmath124 sigma contours of @xmath125 , with @xmath126-sigma defined by @xmath127 $ ] . it is clear from the right hand panel that fixing @xmath49 and @xmath128 , but varying @xmath19 , and therefore @xmath32 , is not a good sequence to use for phenomenology since there is very little difference in the @xmath15 s as @xmath19 varies . the fig . [ fig : contourh ] @xmath129 contour map shows that indeed the data does not determine the hubble parameter very well . = 4.5 in = 4.5 in with the most recent experiments ( @xmath130 ) , the computer power required to calculate the likelihood over a sufficiently wide model space is becoming prohibitive . the @xmath90-eigenmodes also provide a form of data compression which can drastically reduce the required analysis time . by rotating to a basis in which some `` canonical theory '' with correlation matrix @xmath131 is diagonalized by the matrix @xmath132 , but only retaining some fraction of the modes , we efficiently remove parts of the data dominated by noise ( modes with very low @xmath133 ) , but retain the gaussian character of the likelihood for the remaining modes . for other theories , the transformed theory matrix @xmath134 will no longer be diagonal , so the full matrix calculation must still be performed , but now on the smaller space of observations restricted to the modes with the highest `` canonical '' @xmath90 . moreover , for these theories , the @xmath90-modes will be somewhat different , so the compression will not be as efficient ( we will have thrown out a bit `` more signal and less noise '' ) . still , we have achieved compression as good as 90% for experiments ( like sk94 ) with two channels and 65% for the full sk94 - 95 dataset , which has already been re - binned to remove some of the redundancy in beam oversampling and channel - to - channel differences . because the cost of the matrix calculations involved scales as @xmath135 , these result in significant speedups : from @xmath136 hour to @xmath137 minutes per point in parameter space for _ sk95 _ ( which is actually significantly worse than the expected @xmath138 speedup due to overhead ) . in fig . [ fig : compress ] , we show @xmath124 sigma likelihood contours for the sk94 - 95 cap dataset , as in fig . [ fig : contour ] . the lower left panel superposes the contours of the @xmath139 case upon those with all 2016 modes included . the similarity of the contours shows that both the amplitude and the index determinations are not compromised by @xmath90 cuts . for the @xmath140 case , contours @xmath141 are very similar as well . thus we can achieve significant degrees of compression without loss of information . in the following , we apply no data compression to the analysis of the dmr and sp data , but for sk95 we present results using the top 700 modes from the canonical standard cdm theory , the @xmath49 model in the @xmath128 gyr sequence . the reason the compression works can be understood by examining the @xmath90 power spectra , shown in fig . [ fig : snmode ] . the curve is the theoretical spectrum @xmath142 given by the eigenmodes , for a `` standard cdm '' model with amplitude @xmath143 , the value suggested by cobe . the points are the observations @xmath89 , with the same binning as the theory curve . ( the bins require a certain signal - to - noise when summed , but a minimum number are required to define a bin so that the error bars are not too large . ) the error bars contain both variances associated with the pixel noise and with the theoretical cosmic variance ( noise - noise , noise - signal and signal - signal terms ) . to be a good fit to the data , the error bars should pass through the theory curve . after the top few hundred modes , the eigenvalues have @xmath133 so we do not expect them to contribute significantly to the likelihood . we emphasize that it is legitimate to use any mode subset : the relative likelihoods we obtain will tell us which theory is preferred for those modes . it is just that we do not want to build any prior prejudice for a theory by compressing the data in a way which may be biased in its favour over the other theories we are testing . thus we choose to go far into the @xmath90 tail , retaining 700 modes . the @xmath90 mode formalism also can be used to design experiments to discriminate particular theories ( knox , these proceedings ) . = 4.5 in = -0.4 in = 4.5 in = -0.4 in combining experiments to get a total likelihood is straightforward . if the pixels are uncorrelated , either because they overlap little on the sky or in @xmath76 , we only need to multiply the individual likelihoods together . this is the case for cobe / dmr , sp94 , sk94 - 95 . if there is significant overlap , then the experiments should ideally be combined and considered to be one larger experiment , with @xmath79 connecting the pixels in one experiment with the pixels in the other , although the cross - pixel @xmath64 will be zero . we have applied this to the sk95+msam dataset , in joint work with charbonneau and knox , but will not describe it here . the upper right panel of fig . [ fig : contour ] shows the likelihood for the @xmath47 , gw , 13 gyr , tilted sequence . each experiment individually constrains the amplitude , @xmath12 , better than the shape , @xmath13 : the window functions cover a narrow range of @xmath76 . note that the sk experiment does better than dmr at determining the slope . the calibration uncertainty for sk95 is the reason that @xmath12 is not more tightly constrained . [ fig : contour ] shows the advantage offered by combining the results of different experiments : the long baseline in @xmath76 helps considerably in localizing the @xmath13 contours . in fig . [ fig : contourh ] , we see that the dmr data does not restrict the value of @xmath19 , and thus not of @xmath32 , whereas the sk data does , yet the combined data focusses the @xmath12 determination , but not the @xmath19 determination . the sk95 data prefers more power than is predicted by the dmr data for `` standard '' models . thus , the @xmath49 , @xmath47 , @xmath144 model has @xmath145 for dmr , while for sk95 it is @xmath146 . increasing @xmath147 to 0.2 brings them closer into line , @xmath148 for dmr , while for sk95 it is @xmath149 ; and this model is much preferred statistically to the 0.0125 one . we could repeat the contour maps for the 11 and 15 gyr vacuum sequences , for the fixed @xmath18 open models , and for the variable @xmath150 sequence , but it is more concise to quote single numbers , our estimates of the individual cosmological parameters . to that end , we marginalize over the other parameters in the sequence , assuming a prior probability for the parameters . if it is uniform we get the results in the left columns of table [ tab : results ] . the idea that first motivated this project was that the sk94 and sp94 data looked sufficiently robust to return to the multiresolution approach combining experiments to get best possible constraints , @xcite , and this would significantly improve the cobe - only errors on @xmath13 . the sk94+sp94+dmr4 column is the culmination of that effort . however , the sk95 data took us to significantly higher @xmath76 and the promise of greater discrimination among models based on how they rise to the doppler peak . notice the rather large shift in @xmath13 when we pass from the 1300 pixel 2-channel sk94 data , which had chopping templates from 3 to 9 , to the sk95 set , which had 10 - 19 projections as well as much more 3 - 9 data , but only for the q - band . a worry is that non - cosmic signals at high @xmath76 might be contaminating the 10 - 19 template projections . c|c|ccc|cc & * dmr4 * & * sk94+sp94*&*sk94 - 95 * & * lss*+dmr4 & * lss*+dmr4 + & & + dmr4&+sp94+dmr4 & & + sk94 - 95+sp94 + + @xmath151=50 & @xmath13 & @xmath152 & @xmath153 & @xmath154 & @xmath155 & @xmath156 + @xmath151=70 & @xmath13 & @xmath157 & @xmath158 & @xmath159 & @xmath160 & @xmath160 + all @xmath151 & @xmath13 & & & @xmath161 & @xmath162 & @xmath163 + @xmath13=1 & @xmath151 & @xmath164 & @xmath165 & @xmath165 & @xmath166 & @xmath167 ( @xmath168 ) + all @xmath13 & @xmath151 & & & @xmath169 & @xmath170 & @xmath171 ( @xmath172 ) + + @xmath151=43 & @xmath13 & @xmath173 & @xmath174 & @xmath175 & @xmath176 & @xmath177 + @xmath151=70 & @xmath13 & @xmath178 & @xmath179 & @xmath180 & @xmath181 & @xmath182 + all @xmath151 & @xmath13 & & & @xmath183 & @xmath184 & @xmath185 + @xmath13=1 & @xmath151 & @xmath186 & @xmath187 & @xmath188 & @xmath189 & @xmath190 ( @xmath191 ) + all @xmath13 & @xmath151 & & & @xmath192 & @xmath193 & @xmath194 ( @xmath195 ) + c|ccccc + & @xmath151 & @xmath13 & @xmath32 & @xmath12 & + 13 gyr @xmath196 & 50 & @xmath197 & 0 & @xmath198 ( 1.24 ) & ( @xmath199 ) + 13 gyr @xmath196 & 50 & @xmath200 & 0 & @xmath201 ( 0.62 ) & ( @xmath202 ) + 15 gyr @xmath203 & 43 & @xmath204 & 0 & @xmath205 ( 0.76 ) & ( @xmath206 ) + 13 gyr @xmath203 & 50 & @xmath197 & 0 & @xmath207 ( 0.90 ) & ( @xmath208 ) + 13 gyr @xmath203 & 60 & @xmath200 & 0.43 & @xmath209 & ( @xmath210 ) + 11 gyr @xmath203 & 59 & @xmath211 & 0 & @xmath212 ( 1.54 ) & ( @xmath213 ) + 13 gyr open & 55 & @xmath204 & @xmath34=0.60 & 1.0 & ( @xmath214 ) + + 15 gyr @xmath203 & 55 & @xmath215 & 0.52 & @xmath216 & ( best fit ) + 13 gyr @xmath203 & 65 & @xmath215 & 0.56 & @xmath217 & ( @xmath218 down ) + 11 gyr @xmath203 & 85 & @xmath204 & 0.69 & @xmath219 & ( @xmath220 down ) + we now wish to add some large scale structure constraints , by constructing prior probabilities that roughly correspond to the restrictions arising from observations of galaxy clustering and cluster abundances . the linear power spectrum for density fluctuations is often characterized by the shape parameter , @xmath221}$ ] , which is 0.48 for the standard cdm model . here @xmath17 . assuming a linear bias model for how the galaxy distribution is amplified over the mass distribution , the clustering data implies @xmath222 . the abundance as a function of x - ray temperature also heavily constrains @xmath12 . values from 0.5 up to 0.7 are obtained for @xmath223 cdm - like models . for @xmath224 , the value is higher , scaling roughly as @xmath225 . there are also many estimates of the combination @xmath226 that are obtained by relating the galaxy flow field to the galaxy density field inferred from redshift surveys , which all take the form @xmath227 \ , \beta_g$ ] , where @xmath228 is the galaxy biasing factor and @xmath229 is a number whose value depends upon data set and analysis procedure : @xcite give @xmath230 for an average of a number of estimates in the literature , and @xmath231 for a determination using a maximum likelihood technique for the iras survey and the mark iii velocity field data set , while a higher ( @xmath232 ) number is obtained using potent on this data set . it is usual to take @xmath233 for galaxies , which gives a @xmath12 consistent with the cluster value , but @xmath228 can depend upon the galaxy types being probed , upon scale , and could be bigger or smaller than @xmath234 . we want to choose priors for @xmath12 , @xmath32 and @xmath235 that reflect these lss ranges , but we certainly do nt want to be too miserly in our choice of allowed ranges . a straight gaussian tends to be overly supportive of the mean , while a tophat error has no probability in the wings . using priors which convolve a gaussian with a tophat and have different upper and lower errors give us the flexibility we require . it is similar to specifying both a statistical and a systematic error . for the exercise shown in the tables , we required that @xmath236 be @xmath237 and @xmath238 be @xmath239 . the latter has a high probability at 0.55 , but little at 0.50 ( although some authors actually prefer this value ) . sample lss+cmb numbers are given in table [ tab : results ] . the tiny error bars when lss constraints are added to the cmb data are amusing , but are far from definitive at this stage . the reason the errors are small is typically that the cmb data pushes for a likelihood peaked at high @xmath240 , and this multiplies the lss likelihood peaking at 0.6 or so . the product of the two has a narrow peak but also a small likelihood . this asymmetry is not as pronounced for the hot / cold hybrid models . table [ tab : bestfits ] gives the parameters for the best fits to the data for the various cases . the associated @xmath15 s are shown in fig . [ fig : clbestfit ] . note that the models which best fit the cmb data for a given age often have positive tilts . while positive scalar tilt is possible in inflation models , it requires special constructs in the inflaton potential in a region corresponding to just where we can observe it with the cmb . more likely are negative tilts . if we restrict our attention to these ( second row ) , then the best fit for the @xmath150 sequence ( 13 gyr , @xmath47 ) is @xmath241 and @xmath242 , high not low . in the lower lss part of table [ tab : bestfits ] , the 13 gyr best fit is one sigma down from the 15 gyr best fit , and the 11 gyr is two sigma down . the analysis shows that @xmath13 lies close to the value predicted by inflation . the @xmath151 limits are suggestive , but better cmb data is needed to strengthen the constraint to usable values . of course when the lss data are included , @xmath243 is suggested for @xmath49 cdm models , though it is not needed for hot / cold hybrids with @xmath244 . table [ tab : results ] shows adding sk95 and sp94 to lss and dmr4 does not add much further discrimination , but this should change dramatically in the next few years , with the advent of long duration balloon experiments , interferometers , map and cobras / samba . bond , j.r . 1996 , _ theory and observations of the cosmic background radiation _ , in `` cosmology and large scale structure '' , les houches session lx , august 1993 , ed . r. schaeffer , elsevier science press .

0.4 cm we describe the bayesian - based signal - to - noise eigenmode method for cosmological parameter estimation , show how it can be used to optimally compress large cmb anisotropy data sets to manageable sizes , and apply it to the dmr 4-year , south pole and saskatchewan data , individually and in combination . a simple prior probability method is used to include large scale structure observations . estimates of the hubble parameter , vacuum energy density , baryon fraction and primordial spectral tilt derived from the combined data are given . epsf # 1#23.6pt # 1@xmath0 ^ # 1#1 # 1#1 # 1#1 # 1#1 c 3mp@xmath1 m # 1to 0pt#1 @xmath2 canadian institute for theoretical astrophysics , toronto , ontario , canada . @xmath3 center for particle astrophysics , uc berkeley , berkeley ca usa .

theoretical studies of phase - segregated bose - einstein condensate ( bec ) mixtures and the experimental realization of such systems have opened up a new avenue for investigating many interesting properties of inhomogeneous becs , in particular the superfluid dynamics of the interface . on the one hand , one has focused on the exploration of hydrodynamic interface instabilities , such as the kelvin - helmholtz instability @xcite , the rayleigh - taylor instability @xcite and the richtmayer - meshkov instability @xcite . on the other hand , one has studied several characteristics specific to interfaces , such as the interfacial tension @xcite and capillary wave excitations , starting by solving approximately the gross - pitaevskii ( gp ) equations in two distinct regimes of weak and strong segregation @xcite . dynamical properties of becs are often studied numerically by solving the time - dependent gp equations with the pseudospectral method or via diagonization of the bdg equations . dispersion relations can to some extent be determined analytically through a variational method applied to the gp action or by solving the gp equations or bdg equations in asymptotic regimes such as the long wavelength limit . however , the analytical studies have either been limited to special cases or have ignored the structure of the interface on length scales smaller than or equal to the healing length . our aim in this paper is to provide a detailed analytical solution of a model which possesses the essential physical ingredients of gp theory , but is different from it and much easier to solve . in ref.@xcite the model defined through the double - parabola approximation ( dpa ) was shown to be a simple but useful approach for treating _ static _ interfacial problems in phase - segregated becs . the dpa has even played a constructive role in conjecturing new exact solutions to the gp equations @xcite . the present paper proposes the _ dynamical _ counterpart of this model . here , the dpa in combination with the bogoliubov - de gennes ( bdg ) equations allow us to derive analytical expressions for ripplon excitations and their dispersion relation . these excitations are nambu - goldstone ( ng ) modes associated with the broken symmetries in this inhomogeneous system . surface phonons ( at an interface ) can be studied in a similar fashion but are not addressed in this paper . a ripplon is a capillary wave traveling along the interface , causing it to undulate perpendicular to its planar projection , and is recognized as the primary lowest - energy excitation within the bogoliubov spectrum . in a uniform external potential the dispersion relation of ripplons , in the long - wavelength limit , @xmath9 , where @xmath10 is the angular frequency and @xmath11 the wave number of the capillary wave , can be obtained by applying a linear approximation to the gp theory leading to young - laplace - like equations . this result coincidences with the dispersion relation of a classical fluid . the secondary low - energy excitations are phonons . while bulk phonons correspond to the propagation of the pressure waves within a condensate in bulk , localized surface phonons with dispersion @xmath12 are predicted at the interface , propagating at a speed lower than the speed of sound . drawing an analogy to mechanical waves , ripplons have an important transverse character , while surface phonons are mainly longitudinal in nature . let us first recall the familiar frame - work . we assume a uniform external potential ( which without loss of generality can be set to zero ) and start from the gp lagrangian density @xcite @xmath13 with hamiltonian density @xmath14 + g_{12 } \lvert\psi_1\rvert^2 \lvert\psi_2\rvert^2.\ ] ] for atomic species @xmath15 , @xmath16 is the wave function of the condensate or order parameter " , @xmath17 the atomic mass , @xmath18 the repulsive intra - species interaction strength , @xmath19 the repulsive inter - species interaction strength and @xmath20 the intra - species ( for @xmath21 ) or inter - species ( for @xmath22 and @xmath23 ) @xmath24-wave scattering length . by introducing the dimensionless quantities , @xmath25 , with @xmath26 the healing length and @xmath27 the number density of condensate @xmath15 in bulk , @xmath28 , @xmath29 , and @xmath30 , with @xmath31 , and @xmath32 the chemical potential of condensate @xmath15 , we scale the lagrangian density in and hamiltonian density in to @xmath33 with @xmath34 + k \lvert\psi_1\rvert^2 \lvert\psi_2\rvert^2,\ ] ] where the bulk pressure @xmath35 is given by @xmath36 , which is independent of the label @xmath15 provided the mixture is at bulk two - phase coexistence , which indeed we must assume in order to ensure the stability of an interface in bulk . next we make a transformation of the dimensionless lagrangian density by writing @xmath37 we then have a lagrangian density in terms of the new order parameters @xmath38 , @xmath39 - \hat{\mathcal{v } } ( \phi_1,\phi_2),\ ] ] in which the potential @xmath40 takes the form @xmath41 + k \lvert\phi_1\rvert^2 \lvert\phi_2\rvert^2.\ ] ] recall that for @xmath42 the two components are immiscible and a phase - segregated bec forms @xcite . we assume that the planar interface ( without excitation ) is located at @xmath43 and the condensates 1 and 2 reside in half - spaces @xmath44 and @xmath45 , respectively . we call the bec mixture _ symmetric _ when the atomic masses are equal , @xmath46 ( a frequently fulfilled condition ) , _ and _ the intra - species interaction strengths are equal @xmath47 ( a special requirement ) . furthermore , for a symmetric mixture at bulk two - phase coexistence also the bulk densities are equal , @xmath48 , and so are the chemical potentials . consequently , @xmath49 for a symmetric mixture . we recall the dimensionless mean - field dpa gp lagrangian density @xcite @xmath50 - \hat{\mathcal{v}}_{\mathrm{dpa}}(\phi_1,\phi_2),\ ] ] where @xmath51 takes the form @xmath52 where @xmath53 . in the labels @xmath15 and @xmath54 comply with the following important convention , which we will henceforth maintain _ throughout this paper _ : @xmath55 this potential is the combination of two paraboloids obtained by the expansions of the potential in gp theory about bulk condensate 1 , @xmath56 , for the half - space @xmath57 , and bulk condensate 2 , @xmath58 , for the half - space @xmath59 . in bulk , time derivatives and spatial gradients in the lagrangian density vanish and the minimum of the potential , @xmath60 , is the same as in the gp theory , see @xcite . by extremizing the action with the lagrangian density with respect to a variation @xmath61 , we obtain the dpa to the time - dependent gp equations , @xmath62 where @xmath63 . these equations are decoupled , which greatly simplifies the problem . although most terms are linear in the order parameters , there is an important nonlinearity through the nontrivial dependence on the phase of the order parameter . the equations can also be obtained directly from the ( exact ) gp equations by expanding the amplitudes to first order in the perturbations about their bulk values . to see this , we make use of the original expansions that define the dpa @xcite @xmath64 we observe that keeping contributions of zeroth and first order in the @xmath65 s and @xmath66 s reproduces identically . alternatively , one can derive these equations hydrodynamically " by combining a bernoulli - type of equation and the continuity equation , obtained by taking the variation of the amplitudes and the phases of the order parameters , respectively @xcite . for a planar interface at @xmath67 , the static interface profiles are real solutions of the form of time - independent stationary states @xmath68 , with @xmath69 , while the ( unperturbed ) system is translationally invariant along @xmath70 and @xmath71 directions . by matching the functions and their first derivatives at @xmath67 and using the boundary conditions @xmath72 and @xmath73 , we obtain @xmath74 note that the second derivative of these profiles is discontinuous at the interface . after inserting the real and time - independent solutions in , we multiply the equation for condensate @xmath15 by @xmath75 and the equation for condensate @xmath54 by @xmath76 , then add them up and integrate the expression over @xmath77 . the result is a constant and represents the constant of motion in dpa , i.e. , @xmath78 the value of the constant must be fixed so that the minimum of @xmath51 , given in , is respected in each of the ( homogeneous ) bulk phases . note that the first term in the lhs of this equation is the sum of gradients squared of the two stationary solutions and is usually called the gradient energy . considering low - energy interfacial excitations such as ripplons and surface phonons , or even more general excitations , we perturb the complex order parameters about their ( real ) stationary state forms @xmath79 in which @xmath80 is assumed to be a small perturbation of the value of @xmath81 . this implies that the phase angle @xmath82 of @xmath38 is small ( i.e. , of the order of im@xmath83 ) in regions where @xmath81 is close to its bulk value ( which is unity ) . in gp theory , by taking this perturbation into the gp equations , we can derive the bdg equations to first order in @xmath84 . alternatively , the bdg equations can be derived by including the variation @xmath85 directly in the gp action , using . for a brief recapitulation of these fundamental equations , see appendix [ sec : gptheory ] . within the dpa , however , a considerable simplification of these bdg equations can be obtained , while keeping as we shall show the essential physics of our problem intact . we derive the dpa to the bdg equations directly as follows . firstly , we insert the perturbed order parameter into the time - dependent dpa to the gp equations . this leads to @xmath86 next , this can be rewritten in terms of the perturbations alone if we make the additional assumption that the phase angle @xmath82 is small . this is valid provided @xmath87 . since condensate @xmath15 is close to its bulk density in the half - space concerned , this assumption is appropriate within the dpa strategy . then , to first order in the @xmath84 , @xmath88 with the definition @xmath89 and @xmath90 is considered small ( compared to 1 ) in the usual dpa spirit . this is a good approximation away from the interface , whereas close to the interface the upper bound is @xmath91 , as can be seen from . consequently , can be written in the more canonical form , @xmath92 which _ defines _ our dpa model at the level of the bdg equations . we will see shortly that the correction term @xmath93 in these equations is crucially important to obtain the correct physics ( including the correct zero modes " ) within the dpa . for completeness we also give the explicit expressions for the modulus and the phase angle of the perturbed wave function to the lowest relevant orders in the perturbation and in the deviations from bulk values , @xmath94 this shows that @xmath95 is of first order in @xmath82 , but of zeroth order in the @xmath65 , whereas @xmath96 is of first order in the @xmath65 ( or smaller ) . we conclude that terms of order @xmath97 can be neglected , whereas terms of order @xmath98 ought to be kept . based on these insights we make the observation that the dpa procedure commutes with the derivation of the bdg equations . indeed , starting from the exact bdg equations ( see appendix [ sec : gptheory ] ) and applying and also leads to as we will now show . the exact bdg equations , derived from gp theory , read @xmath99 \delta\phi_j + \phi_{j0}^2 \delta\phi_j^{\ast } + k \phi_{j0}\phi_{j'0 } ( \delta\phi_{j ' } + \delta\phi_{j'}^{\ast } ) , \ ; j=1,2 \ ; ( j \ne j')\ ] ] if we consider the half - space in which condensate @xmath15 is close to its bulk state , we can apply and . applying the dpa we can then safely neglect the second - order quantities @xmath100 and @xmath101 . this already reduces the equation to @xmath102 \delta\phi_j + \phi_{j0}^2 \delta\phi_j^{\ast } , \ ; \mbox{with } \ ; \phi_{j0 } = 1 + \epsilon_{j0}\ ] ] next we apply and expand , to obtain @xmath103 \delta\phi_j + ( 1 + 2\epsilon_{j0 } ) \delta\phi_j^{\ast}\ ] ] subsequently we regroup terms and neglect @xmath97 but keep @xmath98 as we have explained . this leads to @xmath104 \delta\phi_j + \delta\phi_j^{\ast}+\epsilon_{j0 } ( \delta\phi_j - \delta\phi_j^{\ast}),\ ] ] which is identical to the first equation in . conversely , considering the half space in which condensate @xmath54 is close to its bulk state , prompts us to neglect the second - order quantities @xmath105 and @xmath106 , so that only the following terms remain , @xmath107 \delta\phi_j\ ] ] after writing @xmath108 and keeping only the leading order , since @xmath84 is already of first order in the dpa , this equation reproduces the second equation in , if we take into account the labeling convention . this completes the alternative derivation of . following the bogoliubov analysis , we now write a general @xmath84 as a linear superposition of different fourier - like modes @xmath109,\ ] ] where the reduced wave vector @xmath110 and reduced frequency @xmath111 for each mode of the surface wave are defined by @xmath112 and @xmath113 . note that @xmath114 . note also that for a symmetric bec mixture at two - phase coexistence the reduced frequencies @xmath115 are the same for both condensates . for an interfacial excitation ( a ripplon or , also , a localized phonon ) propagating along a direction , say @xmath70 , parallel to the topological defect ( i.e. , the interface midplane ) , the problem simplifies to a two - dimensional one , with @xmath116 , @xmath117 and @xmath118,\ ] ] we define the interface displacement @xmath119 and its dimensionless counterparts @xmath120 through the position along @xmath77 where the order parameters intersect " , @xmath121 note that @xmath122 in the absence of an excitation . the converse need not be true . for example , interface breather modes are excitations for which we can fix @xmath122 for all times . for a ripplon mode , the interface displacement is a transverse wave traveling along @xmath70 . we consider a ripplon mode of fixed wavelength and study its properties . for concreteness , let @xmath123 where @xmath124 is the amplitude of the interface displacement @xmath125 . importantly , we assume the excitation is small in the sense that @xmath126 . this means that the healing lengths are large compared to the interface displacement and we anticipate that interface structure plays an important physical role in this regime . for the order parameters associated with this ripplon the following form is proposed , @xmath127 featuring the rigid - shift displacement @xmath128 and its dimensionless counterparts @xmath129 . these are of the same form as , but the amplitudes may differ , i.e. , @xmath130 further , the real function @xmath131 incorporates amplitude fluctuations beyond the bare rigid shift of the static profile already incorporated in the argument of @xmath132 . the presence of @xmath131 can cause a ( condensate - dependent ) additional shift and also a deformation of the interface and this is the physical reason for a possible difference between the functions @xmath133 and @xmath125 . finally , the real function @xmath82 contains the phase fluctuations ( relative to @xmath134 ) . for the specific interface displacement consistency of the transverse interface displacement velocity with the phase fluctuation of each component , @xmath135 and satisfaction of the hydrodynamic equation of continuity , for each component , @xmath136 imply the following forms to first order in the rigid - shift displacement amplitude @xmath137 , @xmath138 where @xmath139 and @xmath140 are ( real ) functions that are localized about the interface , since a ripplon is a localised ng mode , whose effect decays when we move away from the interface into the bulk . these functions are related to one another through the continuity equation , which , after scaling the variables , takes the form @xmath141 to first order in the amplitude @xmath137 . these yet unknown functions @xmath139 and @xmath140 , which in general also depend on @xmath142 and @xmath115 , contain the interesting physics of the excitation . for @xmath143 the interface is merely rigidly shifted with respect to its unperturbed state . for @xmath144 the interface structure may undergo a change further , the properties and form of @xmath145 are of course crucial for the existence of a superfluid velocity profile . note that far away from the interface ( in the far field " ) , where condensate @xmath15 is present in bulk , we can approximate @xmath146 and the equation of continuity simplifies to @xmath147 we now expand @xmath38 in to first order in @xmath148 and match it with @xmath38 in using the term corresponding to wave number @xmath11 in @xmath84 given in . this leads to the following results @xmath149 where we have included the subscript @xmath11 in the functions @xmath150 and @xmath151 to anticipate their explicit dependence on the wave number . note that the separation of the derivative part and the part containing @xmath152 in the equation for @xmath153 is not unique and depends on the precise form of the proposed order parameter function . in contrast , the expression for @xmath154 is generally valid , since it is merely a tautology featuring the yet unknown function @xmath155 . the continuity conditions , at the interface ( @xmath67 ) , for these functions and their first derivatives are a nontrivial problem in general . we will see in the next subsection how these conditions naturally manifest themselves in the dpa model . we can already anticipate that a discontinuous second derivative in @xmath81 at the interface , typical for dpa profiles , generates a discontinuous first derivative in @xmath156 . in fact , even a discontinuity in @xmath156 at the interface is possible , and this seems to arise naturally in the strong segregation limit , in which the exact gp interface profile ( as well as its dpa counterpart ) displays a discontinuous first derivative . we conclude that continuity conditions on @xmath156 have to be examined carefully case by case . since the structure of the interface is by and large determined by the properties of @xmath81 and its first derivative , it is apt to denote the boundary conditions on @xmath156 by structural " boundary conditions . a different situation arises when it comes to @xmath157 . in view of , the function @xmath155 is entangled with the phase that defines the velocity potential of the superfluid . the continuity , at the interface , of the first derivative of @xmath155 , is a prerequisite for the fluid mechanical continuity of the transverse velocity of the interface displacement . therefore , we may denote the requirement of continuity of the first derivative of @xmath157 at the interface by hydrodynamical " boundary condition . naturally , we require @xmath157 itself to be continuous also ( since a delta - function - like singularity in the transverse velocity would be unacceptable ) . by substituting @xmath158 with the bogoliubov form corresponding to fixed wave number @xmath11 in into the dpa for the bdg equations , we obtain the following dpa for the bdg equations in terms of @xmath156 and @xmath157 , @xmath159 where @xmath160 . to test the usefulness of these equations , we first apply them to obtain the zero modes within dpa . in the limit @xmath161 and @xmath162 the dpa bdg equations reduce to @xmath163 we now show that these equations are solved by the functions that are precisely the dpa applied to the exact zero modes of the bdg equations . to see this , consider the exact zero modes given in and of appendix a. then write , as usual in the dpa formalism , @xmath164 where the expressions for the supposedly small ( compared to unity ) quantities @xmath90 and @xmath165 can be read off from . at the interface @xmath90 and @xmath165 can not both be small , but fortunately they are well bounded in absolute value by @xmath166 and @xmath167 , respectively . now recall that in dpa calculations we neglect contributions of second and higher order in the @xmath65 s and @xmath66 s . we concentrate first on the nontrivial second equation of , and test a solution of the form @xmath81 . we get @xmath168 where we have neglected the term of second order in @xmath90 . the remaining three equations are trivial to test and the solutions are easily identified . altogether , we find that the dpa bdg equations are solved by zero modes of the form @xmath169 and @xmath170 provided the dpa solutions of the static order parameters are used and calculations are carried out to , and including , first - order terms in the deviations @xmath65 and @xmath66 from bulk values . returning now to the non - zero modes , we emphasize that the nontrivial second differential equation of embodies the equation of continuity , to first order in @xmath90 . this can be seen by using the forms and by using also @xmath171 , which is valid to first order in @xmath90 , as follows from for a static solution . the second equation of therefore represents _ the dpa to the continuity equation _ and is valid for all distances , at and away from the interface . to make further progress towards the general solutions of the dpa bdg equations , we recall that the correction term in the second equation of contains the factor @xmath172 , which is itself already a first - order correction in the dpa strategy consequently , here it is sufficient to take into account only the contribution to @xmath157 calculated in the absence of the correction term . we will deal with this systematically in the frame - work of perturbation theory . firstly , let us denote by @xmath173 and @xmath174 the zeroth - order " solutions of the dpa bdg equations in the absence of the term @xmath175 in . these equations are solved by the exponentially localized functions @xmath176 in which the exponents are @xmath177 note that the functions @xmath178 and @xmath179 need no superscript because no approximation is made in their differential equations . we note that the following relations hold among the coefficients ( since exponentials with different decay are , in general , independent ) , @xmath180 there remain 4 undetermined coefficients ( the @xmath181 and the @xmath182 ) for each bec component . it is instructive to give also the zeroth - order solutions in terms of the @xmath183 and @xmath184 , because some of these are just single exponentials , @xmath185 where the @xmath182 are the same as in and @xmath186 now we are prepared to return to the problem of solving the first two dpa bdg equations in considering the term @xmath175 perturbatively . note that this term is small in the weak - segregation limit , where @xmath187 is small , but we are interested in using the approximation for all @xmath187 . therefore we assume @xmath188 to be small and perform a perturbation calculation to first order in the amplitude of this quantity , which we call @xmath189 , i.e. , @xmath190 we therefore expand , @xmath191 our main task now is to find the solutions of the _ inhomogeneous _ system @xmath192 note that we have used the explicit form of @xmath81 in dpa and @xmath193 is given by the second equation of . the solutions of these first - order " equations consist of a particular solution complemented by an arbitrary solution of the _ homogeneous _ system , which is by definition the system but without the last term . summarizing , we can write the general solution to the first two inhomogeneous dpa bdg equations in formally as follows , to first order in @xmath189 , @xmath194 the superscript @xmath195 refers to the particular solution of the system and the superscript @xmath196 refers to an arbitrary solution of the homogeneous system , which coincides formally with an arbitrary solution of the zeroth - order system . after inspection of the system we can write the particular solutions in the form @xmath197 where the ( finite ) coefficients @xmath198 and @xmath199 are found to satisfy the following equations @xmath200 this leads to the following expressions , which are useful for @xmath201 , @xmath202 the arbitrary homogeneous solutions present in the first - order correction " are of the form @xmath203 note that without loss of generality , we can choose some of the coefficients to be equal to zero . in particular , we have the freedom to choose one of the coefficients in @xmath153 or @xmath154 to set the scale of the excitation . if we choose to fix @xmath204 , we do not need to introduce any first - order correction to it and can thus set @xmath205 . this entails also @xmath206 in view of . since , furthermore , @xmath207 is related to @xmath208 through the same constraint , we observe that the presence of the homogeneous solutions generates only one additional unknown , say @xmath208 . we now proceed to the discussion of the full solution of our system of 4 differential equations . even though the two pairs of equations are independent , their solutions get coupled through suitable boundary conditions , which reduce the number of undetermined coefficients . indeed , we must invoke several continuity conditions at the interface . firstly , for @xmath209 ( excluding the case of total segregation ) it is reasonable to impose _ continuity _ of all solutions at the interface , since the zero modes are continuous at the interface , in gp theory as well as in the dpa , and we aim at building a dynamical dpa with no more singularities than those present in the static dpa . this leads to the following relations , @xmath210 while the coefficients @xmath181 and @xmath211 of the zeroth - order " homogeneous solutions satisfy . next , we require continuity of the hydrodynamic " amplitude , i.e. , the first derivative of @xmath157 , at the interface . this gives @xmath212 now we are left with ( only ) one undetermined coefficient per bec component , assuming that we fix @xmath204 . upon inspection of our system of differential equations with these 3 boundary conditions , we conclude that it is not necessary to include an extra homogeneous solution in the first - order perturbation scheme . the boundary conditions can all be satisfied using just the particular solution as the perturbation . so we can simplify the analysis and choose @xmath213 , which implies also @xmath214 . we are now in a position to generate all the modes analytically , within the dpa , for values of the reduced wave numbers @xmath142 and reduced frequencies @xmath115 that are arbitrarily chosen , as long as the exponents @xmath215 and @xmath216 , defined in , are real numbers . clearly , this mathematical set of modes is larger than the physical subset that satisfies the yet undetermined dispersion relation ( the physical constraint expressing the dependence of the frequency on the wave number ) . we will turn to the derivation of the dispersion relation later , but for now , we consider the frequencies and wave numbers as independent parameters and present figures illustrating functions @xmath156 and @xmath157 that satisfy all our approximations so far . the functions are of the form , with 8 coefficients that must satisfy the 7 equations , , , and . there remains one undetermined coefficient that , without loss of generality , can be attributed an arbitrary value , which sets the scale for the mode . fig.1 shows arbitrary examples of @xmath217-modes for the symmetric case @xmath218 , implying @xmath219 . for this case the modes satisfy reflection antisymmetry @xmath220 . for the wave number the choice @xmath221 has been made . furthermore , the chosen reduced frequencies are @xmath222 . the freedom to choose the coefficient @xmath204 has been exploited by fixing it to @xmath223 , with @xmath224 , in accord with the amplitude implied by . four segregation strengths are presented : weak segregation ( @xmath225 ) , intermediate segregation ( @xmath226 and @xmath227 ) and strong segregation ( @xmath228 ) . note that the curve for @xmath227 is nearly , but not fully , symmetric about the interface . the reason for this ( near-)symmetry is that , for @xmath227 , the amplitudes of the static dpa order parameters are equal to @xmath229 at the interface since @xmath230 . -mode for various condensate segregation strengths . calculations are shown for the symmetric case @xmath231 . the curve with the lowest peak corresponds to reduced interaction strength @xmath225 ( weak segregation ) , the two intermediate peaks to @xmath226 and @xmath227 and the highest peak to @xmath228 ( strong segregation ) . note that @xmath217 is continuous , but its first derivative is discontinuous , at the interface , which is located at @xmath232.,scaledwidth=77.0% ] for the same choices of parameters , fig.2 shows examples of @xmath233-modes . for the symmetric case @xmath218 the modes satisfy reflection antisymmetry @xmath234 . figs.3 and 4 show examples of @xmath217-modes and @xmath233-modes , respectively , at constant interaction strength @xmath235 and at constant reduced wave number , but with varying reduced frequency . -mode for various condensate segregation strengths . the curve with the mildest slope at the interface corresponds to reduced interaction strength @xmath225 ( weak segregation ) , the intermediate curves to @xmath226 and @xmath227 and the curve with the steepest slope at the interface to @xmath228 ( strong segregation ) . note that @xmath233 and its first derivative are continuous at the interface , which is located at @xmath232.,scaledwidth=77.0% ] -mode for various reduced excitation frequencies and for @xmath227 . the lower curve corresponds to the lowest frequency @xmath236 , the intermediate curves to @xmath237 and @xmath238 and the upper curve to @xmath239 . note that the curves are nearly , but not fully , symmetric about the interface.,scaledwidth=77.0% ] -mode for various reduced excitation frequencies and for @xmath227 . the lower curve corresponds to the lowest frequency @xmath236 , the intermediate curves to @xmath237 and @xmath238 and the upper curve to @xmath239.,scaledwidth=77.0% ] we now address the long - wavelength limit . we assume that when @xmath11 tends to zero , also @xmath10 tends to zero , anticipating the existence of a dispersion relation . in fact , as is appropriate for ripplons , we assume that @xmath10 tends to zero more rapidly than linearly in @xmath11 , and we eventually attempt to capture the correct exponent from our calculations . to make progress we expand all the auxiliary quantities in @xmath240 and @xmath241 to the lowest relevant orders , suppressing momentarily the subscript @xmath15 , @xmath242 these expressions allow us to check explicitly the zero modes within our dpa strategy . considering the limit @xmath243 and @xmath244 , from we readily obtain @xmath245 with @xmath246 finite ( or zero ) . furthermore , @xmath247 must vanish , in view of and , and the fact that @xmath248 is finite ( or zero ) . so , @xmath249 consequently , using also we obtain @xmath250 furthermore , regardless of the ( finite or zero ) value of @xmath251 , implies @xmath252 however , the determination of @xmath253 is not straightforward and requires a careful examination of the zero - mode limit . using we obtain , using the symbol @xmath254 " for asymptotically equal to " ( whereas @xmath255 " is used for proportional to " ) , @xmath256 which tends to zero , regardless of the ( finite or zero ) value of @xmath246 , provided @xmath257 in the long wavelength limit . the information gathered so far already narrows down considerably the form of the zero modes . we get @xmath258 and @xmath259 since @xmath260 must be proportional to the dpa order parameter and @xmath261 to its derivative , owing to and , respectively , we must ask @xmath262 and , moreover , @xmath263 must vanish at least as fast as @xmath264 , for @xmath244 , in order to avoid a divergence of @xmath246 , in view of . next , implies @xmath265 and leads to @xmath266 the foregoing results allow us to retrieve the correct zero modes within the dpa , @xmath267 note that some of the amplitudes can be zero depending on which kind of zero modes are excited . for a uniform translation along @xmath77 , @xmath268 , and for a uniform phase rotation @xmath269 . the coefficients that vanish in the zero - mode limit will henceforth be regarded as dynamical amplitudes , and the non - vanishing coefficients will be regarded as static amplitudes . we now combine the relations among the coefficients , and , with the expansions - to lowest orders in @xmath240 and @xmath241 and substitute them in . with the help of the auxiliary expansions @xmath270 this leads to @xmath271 we now note that @xmath187 can be expanded in @xmath189 as follows @xmath272 applying this in leads to @xmath273 with @xmath274 since the next - to - leading term in this expression is of order @xmath275 , we obtain a good approximation by keeping only the leading term . this leads to the important result @xmath276 we conclude that also @xmath277 is a dynamical amplitude , since @xmath246 vanishes , provided @xmath257 in the long wavelength limit . in hindsight , this is not surprising because our excitation does not contain a uniform phase rotation in the zero - mode limit , but only a translational mode with non - zero ( fixed ) amplitude @xmath124 and diverging wavelength . in sum , the zero modes for our specific ripplon problem reduce to the simple forms @xmath278 where we have exploited the freedom to fix the static amplitude @xmath248 with the help of the general form for the case @xmath279 , i.e. , in the absence of structure fluctuations . the relations among coefficients derived so far allow one to express all the coefficients in terms of @xmath204 , whose static limit @xmath248 is known . we thus obtain to first order in @xmath189 , using @xmath280 , the following explicit forms for the long - wavelength limit of all the relevant functions contained in , @xmath281 where the overall proportionality factor is an unimportant unknown function of @xmath115 and @xmath282 which must approach unity in the long wavelength limit . next , limiting ourselves to the leading terms , we obtain the _ asymptotic forms _ , for @xmath283 , in a form that allows one to identify the dpa versions of the functions @xmath151 and @xmath150 , defined through , that pertain to the phase fluctuations and shape fluctuations , respectively , of the dynamically perturbed interface . we obtain , after summation according to , and recalling @xmath284 , @xmath285 note that the first term in the first right - hand - side in @xmath157 is the dominant one in the bulk region far from the interface . this term expresses the slow decay ( decay length @xmath286 wavelength ) of the @xmath287 modes . this is exactly what one should expect in view of a qualitative analysis ( not performed here ) of the ripplon excitation in gp bdg theory ( beyond dpa ) in the distant field , far away from the interface . note also the important last term in the right - hand - side in @xmath179 . it gives rise to a position dependence of the function @xmath288 , essential for obtaining a continuous superfluid velocity across the interface . from these expressions we can extract the leading term(s ) of the functions @xmath152 and @xmath155 within the dpa , using the correspondence . they take the following form @xmath289 note that the dpa has allowed us to obtain explicit expressions for these functions at all distances @xmath77 . we now consider the asymptotic solutions to be the physical dpa modes that we wish to study further . in particular , we consider the dpa ripplons to be perturbations based on these modes , which are converted to the functions @xmath183 and @xmath184 through , and then substituted in . our goal is to derive a dispersion relation for ripplons that are based on the modes . note that the expressions satisfy the dpa bdg equations order by order in the expansion for long wavelengths . however , after expanding and truncating to obtain the final asymptotic forms , the dpa bdg equations are no longer satisfied for arbitrary frequencies and wave numbers . it is this property of that allows us to obtain a _ constraint _ on the physically acceptable combinations of @xmath10 and @xmath11 for our model ripplon , as follows . we suspect that a dispersion relation @xmath290 might be obtained directly from a judicious combination of dpa bdg equations , integrated over space . using the first and third equations of we derive the identity , @xmath291 we note that the right - hand - side of this identity , which carries the dimension of length , equals minus the reduced ( i.e. , scaled with the bulk pressure ) excess energy per unit area associated with the capillary wave perturbation @xcite , @xmath292 is determined making use of dispersion relation . in general the reduced frequency increases when the segregation strength @xmath235 increases , since then the interfacial tension increases . the upper curves correspond to strong segregation and the lower ones to weak segregation . it is conspicuous that the relative importance of the finite - wavelength correction diminishes as the weak segregation limit ( @xmath293 ) is approached.,scaledwidth=88.0% ] -mode , according to , and using the approximation given in the second line for @xmath156 , for various condensate segregation strengths . a symmetric bec mixture is assumed , so @xmath231 and consequently @xmath294 , with @xmath295 . the reduced wave number is fixed to @xmath296 and the reduced frequency @xmath297 is determined using dispersion relation . the curve with the lowest peak corresponds to reduced interaction strength @xmath225 ( weak segregation ) , the two intermediate curves to @xmath226 and @xmath227 and the curve with the highest peak to @xmath228 ( strong segregation ) . note that @xmath217 is continuous , but its first derivative is discontinuous , at the interface , which is located at @xmath232 . also note that for @xmath227 the curve is symmetric about the interface . this is a consequence of the symmetry of the analytic forms for @xmath298 in for @xmath299.,scaledwidth=88.0% ] -mode , according to , and using the approximation given in the second line for @xmath179 , for various condensate segregation strengths . a symmetric bec mixture is assumed , so @xmath231 and consequently @xmath300 , with @xmath295 . the reduced wave number is fixed to @xmath296 and the reduced frequency @xmath297 is determined using dispersion relation . the curve with the mildest slope at the interface corresponds to reduced interaction strength @xmath225 ( weak segregation ) , the two intermediate curves to @xmath226 and @xmath227 and the curve with the steepest slope at the interface to @xmath228 ( strong segregation ) . note that @xmath233 and its first derivative are continuous at the interface.,scaledwidth=88.0% ] and @xmath301 , again according to , for @xmath302 . an asymmetric bec mixture is assumed , with @xmath303 . the wave number is fixed through @xmath296 and the reduced frequencies are determined using dispersion relation and the identity @xmath304.,scaledwidth=88.0% ] and @xmath305 , according to , for @xmath302 . an asymmetric bec mixture is assumed , with @xmath303 . the wave number is fixed through @xmath296 and the reduced frequencies are determined using dispersion relation and the identity @xmath304 . , scaledwidth=88.0% ] now , for the family of modes that define the dpa ripplons that we are interested in , the identity is not satisfied for arbitrary @xmath10 and @xmath11 , but only for those pairs that satisfy a constraint . indeed , if we use the asymptotic solutions , the identity readily leads to the following relationship between the frequencies and the wave numbers , @xmath306 which reduces , to leading order for long wavelengths , to @xmath307 or , after unscaling the variables , and using , @xmath308 this is a physically well behaved dispersion relation . firstly , in the context of bec mixtures and also far beyond , it is well established that the frequency of capillary waves possesses a leading @xmath309 dependence on the wave number . secondly , specifically for bec mixtures , the ripplon dispersion takes the form @xmath310 in the long wavelength limit @xcite . note that this is formally identical to the dispersion relation for surface waves in classical fluids , known since long @xcite . here , @xmath311 is the sum of the condensate mass densities in bulk and @xmath312 is the interfacial tension , which takes the following simple analytic form within the static dpa model studied previously @xcite , @xmath313 the interfacial tension implied by is identical to this expression . we conclude that the dynamic dpa is fully consistent with the static dpa . a higher - order correction to the dispersion relation can be derived from . the correction in @xmath314 is of order @xmath315 . for the symmetric case ( @xmath316 and @xmath317 ) the expression reduces to @xmath318 with @xmath319 . from this expression the character of the first correction to the long wavelength behavior can be clearly seen . note that the correction vanishes in the weak segregation limit , relative to the leading term , since it contains an extra factor @xmath224 . this is a remarkable prediction , which may well survive beyond the dpa , because the dpa is a fairly good approximation for weak segregation . the dispersion relation including the finite - wavelength correction is shown in fig.5 for various segregation strengths . it is straightforward to derive the correction for the general ( asymmetric ) case . the result , after unscaling the variables , is @xmath320 in scaled variables this can be written in the following handy form , @xmath321 with @xmath322 . in the figures 6 - 9 we show examples of @xmath217-modes and @xmath233-modes that are analytically calculated using the asymptotic expressions valid for long wavelengths . the dispersion relation has been used to determine the frequencies for given wave numbers . we are now in a position to present the explicit analytic form of the ( moduli of the ) condensate order parameters undergoing the capillary wave perturbation of the interface . recalling , we have obtained latexmath:[\[\label{eq : perturbedphi1drecall } to first order in the amplitude @xmath148 and for long wavelengths . the second term in the right - hand - side is the rigid shift contribution and the last term is the effect of the function @xmath150 . we now regroup terms to obtain @xmath324 the last of these forms reveals that a deformation of the interface occurs , caused by two condensate - specific shifts that are independent of @xmath77 and therefore still rigid . the two shifts are in phase with each other , but may have different amplitudes ( except for a symmetric mixture in which case the shift amplitudes are equal ) . therefore , the leading - order interface deformation is , _ i ) _ a simple _ enhanced rigid shift _ for symmetric mixtures , and _ ii ) _ an enhanced rigid shift combined with a _ modulation _ of the mutual penetration of the condensates . in other words , in the generic case the bare capillary wave is enhanced and decorated with a modulation of the segregation , apparent as a modulation of the overlap of the order parameters at the interface . this overlap is in principle measurable because it corresponds to the first derivative , with respect to the reduced interaction @xmath235 , of the grand potential @xcite . we proceed to calculate the interface displacement @xmath119 using the foregoing result and the intersection criterion , again to first order in @xmath148 . we obtain @xmath325 this expression highlights the symmetric part , corresponding to the average enhanced rigid shift , and the antisymmetric part , associated with the difference of the enhanced rigid shifts of the individual condensates . regrouping the contributions and reading off the derivatives at @xmath67 from leads to the compact result , @xmath326 where we have used @xmath327 , with @xmath328 the sound velocity of condensate @xmath15 in bulk , @xmath329 in the symmetric case ( @xmath46 and @xmath330 ) there is only one sound velocity , @xmath8 , and the interface displacement simplifies to @xmath331 ; rigid " , innermost curve ) rigid - shift displacement depends mainly on the ratio of phase velocity and sound velocity and increases weakly with increasing healing length asymmetry . , scaledwidth=88.0% ] .,scaledwidth=88.0% ] we already mentioned that , for generic asymmetric mixtures , the interface deformation comprises a modulation of the interface midpoint density . this phenomenon implies a periodic variation of the extent to which phase segregation takes place . a decrease in the midpoint density is an apparent strengthening of segregation between the condensates , while a midpoint density increase corresponds to more mutual penetration and thus a weakening of segregation . performing a first - order taylor expansion of the common midpoint densities about the unperturbed midpoint value @xmath332 , we find that the segregation modulation is described by the following result , recalling @xmath333 , @xmath334 is the unperturbed midpoint value . note that the modulation is antisymmetric for reflection about @xmath67 ( exchange of condensates ) and vanishes in the symmetric case @xmath335 of equal sound velocities . it is noteworthy that the modulation vanishes in the weak segregation limit . the predicted exponent , i.e. , the power of @xmath336 , takes the value @xmath229 , which is the same exponent as for the interfacial tension . figures 11 and 12 illustrate the interfacial order parameter profiles with quarter period time intervals during the passing of the capillary wave . fig.11 deals with a symmetric case , for which the interface deformation consists merely of an enhancement of the rigid shift ( solid lines ) as compared with the bare rigid shift associated with @xmath337 ( dashed lines ) . fig.12 exemplifies the generic case @xmath338 , for which also a modulation of the midpoint density occurs . finally , fig.13 illustrates the interfacial density modulation for asymmetric mixtures , which increases with increasing asymmetry . for completeness we verify whether the interface displacement velocity is consistent with the phase fluctuation of each component , as expressed by . for this it suffices to use for the functions @xmath155 and just the leading term in . we obtain , on the one hand , @xmath339 and , on the other hand , @xmath340 note that all these velocities should be equal and independent of the condensate labels @xmath15 and @xmath54 . this is indeed the case . in this paper we have developed a dynamic double parabola approximation , starting from the static dpa model previously studied . to this end , we first derived a dpa to the time - dependent gp equations . from these we derived dpa bdg equations and so defined a model for studying ripplon excitations . we have also shown that these approximations commute : applying a dpa to the original bdg equations derived from the time - dependent gp equations , leads to the same model . within this model we established explicit mathematical dpa solutions for ripplons for all distances near to and far from the interface and checked that the formalism allows one to obtain correct zero modes consistent with exact relations in gp theory . the accuracy of the dpa relies on the assumption that both @xmath341 and @xmath342 be small compared to unity . these requirements are , strictly speaking , mutually incompatible since the first parameter is only small for strong segregation and the second only for weak segregation . however , the fact that both parameters are bounded from above by unity renders the dpa useful in all segregation regimes . the usefulness of the dpa was already established for calculating static properties analytically @xcite and it emerges here once more in the context of time - dependent properties and phenomena . using the dpa solutions for the modes in the long wavelength limit , we derived a family of modes and used that to define a physical dpa ripplon . for this type of perturbation we derived a dispersion relation @xmath4 directly from the dpa bdg equations combined in integral form and used as a constraint on allowed values for frequencies and wave numbers . in the long wavelength limit the dispersion relation carries the correct wave number dependence @xmath309 and satisfies the correct mean - field scaling behavior in the weak segregation limit , reproducing the exponent @xmath343 describing the dependence of the square root of the interfacial tension on the interaction strength difference @xmath336 . the leading term in the dispersion relation features the static interfacial tension derived earlier within the dpa @xcite . therefore , the dynamic dpa fully embodies the static dpa . the next - to - leading term in the dispersion relation , of order @xmath5 , is also derived analytically for all interaction strengths @xmath235 . this term features its own exponent @xmath344 , implying that the correction vanishes faster than the leading term for @xmath345 . the correction term has to our knowledge not received much attention in the literature , the only exception we know of being a result derived in the strong segregation limit by mishonov @xcite . comparing our correction factor in @xmath314 , which is 1 + @xmath346 in , for @xmath347 , with mishonov s correction factor in @xmath314 , which we estimated to be @xmath348 , would indicate that his correction is significantly larger than ours . this discrepancy merits further investigation . the deviation from the @xmath309 dispersion at finite wavelength is a prediction that can be verified experimentally . state - of - the - art experiments of bec in ( quasi-)uniform optical - box traps @xcite would be the appropriate arena for testing this result . the dynamic dpa model further predicts a small structural deformation of the interface due to the passing of the capillary wave , which can be calculated analytically . its magnitude is of order @xmath349 , with @xmath7 the phase velocity and @xmath8 the sound velocity . the deformation consists of an enhancement of the amplitude of the wave for all types of mixtures . for generic asymmetric mixtures consisting of condensates with unequal healing lengths @xmath338 an additional modulation is predicted of the common value of the condensate densities at the interface . this oscillation of the midpoint density is reminiscent of what would happen in the presence of a longitudinal breather mode . this brings us to new applications of the dynamic dpa to other excitations , e.g. , interface phonons , which are postponed to future work . n.v.t and t.h.p are supported by the vietnam national foundation for science and technology development ( nafosted ) and j.o.i . and c .- y.l . by fwo flanders under grant nr.fwo.103.2013.09 within the framework of the fwo - nafosted cooperation . j.o.i . and . are furthermore supported by ku leuven grant ot/11/063 . n.v.t is also supported by the ministry of education and training of vietnam ( grant b2016-sp2 - 04 ) . the authors thank hans hooyberghs , mehran kardar , todor mishonov , lev pitaevskii and bert van schaeybroeck for extensive discussions . in our derivations we use the following results from the gp theory . the gp potential is @xmath350 + k \lvert\phi_1\rvert^2 \lvert\phi_2\rvert^2.\ ] ] the gp equations are @xmath351 \phi_j \ ; , \ ; j=1,2 \ ; ( j \ne j'),\ ] ] the bdg equations are @xmath352 \delta\phi_j + \phi_{j0}^2 \delta\phi_j^{\ast } + k \phi_{j0}\phi_{j'0 } ( \delta\phi_{j ' } + \delta\phi_{j'}^{\ast } ) , \ ; j=1,2 \ ; ( j \ne j'),\ ] ] or , by substituting @xmath158 with bogoliubov form , @xmath353 or , in terms of @xmath153 and @xmath154 . @xmath354 where @xmath355 and @xmath356 . the solutions in the limit @xmath161 and @xmath162 are called the zero modes . it is easy to obtain the @xmath77-dependence of these modes . they solve @xmath357 the second of these equations is just the static gp equation and is solved by @xmath358 inserting this result in the second equation and then taking the @xmath77-derivative of it , suffices to show that the first equation is solved by @xmath359 h. takeuchi , n. suzuki , k. kasamatsu , h. saito and m. tsubota , phys . a * 81 * , 094517 ( 2010 ) . k. sasaki , n. suzuki and h. saito , phys . a * 83 * , 053606 ( 2011 ) . a. bezett , v. bychkov , e. lundh , d. kobyakov and m. marklund , phys . a * 82 * , 043608 ( 2010 ) . b. van schaeybroeck , phys . rev . a * 78 * , 024624 ( 2008 ) j. o. indekeu , c .- y . lin , n. v. thu , b. van schaeybroeck and t. h. phat , phys . a. * 91 * , 033615 ( 2015 ) . i. e. mazets , phys . rev . a * 65 * , 033618 ( 2002 ) . r. a. barankov , phys . a * 66 * , 013612 ( 2002 ) . d. kobyakov , v. bychkov , e. lundh , a. bezett , v. akkerman and m. marklund , phys . a * 83 * , 043623 ( 2011 ) . c. pethick and h. smith , `` bose - einstein condensation in dilute gases '' , cambridge university press , cambridge ( 2002 ) . l. pitaevskii and s. stringari , bose - einstein condensation " , oxford university press ( 2003 ) . p. ao and s.t . chui , phys . a * 58 * , 4836 ( 1998 ) . f. dalfovo , s. giorgini , l. p. pitaevskii , and s. stringari rev . mod . phys . * 71 * , 463 ( 1999 ) . h. takeuchi and k. kasamatsu , phys . a * 88 * , 043612 ( 2013 ) . see e.g. chap . ix , art . 266 in , h. lamb , hydrodynamics " , dover publications ( new york ) ( 1945 ) . b. van schaeybroeck and j.o . indekeu , critical wetting , first - order wetting and prewetting phase transitions in binary mixtures of bose - einstein condensates " , phys . rev . a * 91 * , 013626 ( 2015 ) t.m . mishonov , capillary waves at the interface of two bose - einstein condensates . long wavelengths asymptotic by trial function approach " , arxiv:1410.6124v2 . gaunt , t.f . schmidutz , i. gotlibovych , r.p . smith , z. hadzibabic , bose einstein condensation of atoms in a uniform potential " , phys . lett . * 110 * , 200406 ( 2013 ) .

the localized low - energy interfacial excitations , or nambu - goldstone modes , of phase - segregated binary mixtures of bose - einstein condensates are investigated analytically by means of a double - parabola approximation ( dpa ) to the lagrangian density in gross - pitaevskii theory for a system in a uniform potential . within this model analytic expressions are obtained for the excitations underlying capillary waves or ripplons " for arbitrary strength @xmath0 of the phase segregation . the dispersion relation @xmath1 is derived directly from the bogoliubov - de gennes equations in limit that the wavelength @xmath2 is much larger than the healing length @xmath3 . the proportionality constant in the dispersion relation provides the static interfacial tension . a correction term in @xmath4 of order @xmath5 is calculated analytically , entailing a finite - wavelength correction factor @xmath6 . this prediction may be tested experimentally using ( quasi-)uniform optical - box traps . explicit expressions are obtained for the structural deformation of the interface due to the passing of the capillary wave . it is found that the amplitude of the wave is enhanced by an amount that is quadratic in the ratio of the phase velocity @xmath7 to the sound velocity @xmath8 . for generic asymmetric mixtures consisting of condensates with unequal healing lengths an additional modulation is predicted of the common value of the condensate densities at the interface .

the iron - based superconductors @xcite have attracted great attention since hosono and co - workers reported the discovery of 26 k superconductivity in fluorine doped lafeaso in 2008 @xcite . unfortunately , there is still no consensus on the superconducting mechanism in them , mainly due to their complicated electronic structures @xcite . there are many families of the iron - based superconductors , such as lao@xmath6f@xmath7feas ( `` 1111 '' ) , ba@xmath6k@xmath7fe@xmath8as@xmath8 ( `` 122 '' ) , nafe@xmath6co@xmath9as ( `` 111 '' ) , and fese@xmath7te@xmath6 ( `` 11 '' ) @xcite . among them , the `` 122 '' family is the most studied one due to the easy growth of sizable high - quality single crystals @xcite . intriguingly , the members of this family do not share a universal superconducting gap structure . while the optimally doped ba@xmath10k@xmath11fe@xmath0as@xmath0 and bafe@xmath12co@xmath13as@xmath0 have nodeless superconducting gaps @xcite , the extremely hole - doped kfe@xmath0as@xmath0 was reported to be a nodal superconductor @xcite . furthermore , the isovalently doped bafe@xmath0(as@xmath6p@xmath9)@xmath0 @xcite and ba(fe@xmath6ru@xmath9)@xmath0as@xmath0 @xcite also manifest nodal superconductivity . so far , the origin of these nodal superconducting gaps is still under debate , particularly in kfe@xmath0as@xmath0 @xcite . the detailed thermal conductivity study provided compelling evidences for a @xmath14-wave gap in kfe@xmath0as@xmath0 @xcite , but the low - temperature angle - resolved photoemission spectroscopy ( arpes ) measurements clearly showed nodal @xmath15-wave gap @xcite . recent arpes and thermal conductivity experiments on highly hole - doped ba@xmath6k@xmath7fe@xmath8as@xmath8 also support nodal @xmath15-wave gap @xcite . kfe@xmath0as@xmath0 has two sister compounds , csfe@xmath0as@xmath0 and rbfe@xmath0as@xmath0 , and both of them are superconducting @xcite . while muon - spin spectroscopy measurements on rbfe@xmath0as@xmath0 polycrystals suggested that rbfe@xmath0as@xmath0 is best described by a two - gap @xmath15-wave model @xcite , recent specific heat and thermal conductivity measurements on csfe@xmath0as@xmath0 single crystals provided clear evidences for nodal superconducting gap in csfe@xmath0as@xmath0 @xcite . to clarify whether the superconducting gap structure of rbfe@xmath0as@xmath0 is indeed different from those of kfe@xmath0as@xmath0 and csfe@xmath0as@xmath0 , more experiments on rbfe@xmath0as@xmath0 single crystals are highly desired . in this paper , we present the low - temperature thermal conductivity of rbfe@xmath0as@xmath0 single crystal down to 100 mk . a significant residual linear term @xmath2 = 0.65 @xmath16 0.03 mw k@xmath3 @xmath4 is obtained in zero magnetic field , and the field dependence of @xmath2 mimics that of csfe@xmath0as@xmath0 . these results clarify that rbfe@xmath0as@xmath0 is also a nodal superconductor . the three compounds kfe@xmath0as@xmath0 , rbfe@xmath0as@xmath0 , and csfe@xmath0as@xmath0 may have a common superconducting gap structure . the rbfe@xmath0as@xmath0 single crystals were grown by self - flux method for the first time , and the process is the same as the growth of csfe@xmath0as@xmath0 single crystals @xcite . the dc magnetization was measured using a superconducting quantum interference device ( mpms , quantum design ) . the specific heat measurement above 1.9 k was performed in a physical property measurement system ( ppms , quantum design ) via the relaxation method , and below 1.9 k it was measured in a small dilution refrigerator integrated into the ppms . for transport measurements , the rbfe@xmath0as@xmath0 single crystal was cleaved to a rectangular shape of dimensions 2.2 @xmath17 1.0 mm@xmath18 in the @xmath19 plane , with 40 @xmath20 m thickness along the @xmath21 axis . contacts were made directly on the sample surfaces with silver paint ( dupont 4929n ) , which were used for both resistivity and thermal conductivity measurements . to avoid degradation , the sample was exposed in air for less than 2 hours . the contacts are metallic with a typical resistance 100 m@xmath22 at 2 k. in - plane thermal conductivity was measured in a dilution refrigerator , using a standard four - wire steady - state method with two ruo@xmath0 chip thermometers , calibrated _ in situ _ against a reference ruo@xmath0 thermometer . magnetic fields were applied along the @xmath21 axis and perpendicular to the heat current . to ensure a homogeneous field distribution in the sample , all fields were applied at a temperature above @xmath23 for transport measurements . as@xmath0 single crystal in @xmath24 20 oe along @xmath21 axis , with zero - field cooling process . ( b ) temperature dependence of specific heat @xmath25 for rbfe@xmath0as@xmath0 single crystal in zero field , plotted as @xmath25 vs @xmath26 . the solid line is the best fit to @xmath27 = @xmath28 + @xmath29@xmath30 + @xmath31@xmath32 from 2.4 to 10 k. ( c ) in - plane resistivity of rbfe@xmath0as@xmath0 single crystal in zero field . the data between 2.2 and 9 k can be fitted to @xmath33 = @xmath5 + @xmath34 , as shown in the inset , which gives @xmath35 = 1.84 @xmath20@xmath22 cm.,width=302 ] as@xmath0 single crystal in magnetic fields up to 1.1 t. ( b ) temperature dependence of the upper critical field @xmath36 , defined by @xmath37 in ( a ) . the solid line is a fit of @xmath36 to the ginzburg - landau equation , which gives @xmath38 0.97 t.,width=302 ] figure 1(a ) shows the low - temperature dc magnetization of rbfe@xmath0as@xmath0 single crystal , measured in @xmath39 = 20 oe along @xmath21 axis , with zero - field cooling process . the @xmath1 2.10 k is defined at the onset of the diamagnetic transition . the magnetization does not saturate down to 1.8 k , where the superconducting volume fraction is already as large as 40% . with decreasing temperature , the superconducting volume fraction should further increase to reach the fully shielded state . in fig . 1(b ) , we present the low - temperature specific heat of rbfe@xmath0as@xmath0 single crystal down to 100 mk in zero field , plotted as @xmath25 vs @xmath26 . a significant jump due to the superconducting transition is observed at @xmath1 2.1 k , which indicates the high quality of our sample . in order to determine the zero - field normal - state sommerfeld coefficient @xmath40 , the specific heat above @xmath23 is fitted to @xmath27 = @xmath28 + @xmath29@xmath30 + @xmath31@xmath32 , with @xmath29 and @xmath31 as the lattice coefficients . the solid line in fig . 1(b ) is the best fit of @xmath25 from 2.4 to 10 k , which gives @xmath40 = 127.3 @xmath16 0.9 mj mol@xmath41 k@xmath3 , @xmath29 = 0.66 @xmath16 0.04 mj mol@xmath41 k@xmath42 , and @xmath31 = 0.0029 @xmath16 0.0005 mj mol@xmath41 k@xmath43 . from the relation @xmath44 = ( 12@xmath45 / 5@xmath29)@xmath46 , where @xmath47 is the molar gas constant and @xmath48 = 5 is the total number of atoms in one unit cell , the debye temperature @xmath44 = 245 k is estimated . this value is comparable to those of kfe@xmath0as@xmath0 and csfe@xmath0as@xmath0 @xcite . the in - plane resistivity of rbfe@xmath0a@xmath0 single crystal in zero filed is plotted in fig . the @xmath1 2.13 k , defined by @xmath33 = 0 , agrees well with the magnetization and specific heat measurements . for the polycrystalline sample of rbfe@xmath0a@xmath0 , @xmath49 2.6 k was defined at the onset of the diamagnetic transition @xcite , which is 0.5 k higher than our single crystal . similarly , @xmath49 2.2 k was defined at the onset of the diamagnetic transition for the csfe@xmath0as@xmath0 polycrystal @xcite , but @xmath49 1.8 k was found in the csfe@xmath0as@xmath0 single crystal @xcite . it is unclear why the @xmath23 shows difference between polycrystalline sample and single crystal for rbfe@xmath0a@xmath0 and csfe@xmath0a@xmath0 . in case that the single crystals have intrinsic @xmath23 , the @xmath23 of ( k , rb , cs)fe@xmath0a@xmath0 series ( 3.8 , 2.1 , and 1.8 k , respectively ) decreases with the increase of the ionic radius of alkali metal . in the inset of fig . 1(c ) , the normal - state @xmath50 below 9 k can be well fitted by @xmath33 = @xmath5 + @xmath34 , with @xmath5 = 1.84 @xmath16 0.01 @xmath20@xmath22 cm and @xmath51 = 0.16 @xmath20@xmath22 cm k@xmath52 . similar non - fermi - liquid behavior of @xmath50 was also observed in kfe@xmath0as@xmath0 and csfe@xmath0as@xmath0 @xcite , which may result from antiferromagnetic spin fluctuations @xcite . the residual resistivity ratio rrr = @xmath33(292k)/@xmath53 310 again reflects the high quality of our rbfe@xmath0as@xmath0 single crystal . figure 2(a ) shows the low - temperature resistivity of rbfe@xmath0as@xmath0 single crystal in magnetic fields up to 1.1 t. in order to estimate the zero - temperature upper critical field @xmath54 , the temperature dependence of @xmath36 is plotted in fig . 2(b ) , defined by @xmath33 = 0 in fig . @xmath38 0.97 t is obtained by fitting the data with the ginzburg - landau equation @xmath55/[1+(t / t_c)^{2}]$ ] @xcite . as@xmath0 single crystal in zero and magnetic fields applied along the @xmath21 axis . the solid line is a fit of the zero - field data between 0.1 and 0.3 k to @xmath56 , giving a residual linear term @xmath2 = 0.65 @xmath16 0.03 mw k@xmath3 @xmath4 . the dashed line is the normal - state wiedemann - franz law expectation @xmath57/@xmath5 , with @xmath57 the lorenz number 2.45 @xmath17 10@xmath58 w @xmath22 k@xmath3 and @xmath5 = 1.84 @xmath20@xmath22 cm.,width=317 ] the low - temperature heat transport measurement is a bulk technique to probe the gap structure of superconductors @xcite . in fig . 3 , the in - plane thermal conductivity of rbfe@xmath0as@xmath0 single crystal in zero and applied field is plotted as @xmath59 vs @xmath26 @xcite . the thermal conductivity at very low temperature can be usually fitted to @xmath59 = @xmath60 , in which the two terms @xmath61 and @xmath62 represent contributions from electrons and phonons , respectively . the power @xmath63 is typically between 2 and 3 , due to specular reflections of phonons at the boundary @xcite . since all the curves in fig 3 are roughly linear , we fix @xmath63 to 2 . the value @xmath64 2 has been previously observed in dirty kfe@xmath0as@xmath0 @xcite , ba(fe@xmath6ru@xmath9)@xmath0as@xmath0 @xcite , and csfe@xmath0as@xmath0 single crystals @xcite . here , we only focus on the electronic term . in zero field , the fitting of the data between 0.1 to 0.3 k gives @xmath2 = 0.65 @xmath16 0.03 mw k@xmath3@xmath4 . if we slightly change the fitting range , we obtain @xmath2 = 0.62 @xmath16 0.03 mw k@xmath3@xmath4 for the range below 0.27 k and @xmath2 = 0.63 @xmath16 0.04 mw k@xmath3@xmath4 for the range below 0.24 k. therefore , the value of @xmath2 basically does not depend on the temperature range chosen for the fit . such a significant @xmath2 is usually contributed by nodal quasiparticles , thus it is a strong evidence for nodal superconducting gap @xcite . previously , @xmath2 = 2.27 @xmath16 0.02 and 3.6 @xmath16 0.5 mw k@xmath3@xmath4 were observed for dirty and clean kfe@xmath0as@xmath0 single crystals , respectively @xcite . for csfe@xmath0as@xmath0 single crystal with @xmath5 = 1.80 @xmath20@xmath22 cm , @xmath2 = 1.27 @xmath16 0.04 mw k@xmath3@xmath4 was found @xcite . the zero - field value of @xmath2 for rbfe@xmath0as@xmath0 is about 5@xmath65 of the normal - state widemann - franz law expectation @xmath66 = @xmath67 = 13.5 mw k@xmath3 @xmath4 , with @xmath57 the lorenz number 2.45 @xmath17 10@xmath58 w @xmath22 k@xmath3 and @xmath5 = 1.84 @xmath20@xmath22 cm . in @xmath39 = 0.9 t , the experimental data roughly satisfy the widemann - franz law , so we take 0.9 t as the bulk @xmath54 . this value is slightly lower than that obtained from resistivity measurements , but it does not affect our discussion of the filed dependence of @xmath2 below . of rbfe@xmath0as@xmath0 as a function of @xmath68 . for comparison , similar data are shown for the clean @xmath15-wave superconductor nb @xcite , the @xmath14-wave cuprate superconductor tl-2201 @xcite , the dirty and clean kfe@xmath0as@xmath0 @xcite , and csfe@xmath0as@xmath0 @xcite.,width=328 ] the field dependence of @xmath2 may provide more information on the superconducting gap structure @xcite . in fig . 4 , we plot the normalized @xmath69 of rbfe@xmath0as@xmath0 together with the typical @xmath15-wave superconductor nb @xcite , the @xmath14-wave cuprate superconductor tl@xmath0ba@xmath0cuo@xmath70 ( tl-2201 ) @xcite , the dirty and clean kfe@xmath0as@xmath0 @xcite , and csfe@xmath8as@xmath8 @xcite . for an @xmath15-wave superconductor with isotropic gap , such as nb , @xmath71 grows exponentially with the field @xcite . for the @xmath14-wave superconductor tl-2201 , @xmath2 increases roughly proportional to @xmath72 at low field @xcite , due to the volovik effect @xcite . from fig . 4 , the normalized @xmath69 curve of rbfe@xmath0as@xmath0 is very close to that of csfe@xmath0as@xmath0 and lies between the dirty and clean kfe@xmath0as@xmath0 . 0.48p2.3cmp0.8cmp1.3cmp1.2cmp1.1cmp0.9 cm & @xmath23 & @xmath5 & @xmath40 & @xmath54 & @xmath2 + & ( k ) & ( @xmath73 cm ) & ( @xmath74 ) & ( t ) & ( @xmath75 ) + kfe@xmath8as@xmath8(clean ) & 3.8 & 0.21 & 94 & 1.60 & 3.60 + kfe@xmath8as@xmath8(dirty ) & 2.5 & 3.32 & 91 & 1.25 & 2.27 + rbfe@xmath8as@xmath8 & 2.1 & 1.84 & 127 & 0.97 & 0.65 + csfe@xmath8as@xmath8 & 1.8 & 1.80 & 184 & 1.40 & 1.27 + as listed in table i , the @xmath5 of dirty and clean kfe@xmath0as@xmath0 differ by 15 times @xcite , while rbfe@xmath0as@xmath0 and csfe@xmath0as@xmath0 have comparable @xmath5 , with values lying between that of dirty and clean kfe@xmath0as@xmath0 . therefore , in ( k , rb , cs)fe@xmath0as@xmath0 serial superconductors , the field dependence of @xmath2 seems to correlate with the impurity level . although reid @xmath76 @xmath77 argued that the @xmath69 of clean kfe@xmath0as@xmath0 is a compelling evidence for @xmath14-wave gap @xcite , recent thermal conductivity measurements on highly hole - doped ba@xmath6k@xmath9fe@xmath0as@xmath0 single crystals support nodal @xmath15-wave gap @xcite . for such a complex nodal @xmath15-wave gap structure , likely with both nodal and nodeless gaps of different magnitudes , it is hard to get a theoretical curve of @xmath69 . one needs to carefully consider the effect of impurity on the behavior of @xmath69 . nevertheless , the evolution of the normalized @xmath69 suggests a common nodal gap structure in ( k , rb , cs)fe@xmath0as@xmath0 serial superconductors . in table i , we also list the @xmath23 , @xmath40 , @xmath54 , and @xmath2 of the ( k , rb , cs)fe@xmath0as@xmath0 serial superconductors @xcite . both @xmath23 and @xmath40 show a systematic change with increasing the ionic radii of alkali metal . the @xmath40 values of rbfe@xmath0as@xmath0 and csfe@xmath0as@xmath0 are very large among all iron - based superconductors , which reflects their abnormally large density of states or effective mass of electrons . this may be explained by recent arpes measurement on csfe@xmath0as@xmath0 single crystals , which suggests that the large separation of feas layers along @xmath21 axis makes the system more two - dimensional and enhances the electronic correlations @xcite . neither the @xmath54 nor @xmath2 shows a systematic change . the @xmath54 of csfe@xmath0as@xmath0 is abnormally high , which may also relate to its much enhanced two dimensionality and electronic correlations . as for the @xmath2 , it depends on the very details of the nodal gap , such as the slope of the gap at the node . for the accidental nodes appearing in the complex fermi surfaces of ( k , rb , cs)fe@xmath0as@xmath0 , the @xmath2 may not necessarily manifest systematic change with the increase of the ionic radii of alkali metal . in summary , we have measured the magnetization , resistivity , low - temperature specific heat and thermal conductivity of rbfe@xmath0as@xmath0 single crystals . a nodal superconducting gap in rbfe@xmath0as@xmath0 is strongly suggested by the observation of a significant residual linear term @xmath2 = 0.65 mw k@xmath3 @xmath4 in zero magnetic field . it is concluded that ( k , rb , cs)fe@xmath0as@xmath0 serial superconductors may have a common nodal gap structure , and the field dependence of @xmath2 seems to evolve with the impurity level . + this work is supported by the natural science foundation of china , the ministry of science and technology of china ( national basic research program no : 2012cb821402 and 2015cb921401 ) , program for professor of special appointment ( eastern scholar ) at shanghai institutions of higher learning . + @xmath78 e - mail : shiyan@xmath79li@fudan.edu.cn h. ding , p. richard , k. nakayama , k. sugawara , t. arakane , y. sekiba , a. takayama , s. souma , t. sato , t. takahashi , z. wang , x. dai , z. fang , g. f. chen , j. l. luo , and n. l. wang , europhys . lett . * 83 * , 47001 ( 2008 ) . x. g. luo , m. a. tanatar , j .- ph . reid , h. shakeripour , n. doiron - leyraud , n. ni , s. l. budko , p. c. canfield , h. q. luo , z. s. wang , h .- h . wen , r. prozorov , and l. taillefer , phys . b * 80 * , 140503(r ) ( 2009 ) . k. hashimoto , a. serafin , s. tonegawa , r. katsumata , r. okazaki , t. saito , h. fukazawa , y. kohori , k. kihou , c. h. lee , a. iyo , h. eisaki , h. ikeda , y. matsuda , a. carrington , and t. shibauchi , phys . rev . b * 82 * , 014526 , ( 2010 ) . k. hashimoto , m. yamashita , s. kasahara , y. senshu , n. nakata , s. tonegawa , k. ikada , a. serafin , a. carrington , t. terashima , h. ikeda , t. shibauchi , and y. matsuda , phys . b * 81 * , 220501(r ) ( 2010 ) . j .- ph . reid , m. a. tanatar , a. juneau - fecteau , r. t. gordon , de cotret s. ren , n. doiron - leyraud , t. saito , h. fukazawa , y. kohori , k. kihou , c. h. lee , a. iyo , h. eisaki , r. prozorov , and louis taillefer , phys . lett . * 109 * , 087001 ( 2012 ) . k. okazaki , y. ota , y. kotani , w. malaen , y. ishida , t. shimojima , t. kiss , s. watanabe , c .- t . chen , k. kihon , c. h. lee , a. iyo , h. eisaki , t. satio , h. fukazawa , y. kohori , k. hashimoto , t. shibauchi , y. matsuda , h. ikeda , h. miynhara , r. arita , a. chiainani , and s. shin , science * 337 * , 1314 ( 2012 ) . m. sutherland , d. g. hawthorn , r. w. hill , f. ronning , s. wakimoto , h. zhang , c. proust , etienne boaknin , c. lupien , louis taillefer , rui xing liang , d. a. bonn , w. n. hardy , robert gagnon , n. e. hussey , t. kimura , m. nohara , and h. takagi , phys . rev . b * 67 * , 174520 ( 2003 ) .

the in - plane thermal conductivity of iron - based superconductor rbfe@xmath0as@xmath0 single crystal ( @xmath1 2.1 k ) was measured down to 100 mk . in zero field , the observation of a significant residual linear term @xmath2 = 0.65 mw k@xmath3 @xmath4 provides clear evidence for nodal superdonducting gap . the field dependence of @xmath2 is similar to that of its sister compound csfe@xmath0as@xmath0 with comparable residual resistivity @xmath5 , and lies between the dirty and clean kfe@xmath0as@xmath0 . these results suggest that the ( k , rb , cs)fe@xmath0as@xmath0 serial superconductors have a common nodal gap structure .

calibration is ubiquitous in all fields of science and engineering . it is an essential step to guarantee that the devices measure accurately what scientists and engineers want . if sensor devices are not properly calibrated , their measurements are likely of little use to the application . while calibration is mostly done by specialists , it often can be expensive , time - consuming and sometimes even impossible to do in practice . hence , one may wonder whether it is possible to enable machines to calibrate themselves automatically with a smart algorithm and give the desired measurements . this leads to the challenging field of _ self - calibration _ ( or _ blind calibration _ ) . it has a long history in imaging sciences , such as camera self - calibration @xcite , blind image deconvolution @xcite , self - calibration in medical imaging @xcite , and the well - known phase retrieval problem ( phase calibration ) @xcite . it also plays an important role in signal processing @xcite and wireless communications @xcite . self - calibration is not only a challenging problem for engineers , but also for mathematicians . it means that one needs to estimate the calibration parameter of the devices to adjust the measurements as well as recover the signal of interests . more precisely , many self - calibration problems are expressed in the following mathematical form , @xmath4 where @xmath5 is the observation , @xmath6 is a partially unknown sensing matrix , which depends on an unknown parameter @xmath7 and @xmath8 is the desired signal . an uncalibrated sensor / device directly corresponds to _ imperfect sensing _ " , i.e. , uncertainty exists within the sensing procedure and we do not know everything about @xmath6 due to the lack of calibration . the purpose of self - calibration is to resolve the uncertainty i.e. , to estimate @xmath7 in @xmath6 and to recover the signal @xmath8 at the same time . the general model is too hard to get meaningful solutions without any further assumption since there are many variants of the general model under different settings . in , @xmath6 may depend on @xmath7 in a nonlinear way , e.g. , @xmath7 can be the unknown orientation of a protein molecule and @xmath8 is the desired object @xcite ; in phase retrieval , @xmath7 is the unknown phase information of the fourier transform of the object @xcite ; in direction - of - arrival estimation @xmath7 represents unknown offset , gain , and phase of the sensors @xcite . hence , it is impossible to resolve every issue in this field , but we want to understand several scenarios of self - calibration which have great potential in real world applications . among all the cases of interest , we assume that @xmath6 _ linearly _ depends on the unknown @xmath7 and will explore three different types of self - calibration models that are of considerable practical relevance . however , even for linear dependence , the problem is already quite challenging , since in fact we are dealing with _ bilinear ( nonlinear ) inverse problems_. all those three models have wide applications in imaging sciences , signal processing , wireless communications , etc . , which will be addressed later . common to these applications is the need for _ fast _ self - calibration algorithms , which ideally should be accompanied by theoretical performance guarantees . we will show under certain cases , these self - calibration problems can be solved by _ linear least squares _ exactly and efficiently if no noise exists , which is guaranteed by rigorous mathematical proofs . moreover , we prove that the solution is also robust to noise with tools from random matrix theory . by assuming that @xmath6 linearly depends on @xmath7 , becomes a bilinear inverse problem , i.e. , we want to estimate @xmath7 and @xmath8 from @xmath5 , where @xmath5 is the output of a bilinear map from @xmath9 bilinear inverse problems , due to its importance , are getting more and more attentions over the last few years . on the other hand , they are also notoriously difficult to solve in general . bilinear inverse problems are closely related to low - rank matrix recovery , see @xcite for a comprehensive review . there exists extensive literature on this topic and it we could not possible do justice to all these contributions . instead we will only highlight some of the works which have inspired us . blind deconvolution might be the most important examples of bilinear inverse problems @xcite , i.e. , recovering @xmath10 and @xmath11 from @xmath12 , where @xmath13 " stands for convolution . if both @xmath10 and @xmath11 are inside known low - dimensional subspaces , the blind deconvolution can be rewritten as @xmath14 , where @xmath15 , @xmath16 and @xmath17 " denotes the fourier transform . in the inspiring work @xcite , ahmed , romberg and recht apply the lifting " techniques @xcite and convert the problem into estimation of rank-1 matrix @xmath18 . it is shown that solving a convex relaxation enables recovery of @xmath18 under certain choices of @xmath19 and @xmath20 . following a similar spirit , @xcite uses lifting " combined with a convex approach to solve the scenarios with sparse @xmath8 and @xcite studies the so called blind deconvolution and blind demixing " problem . the other line of blind deconvolution follows a nonconvex optimization approach @xcite . in @xcite , ahmed , romberg and krahmer , using tools from generic chaining , obtain local convergence of a sparse power factorization algorithm to solve this blind deconvolution problem when @xmath21 and @xmath8 are sparse and @xmath19 and @xmath20 are gaussian random matrices , . under the same setting as @xcite , lee et al . @xcite propose a projected gradient descent algorithm based on matrix factorizations and provide a convergence analysis to recover sparse signals from subsampled convolution . however , this projection step can be hard to implement . as an alternative , the expensive projection step is replaced by a heuristic approximate projection , but then the global convergence is not fully guaranteed . both @xcite achieve nearly optimal sampling complexity . @xcite proves global convergence of a gradient descent type algorithm when @xmath19 is a deterministic fourier type matrix and @xmath20 is gaussian . results about identifiability issue of bilinear inverse problems can be found in @xcite . another example of self - calibration focuses on the setup @xmath22 , where @xmath23 . the difference from the previous model consists in replacing the subspace assumption by multiple measurements . there are two main applications of this model . one application deals with blind deconvolution in an imaging system which uses randomly coded masks @xcite . the measurements are obtained by ( subsampled ) convolution of an unknown blurring function @xmath24 with several random binary modulations of one image . both @xcite and @xcite developed convex relaxing approaches ( nuclear norm minimization ) to achieve exact recovery of the signals and the blurring function . the other application is concerned with calibration of the unknown gains and phases @xmath24 and recovery of the signal @xmath8 , see e.g. @xcite . cambareri and jacques propose a gradient descent type algorithm in @xcite and show convergence of the iterates by first constructing a proper initial guess . an empirical study is given in @xcite when @xmath8 is sparse by applying an alternating hard thresholding algorithm . recently , @xcite study the blind deconvolution when inputs are changing . more precisely , the authors consider @xmath25 where each @xmath26 belongs to a different known subspace , i.e. , @xmath27 . they employ a similar convex approach as in @xcite to achieve exact recovery with number of measurements close to the information theoretic limit . an even more difficult , and from a practical viewpoint highly relevant , scenario focuses on _ self - calibration from multiple snapshots _ @xcite . here , one wishes to recover the unknown gains / phases @xmath23 and a signal matrix @xmath28 $ ] from @xmath29 . for this model , the sensing matrix @xmath20 is fixed throughout the sensing process and one measures output under different snapshots @xmath30 . one wants to understand under what conditions we can identify @xmath24 and @xmath30 jointly . if @xmath20 is a fourier type matrix , this model has applications in both image restoration from multiple filters @xcite and also network calibration @xcite . we especially benefitted from work by gribonval and coauthors @xcite , as well as by balzano and nowak @xcite . the papers @xcite study the noiseless version of the problem by solving a linear system and @xcite takes a total least squares approach in order to obtain empirically robust recovery in the presence of noise . if each @xmath31 is sparse , this model becomes more difficult and gribonval et al . @xcite give a thorough numerical study . very recently , @xcite gave a theoretic result under certain conditions . this calibration problem is viewed as a special case of the dictionary learning problem where the underlying dictionary @xmath32 possesses some additional structure . the idea of transforming a blind deconvolution problem into a linear problem can also be found in @xcite , where the authors analyze a certain non - coherent wireless communication scenario . in our work , we consider three different models of self - calibration , namely , @xmath22 , @xmath33 and @xmath34 . detailed descriptions of these models are given in the next section . we do not impose any sparsity constraints on @xmath8 or @xmath31 . we want to find out @xmath31 ( or @xmath8 ) and @xmath24 when @xmath35 ( or @xmath5 ) and @xmath36 ( or @xmath20 ) are given . roughly , they correspond to the models in @xcite respectively . though all of the three models belong to the class of bilinear inverse problems , we will prove that simply solving _ linear least squares _ will give solutions to all those models exactly and robustly for invertible @xmath24 and for several useful choices of @xmath20 and @xmath37 moreover , the sampling complexity is nearly optimal ( up to poly - log factors ) with respect to the information theoretic limit ( degree of freedom of unknowns ) . as mentioned before , our approach is largely inspired by @xcite and @xcite ; there the authors convert a bilinear inverse problem into a linear problem via a proper transformation . we follow a similar approach in our paper . the paper @xcite provides an extensive empirical study , but no theoretical analysis . nowak and balzano , in @xcite provide numerical simulations as well as theoretical conditions on the number of measurements required to solve the noiseless case . our paper goes an important step further : on the one hand we consider more general self - calibration settings . and on the other hand we provide a rigorous theoretical analysis for recoverability and , perhaps most importantly , stability theory in the presence of measurement errors . owing to the simplicity of our approach and the structural properties of the underlying matrices , our framework yields self - calibration algorithms that are numerically extremely efficient , thus potentially allowing for deployment in applications where real - time self - calibration is needed . we introduce notation which will be used throughout the paper . matrices are denoted in boldface or a calligraphic font such as @xmath38 and @xmath39 ; vectors are denoted by boldface lower case letters , e.g. @xmath40 the individual entries of a matrix or a vector are denoted in normal font such as @xmath41 or @xmath42 for any matrix @xmath38 , @xmath43 denotes its operator norm , i.e. , the largest singular value , and @xmath44 denotes its the frobenius norm , i.e. , @xmath45 . for any vector @xmath46 , @xmath47 denotes its euclidean norm . for both matrices and vectors , @xmath48 and @xmath49 stand for the transpose of @xmath38 and @xmath46 respectively while @xmath50 and @xmath51 denote their complex conjugate transpose . we equip the matrix space @xmath52 with the inner product defined by @xmath53 a special case is the inner product of two vectors , i.e. , @xmath54 for a given vector @xmath55 , @xmath56 represents the diagonal matrix whose diagonal entries are given by the vector @xmath55 . @xmath57 is an absolute constant and @xmath58 is a constant which depends linearly on @xmath59 , but on no other parameters . @xmath60 and @xmath61 always denote the @xmath62 identity matrix and a column vector of @xmath63 " in @xmath64 respectively . and @xmath65 and @xmath66 stand for the standard orthonormal basis in @xmath67 and @xmath68 respectively . @xmath13 " is the circular convolution and @xmath69 " is the kronecker product . the paper is organized as follows . the more detailed discussion of the models under consideration and the proposed method will be given in section [ s : model ] . section [ s : main ] presents the main results of our paper and we will give numerical simulations in section [ s : numerics ] . section [ s : proof ] contains the proof for each scenario . we collect some useful auxiliary results in the appendix . this section is devoted to describing three different models for self - calibration in detail . we will also explain how those bilinear inverse problems are reformulated and solved via linear least squares . [ [ self - calibration - from - repeated - measurements ] ] self - calibration from repeated measurements + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + suppose we are seeking for information with respect to an unknown signal @xmath70 with several randomized linear sensing designs . throughout this procedure , the calibration parameter @xmath24 remains the same for each sensing procedure . how can we recover the signal @xmath70 and @xmath24 simultaneously ? let us make it more concrete by introducing the following model , @xmath71 where @xmath72 is a diagonal matrix and each @xmath73 is a measurement matrix . here @xmath35 and @xmath36 are given while @xmath24 and @xmath70 are unknown . for simplicity , we refer to the setup as _ self - calibration from repeated measurements _ " . this model has various applications in self - calibration for imaging systems @xcite , networks @xcite , as well as in blind deconvolution from random masks @xcite . [ [ blind - deconvolution - via - diverse - inputs ] ] blind deconvolution via diverse inputs + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + suppose that one sends several different signals through the same unknown channel , and each signal is encoded differently . namely , we are considering @xmath74 how can one estimate the channel and each signal jointly ? in the frequency domain , this _ blind deconvolution via diverse inputs _ " @xcite problem can be written as ( with a bit abuse of notation ) , @xmath75 where @xmath72 and @xmath73 are the fourier transform of @xmath10 and @xmath76 respectively . we aim to recover @xmath30 and @xmath24 from @xmath77 [ [ self - calibration - from - multiple - snapshots ] ] self - calibration from multiple snapshots + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + suppose we take measurements of several signals @xmath30 with the same set of design matrix @xmath32 ( i.e. , each sensor corresponds one row of @xmath20 and has an unknown complex - valued calibration term @xmath78 ) . when and how can we recover @xmath24 and @xmath30 simultaneously ? more precisely , we consider the following model of _ self - calibration model from multiple snapshots _ : @xmath79 here @xmath23 is an unknown diagonal matrix , @xmath80 is a sensing matrix , @xmath30 are @xmath81 unknown signals and @xmath82 are their corresponding observations . this multiple snapshots model has been used in image restoration from multiple filters @xcite and self - calibration model for sensors @xcite . throughout our discussion , we assume that @xmath24 is _ invertible _ , and we let @xmath83 . here , @xmath23 stands for the calibration factors of the sensor(s ) @xcite and hence it is reasonable to assume invertibility of @xmath24 . for , if a sensor s gain were equal to zero , then it would not contribute any measurements to the observable @xmath5 , in which case the associated entry of @xmath5 would be zero . but then we could simply discard that entry and consider the correspondingly reduced system of equations , for which the associated @xmath24 is now invertible . one simple solution is to minimize a _ nonlinear _ linear squares objective function . let us take as an example ( the others and have quite similar formulations ) , @xmath84 the obvious difficulty lies in the _ biconvexity _ of , i.e. , if either @xmath24 or @xmath8 is fixed , minimizing over the other variable is a convex program . in general , there is no way to guarantee that any gradient descent algorithm / alternating minimization will give the global minimum . however , for the three models described above , there is one shortcut towards the exact and robust recovery of the solution via _ linear _ least squares if @xmath24 is invertible . we continue with when @xmath85 , i.e. , @xmath86 where @xmath87 with @xmath88 the original measurement equation turns out to be a _ linear _ system with unknown @xmath89 and @xmath8 . the same idea of _ linearization _ can be also found in @xcite . in this way , the ground truth @xmath90 lies actually inside the _ null space _ of this linear system . two issues arise immediately : one the one hand , we need to make sure that @xmath91 spans the whole null space of this linear system . this is equivalent to the identifiability @xcite because if @xmath92 or @xmath93 for @xmath94 is the unique solution ( up to a scalar ) to , then @xmath95 spans the null space of . on the other hand , we also need to avoid the trivial scenario @xmath96 , which does not bear any physical meanings . to resolve the latter issue , we add an extra linear constraint to exclude the trivial scenario , see also @xcite . therefore , we hope that without noise , it suffices to solve the following linear system to recover @xmath97 up to a scalar , i.e. , @xmath98 where the scalar @xmath99 can be any nonzero number . as mentioned previously , the last constraint is to avoid the trivial solution . in the presence of additive noise , we replace the linear system above by a linear least squares problem @xmath100 with respect to @xmath89 and @xmath8 , or equivalently , @xmath101 where @xmath102 , @xmath103 , and @xmath104 is the matrix on the left hand side of . following from the same idea , and can also be reformulated into linear systems and solved via linear least squares . the matrix @xmath104 and the vector @xmath46 take a slightly different form for those cases , see and , respectively . note that solving may not be the optimal choice to recover the unknowns from the perspective of statistics since the noisy perturbation actually enters into @xmath104 instead of @xmath105 . more precisely , the noisy perturbation @xmath106 to the left hand side of the corresponding linear system for , and , is always in the form of @xmath107 the size of @xmath106 depends on the models . hence total least squares @xcite could be a better alternative while it is more difficult to analyze and significantly more costly to compute . since computational efficiency is essential for many practical applications , a straightforward implementation of total least squares is of limited use . instead one should keep in mind that the actual perturbation enters only into @xmath108 , while the other matrix blocks remain unperturbed . constructing a total least squares solution that obeys these constraints , doing so in a numerically efficient manner and providing theoretical error bounds for it , is a rather challenging task , which we plan to address in our future work . numerical simulations imply that the performance under noisy measurements depends on the choice of @xmath109 , especially how much @xmath109 and @xmath110 are correlated . one extreme case is that @xmath111 , which is not able to avoid @xmath112 . it might be better to add a constraint like @xmath113 . however , this will lead to a nonlinear problem which may not be solved efficiently . we present our theoretical findings for the three models , and respectively for different choices of @xmath36 or @xmath114 in one of our choices the @xmath36 are gaussian random matrices . the rationale for this choice is that , while a gaussian random matrix is not useful or feasible in most applications , it often provides a benchmark for theoretical guarantees and numerical performance . our other choices for the sensing matrices are structured random matrices , such as e.g. the product of a deterministic partial ( or a randomly subsampled ) fourier matrix or a hadamard matrix with a diagonal binary random matrix . these matrices bring us closer to what we encounter in real world applications . indeed , structured random matrices of this type have been deployed for instance in imaging and wireless communications , see e.g. @xcite . by solving simple variations ( for different models ) of , we can guarantee that the ground truth is recovered exactly up to a scalar if no noise exists and robustly if noise is present . the number of measurements required for exact and robust recovery is nearly optimal , i.e. , close to the information - theoretic limit up to a poly - log factor . however , the error bound for robust recovery is not optimal . it is worth mentioning that once the signals and calibration parameter @xmath24 are identifiable , we are able to recover both of them exactly in absence of noise by simply solving a linear system . however , identifiability alone can not guarantee robustness . for model we will focus on three cases : a. @xmath36 is an @xmath115 complex gaussian random matrix , i.e. , each entry in @xmath36 is given by @xmath116 . b. @xmath36 is an @xmath115 tall " random dft / hadamard matrix with @xmath117 , i.e. , @xmath118 where @xmath119 consists of the first @xmath120 columns of an @xmath121 dft / hadamard matrix and each @xmath122 is a diagonal matrix with entries taking on the value @xmath123 with equal probability . in particular , there holds , @xmath124 c. @xmath36 is an @xmath115 fat " random partial dft matrix with @xmath125 , i.e. , @xmath118 where @xmath119 consists of @xmath126 columns of an @xmath62 dft / hadamard matrix and each @xmath122 is a diagonal matrix , which is defined the same as case ( b ) , @xmath127 our main findings are summarized as follows : [ thm : main1 ] consider the self - calibration model given in , where @xmath104 is as in . then , for the solution @xmath46 of and @xmath128 , there holds @xmath129 a. with probability @xmath130 if @xmath36 is gaussian and @xmath131 ; b. with probability @xmath132 if each @xmath36 is a tall " @xmath133 random hadamard / dft matrix and @xmath134 ; c. with probability @xmath132 if each @xmath36 is a fat " @xmath135 random hadamard / dft matrix and @xmath136 . here @xmath137 and @xmath138 . our result is nearly optimal in terms of required number of measurements , because the number of constraints @xmath139 is required to be slightly greater than @xmath140 , the number of unknowns . theorem [ thm : main1 ] can be regarded as a generalized result of @xcite , in which @xmath24 is assumed to be positive and @xmath36 is gaussian . in our result , we only need @xmath24 to be an invertible complex diagonal matrix and @xmath36 can be a gaussian or random fourier type matrix . the approaches are quite different , i.e. , @xcite essentially uses nonconvex optimization by first constructing a good initial guess and then applying gradient descent to recover @xmath24 and @xmath8 . our result also provides a provable fast alternative algorithm to the blind deconvolution via random masks " in @xcite where a sdp - based approach is proposed . we now analyze model . following similar steps that led us from to , it is easy to see that the linear system associated with is given by @xmath141 [ s : model2-setup ] we consider two scenarios : a. @xmath36 is an @xmath115 complex gaussian random matrix , i.e. , each entry in @xmath36 yields @xmath116 . b. @xmath36 is of the form @xmath142 where @xmath143 is a random partial hadamard / fourier matrix , i.e. , the columns of @xmath144 are uniformly sampled without replacement from an @xmath121 dft / hadamard matrix ; @xmath145 is a diagonal matrix with @xmath146 being i.i.d . bernoulli random variables . [ thm : main2 ] consider the self - calibration model given in , where @xmath104 is as in . let @xmath147 and @xmath148 . then , for the solution @xmath46 of and @xmath128 , there holds @xmath149 a. with probability at least @xmath150 if @xmath36 is an @xmath115 @xmath151 complex gaussian random matrix and @xmath152 b. with probability at least @xmath153 if @xmath36 yields and @xmath154 note that if @xmath155 i.e. , in the noiseless case , we have @xmath156 if @xmath157 here @xmath139 is the number of constraints and @xmath158 is the degree of freedom . therefore , our result is nearly optimal in terms of information theoretic limit . compared with a similar setup in @xcite , we have a more efficient algorithm since @xcite uses nuclear norm minimization to achieve exact recovery . however , the assumptions are slightly different , i.e. , we assume that @xmath24 is invertible and hence the result depends on @xmath24 while @xcite imposes incoherence " on @xmath7 by requiring @xmath159 relatively small , where @xmath17 denotes fourier transform . we again follow a by now familiar procedure to derive the linear system associated with , which turns out to be @xmath160 for this scenario we only consider the case when @xmath20 is a complex gaussian random matrix . [ thm : main3 ] consider the self - calibration model given in , where @xmath104 is as in . let @xmath147 and @xmath148 . then , for the solution @xmath46 of and @xmath161 there holds @xmath162 with probability at least @xmath163 if @xmath20 is a complex gaussian random matrix and @xmath164 where @xmath165 is the gram matrix of @xmath166 in particular , if @xmath167 and @xmath168 , becomes@xmath169 when @xmath170 , theorem [ thm : main3 ] says that the solution to is uniquely determined up to a scalar if @xmath171 , which involves the norm of gram matrix @xmath165 . this makes sense if we consider two extreme cases : if @xmath172 are exactly the same , then we have @xmath173 and @xmath174 and if @xmath172 are orthogonal to one another , @xmath175 . balzano and nowak @xcite show exact recovery of this model when @xmath20 is a deterministic dft matrix ( discrete fourier matrix ) and @xmath30 are generic signals drawn from a probability distribution , but their results do not include stability theory in the presence of noise . for theorem [ thm : main2 ] and theorem [ thm : main3 ] it does not come as a surprise that the error bound depends on the norm of @xmath30 and @xmath24 as well as on how much @xmath110 and @xmath109 are correlated . we can not expect a relatively good condition number for @xmath104 if @xmath176 varies greatly over @xmath177 . concerning the correlation between @xmath110 and @xmath109 , one extreme case is @xmath178 , which does not rule out the possibility of @xmath112 . hence , the quantity @xmath179 affects the condition number . this section is devoted to numerical simulations . four experiments for both synthetic and real data will be presented to address the effectiveness , efficiency and robustness of the proposed approach . for all three models presented , and , the corresponding linear systems have simple block structures which allow fast implementation via the conjugate gradient method for non - hermitian matrices @xcite . in our simulations , we do not set up @xmath104 explicitly . the iteration stops if either the number of iterations reaches at most @xmath180 or the residual of the corresponding normal equation is smaller than @xmath181 . therefore , we are able to deal with problems from imaging processing without much computational issue . throughout our discussion , the @xmath182 ( signal - to - noise ratio ) in the scale of db is defined as @xmath183 we measure the performance with @xmath184 where @xmath185 which is actually equal to @xmath186 here @xmath187 and @xmath70 are the ground truth . in particular , for image examples , we only measure the relative error with respect to the recovered image @xmath188 , i.e. , @xmath189 suppose we have a target image @xmath8 and try to estimate @xmath8 through multiple measurements . however , the sensing process is not perfect because of the missing calibration of the sensors . in order to estimate both the unknown gains and phases as well as the target signal , a randomized sensing procedure is used by employing several random binary masks . we assume that @xmath190 where @xmath119 is a tall " low - frequency dft matrix , @xmath191 is a diagonal @xmath123-random matrix and @xmath70 is an image of size @xmath192 we set @xmath193 and @xmath194 with @xmath195 ; the oversampling ratio is @xmath196 . we compare two cases : ( i ) @xmath7 is a sequence distributed uniformly over @xmath197 $ ] with @xmath198 , and ( ii ) @xmath7 is a steinhaus sequence ( uniformly distributed over the complex unit circle ) with @xmath199 . we pick those choices of @xmath109 because we know that the image we try to reconstruct has only non - negative values . thus , by choosing @xmath109 to be non - negative , there are fewer cancellation in the expression @xmath200 , which in turn leads to a smaller condition number and better robustness . the corresponding results of our simulations are shown in figure [ fig : pos - repeat-1 ] and figure [ fig : pos - repeat-2 ] , respectively . in both cases , we only measure the relative error of the recovered image . 0.5 cm in figure [ fig : pos - repeat-1 ] , we can see that both uncalibrated / calibrated image are quite good . here the uncalibrated image is obtained by first applying the inverse fourier transform and the inverse of the mask to each @xmath201 and then taking the average of @xmath202 samples . we explain briefly why the uncalibrated image still looks good . note that @xmath203 where @xmath204 is the pseudo inverse of @xmath205 here @xmath206 is actually a diagonal matrix with random entries @xmath207 . as a result , each @xmath208 is the sum of @xmath126 rank-1 matrices with random @xmath209 coefficients and is relatively small due to many cancellations inside it . moreover , @xcite showed that most 2-d signals can be reconstructed within a scale factor from only knowing the phase of its fourier transform , which applies to the case when @xmath7 is positive . however , when the unknown calibration parameters are complex variables ( i.e. , we do not know much about the phase information ) , figure [ fig : pos - repeat-2 ] shows that the uncalibrated recovered image is totally meaningless . our approach still gives a quite satisfactory results even at a relatively low snr of 5db . the second experiment is about blind deconvolution in random mask imaging @xcite . suppose we observe the convolution of two components , @xmath210 where both , the filter @xmath21 and the signal of interests @xmath70 are unknown . each @xmath191 is a random @xmath123-mask . the blind deconvolution problem is to recover @xmath211 . moreover , here we assume that the filter is actually a low - pass filter , i.e. , @xmath212 is compactly supported in an interval around the origin , where @xmath17 is the fourier transform . after taking the fourier transform on both sides , the model actually ends up being of the form with @xmath213 where @xmath119 is a fat " partial dft matrix and @xmath7 is the nonzero part of @xmath212 . in our experiment , we let @xmath70 be a @xmath214 image and @xmath215 be a 2-d gaussian filter of size @xmath216 as shown in figure [ fig : random - mask-1 ] . 0.5 cm 0.5 cm in those experiments , we choose @xmath217 since both @xmath7 and @xmath70 are nonnegative . figure [ fig : random - mask-2 ] shows the recovered image from @xmath218 sets of noiseless measurements and the performance is quite satisfactory . here the oversampling ratio is @xmath219 we can see from figure [ fig : random - mask-3 ] that the blurring effect has been removed while the noise still exists . that is partially because we did not impose any denoising procedure after the deconvolution . we choose @xmath36 to be random hadamard matrices with @xmath220 and @xmath221 and @xmath222 with @xmath187 being a positive / steinhaus sequence , as we did previously . each @xmath31 is sampled from standard gaussian distribution . we choose @xmath223 if @xmath187 is uniformly distributed over @xmath224 $ ] and @xmath225 for steinhaus @xmath226 100 simulations are performed for each level of @xmath182 . the test is also given under different choices of @xmath202 . the oversampling ratio @xmath227 is @xmath228 , @xmath229 and @xmath230 for @xmath231 respectively . from figure [ fig : diverse ] , we can see that the error scales linearly with snr in db . the performance of steinhaus @xmath187 is not as good as that of positive @xmath187 for snr @xmath232 . that is because @xmath200 is much larger when @xmath187 and @xmath109 are both nonnegative , which gives better condition number . where @xmath233 , @xmath222 and each @xmath36 is a random hadamard matrix @xmath187 is a steinhaus sequence.,width=257 ] where @xmath233 , @xmath222 and each @xmath36 is a random hadamard matrix . @xmath187 is a steinhaus sequence.,width=257 ] we make a comparison of performances between @xmath187 as a positive or steinhaus sequence with a gaussian random matrix @xmath20 . each @xmath31 is sampled from the standard gaussian distribution and hence the underlying gram matrix @xmath165 is quite close to @xmath234 ( this closeness could be easily made more precise , but we refrain doing so here ) . the choice of @xmath109 and oversampling ratio are the same as those in section [ s : numerics - diverse ] . from figure [ fig : multiple ] , we see that the performance for positive @xmath187 is better than that for the steinhaus case , especially in the lower snr regime ( snr @xmath232 ) . the reason is the low correlation between @xmath109 and @xmath110 when @xmath187 is steinhaus and @xmath235 where @xmath233 , @xmath222 and @xmath20 is a gaussian random matrix . left : @xmath187 is a random vector with each entry uniformly distributed over @xmath224 $ ] ; right : @xmath187 is a random vector with each entry uniformly distributed over unit circle.,width=283 ] where @xmath233 , @xmath222 and @xmath20 is a gaussian random matrix . left : @xmath187 is a random vector with each entry uniformly distributed over @xmath224 $ ] ; right : @xmath187 is a random vector with each entry uniformly distributed over unit circle.,width=283 ] for each subsection , we will first give the result of noiseless measurements . we then prove the stability theory by using the result below . @xcite[prop : perturb ] suppose that @xmath236 is a consistent and overdetermined system . denote @xmath237 as the least squares solution to @xmath238 with @xmath239 . if @xmath240 , there holds , @xmath241 to apply the proposition above , it suffices to bound @xmath242 and @xmath243 . let us start with when @xmath85 and denote @xmath244 and @xmath245 , @xmath246 then we rewrite @xmath247 as @xmath248 where @xmath249 by definition , @xmath250 our goal is to find out the smallest and the largest eigenvalue of @xmath104 . actually it suffices to understand the spectrum of @xmath251 . obviously , its smallest eigenvalue is zero and the corresponding eigenvector is @xmath252 let @xmath253 be the @xmath254-th column of @xmath255 and we have @xmath256 under all the three settings in section [ s : model1-setup ] . hence , @xmath257 it is easy to see that @xmath258 and the null space of @xmath259 is spanned by @xmath260 . @xmath259 has an eigenvalue with value @xmath63 of multiplicity @xmath261 and an eigenvalue with value @xmath228 of multiplicity @xmath63 . more importantly , the following proposition holds and combined with proposition [ prop : perturb ] , we are able to prove theorem [ thm : main1 ] . [ prop : main1 ] there holds @xmath262 a. with probability @xmath130 if @xmath36 is gaussian and @xmath131 ; b. with probability @xmath132 if each @xmath36 is a tall " @xmath133 random hadamard / dft matrix and @xmath134 ; c. with probability @xmath132 if each @xmath36 is a fat " @xmath135 random hadamard / dft matrix and @xmath136 . [ rmk : prop1 ] proposition [ prop : main1 ] actually addresses the identifiability issue of the model in absence of noise . more precisely , the invertibility of @xmath263 is guaranteed by that of @xmath24 . by weyl s theorem for singular value perturbation in @xcite , @xmath264 eigenvalues of @xmath251 are greater than @xmath265 hence , the rank of @xmath266 is equal to @xmath267 if @xmath202 is close to the information theoretic limit under the conditions given above , i.e. , @xmath268 . in other words , the null space of @xmath266 is completely spanned by @xmath269 . note that proposition [ prop : main1 ] gives the result if @xmath85 . the noisy counterpart is obtained by applying perturbation theory for linear least squares . let @xmath270 where @xmath271 is the noiseless part and @xmath106 is defined in . without loss of generality , we assume that @xmath272 according to proposition [ prop : perturb ] , it suffices to estimate the condition number @xmath273 of @xmath271 and @xmath274 . note that @xmath275 where @xmath276 from proposition [ prop : main1 ] and theorem 1 in @xcite , we know that @xmath277 where @xmath278 and @xmath279 following from , we have @xmath280 on the other hand , @xmath281 follows from proposition [ prop : main1 ] . in other words , we have found the lower and upper bounds for @xmath282 or equivalently , @xmath283 now we proceed to the estimation of @xmath284 let @xmath285 be a unit vector , where @xmath286 with @xmath287 and @xmath288 are the eigenvectors of @xmath251 . then the smallest eigenvalue of @xmath289 defined in has a lower bound as follows : @xmath290 which implies @xmath291 . combined with @xmath292 , @xmath293 therefore , with , the condition number of @xmath294 is bounded by @xmath295 from , @xmath243 is bounded by @xmath296 applying proposition [ prop : perturb ] gives the following upper bound of the estimation error @xmath297 which gives theorem [ thm : main1 ] . * [ proof of proposition [ prop : main1](a ) ] * from now on , we assume @xmath298 , i.e. , the @xmath254-th column of @xmath255 , obeys a complex gaussian distribution , @xmath299 . let @xmath300 be the @xmath254-th column of @xmath301 ; it can be written in explicit form as @xmath302 denoting @xmath303 , we obtain @xmath304 obviously each @xmath305 is independent . in order to apply theorem [ thm : bern1 ] to estimate @xmath306 , we need to bound @xmath307 and @xmath308 . due to the semi - definite positivity of @xmath309 , we have @xmath310 and hence @xmath311 this implies @xmath312 now we consider @xmath313 by computing @xmath314 and @xmath315 , i.e. , the @xmath316-th and @xmath317-th block of @xmath318 , @xmath319 { \boldsymbol{e}}_i{\boldsymbol{e}}_i^ * , \\ ( { \mathcal{z}}_{l , i}{\mathcal{z}}_{l , i}^*)_{2,2 } & = & \frac{1}{m } ( { \boldsymbol{a}}_{l , i}{\boldsymbol{a}}_{l , i}^ * - { \boldsymbol{i}}_n){\boldsymbol{v}}{\boldsymbol{v}}^*({\boldsymbol{a}}_{l , i}{\boldsymbol{a}}_{l , i}^ * - { \boldsymbol{i}}_n ) + \frac{1}{m^2}({\boldsymbol{a}}_{l , i}{\boldsymbol{a}}_{l , i}^ * - { \boldsymbol{i}}_n)^2.\end{aligned}\ ] ] following from , , and lemma [ lem : pos ] , there holds @xmath320 by applying the matrix bernstein inequality ( see theorem [ berngaussian ] ) we obtain @xmath321 with probability @xmath322 . in particular , by choosing @xmath323 , i.e , @xmath324 the inequality above holds with probability @xmath325 * [ proof of proposition [ prop : main1](b ) ] * each @xmath301 is independent by its definition in if @xmath118 where @xmath119 is an @xmath115 partial dft / hadamard matrix with @xmath117 and @xmath326 and @xmath327 is a diagonal random binary @xmath123 matrix . let @xmath328 ; in explicit form @xmath329 where @xmath330 follows from the assumption . first we take a look at the upper bound of @xmath331 it suffices to bound @xmath332 since @xmath333 and @xmath259 is positive semi - definite . on the other hand , due to lemma [ lem : pos ] , we have @xmath334 and hence we only need to bound @xmath335 . for @xmath335 , there holds @xmath336 also for any pair of @xmath337 , @xmath338 can be rewritten as @xmath339 where @xmath340 is the @xmath254-th column of @xmath341 and @xmath342 . then there holds @xmath343 where the third inequality follows from lemma [ lem : rade ] . applying lemma [ lem : pos ] to @xmath344 , @xmath345 with probability at least @xmath346 . denote the event @xmath347 by @xmath348 . now we try to understand @xmath349 the @xmath316-th and @xmath317-th block of @xmath350 are given by @xmath351 by using , and @xmath352 , we have @xmath353 combining , , and lemma [ lem : pos ] , @xmath354 by applying with @xmath323 and @xmath355 over event @xmath348 , we have @xmath356 with probability @xmath132 if @xmath134 . * [ proof of proposition [ prop : main1](c ) ] * each @xmath301 is independent due to . let @xmath328 ; in explicit form @xmath357 here @xmath213 where @xmath119 is a fat " @xmath358 partial dft / hadamard matrix satisfying @xmath359 and @xmath191 is a diagonal @xmath360-random matrix . there holds @xmath361 where @xmath362 and @xmath363 for each @xmath364 , @xmath365 hence , there holds , @xmath366 with probability at least @xmath367 , which follows exactly from and lemma [ lem : pos ] . now we give an upper bound for @xmath368 . the @xmath316-th and @xmath317-th block of @xmath350 are given by @xmath369 by using , and @xmath370 , we have @xmath371 for @xmath372 , we have @xmath373 where @xmath374 note that @xmath375 , and there holds , @xmath376 combining , , , and lemma [ lem : pos ] , @xmath377 by applying with @xmath323 , we have @xmath378 with probability @xmath132 if @xmath136 . we start with by setting @xmath85 . in this way , we can factorize the matrix @xmath104 ( excluding the last row ) into @xmath379 where @xmath380 is the normalized @xmath31 , @xmath177 . we will show that the matrix @xmath38 is of rank @xmath381 to guarantee that the solution is unique ( up to a scalar ) . denote @xmath382 and @xmath383 with @xmath384 where @xmath66 is a standard orthonormal basis in @xmath68 . the proof of theorem [ thm : main2 ] relies on the following proposition . we defer the proof of proposition [ prop : main2 ] to the sections [ s : model2-a ] and [ s : model2-b ] . [ prop : main2 ] there holds , @xmath385 a. with probability at least @xmath150 with @xmath386 if @xmath36 is an @xmath115 @xmath151 complex gaussian random matrix and @xmath387 b. with probability at least @xmath388 with @xmath386 if @xmath36 yields and @xmath389 [ rmk : prop2 ] note that @xmath259 has one eigenvalue equal to 0 and all the other eigenvalues are at least 1 . hence the rank of @xmath259 is @xmath390 . similar to remark [ rmk : prop1 ] , proposition [ prop : main2 ] shows that the solution @xmath391 to is uniquely identifiable with high probability when @xmath392 and @xmath393 [ * proof of theorem [ thm : main2 ] ] * from , we let @xmath394 where @xmath271 is the noiseless part of @xmath104 . now , gives @xmath395 define @xmath396 and @xmath397 . from proposition [ prop : main2 ] and theorem 1 in @xcite , we know that the eigenvalues @xmath398 of @xmath399 fulfill @xmath400 and @xmath401 for @xmath402 since @xmath403 ; and the eigenvalues of @xmath259 are 0 , 1 and 2 with multiplicities 1 , @xmath404 , @xmath63 respectively . the key is to obtain a bound for @xmath405 from , @xmath406 where @xmath407 on the other hand , gives @xmath408 since @xmath409 . for @xmath410 , @xmath411 denote @xmath412 such that @xmath413 and @xmath414 where @xmath415 . by using the same procedure as , @xmath416 where @xmath417 with @xmath418 . since @xmath419 the smallest eigenvalue of @xmath420 yields , @xmath421 combining and leads to @xmath422 applying proposition [ prop : perturb ] and @xmath423 , we have @xmath424 which finishes the proof of theorem [ thm : main2 ] . in this section , we will prove that proposition [ prop : main2](a ) if @xmath425 where @xmath298 is the @xmath254-th column of @xmath255 . before moving to the proof , we compute a few quantities which will be used later . define @xmath300 as the @xmath426-th column of @xmath427 @xmath428 where @xmath429 and @xmath430 are standard orthonormal basis in @xmath67 and @xmath68 respectively ; @xmath69 " denotes kronecker product . by definition , we have @xmath431 and all @xmath300 are independent from one another . @xmath432 and its expectation is equal to @xmath433 it is easy to verify that @xmath434 . [ * proof of proposition [ prop : main2](a ) ] * the tool is to use apply matrix bernstein inequality in theorem [ berngaussian ] . note that @xmath435 let @xmath436 and we have @xmath437 since @xmath438 follows from lemma [ lem : pos ] . therefore , the exponential norm of @xmath439 is bounded by @xmath440 and as a result @xmath441 now we proceed by estimating the variance @xmath442 . we express @xmath305 as follows : @xmath443 the @xmath316-th and the @xmath317-th block of @xmath318 are @xmath444 and @xmath445.\ ] ] following from , and , we have @xmath446 due to lemma [ lem : pos ] , there holds , @xmath447 by applying , @xmath448 with @xmath449 , there holds @xmath450 with probability at least @xmath150 if @xmath451 we prove proposition [ prop : main2 ] based on assumption . denote @xmath253 and @xmath452 as the @xmath254-th column of @xmath255 and @xmath453 and obviously we have @xmath454 denote @xmath455 and let @xmath301 be the @xmath364-th block of @xmath50 in , i.e. , @xmath456 with @xmath330 , we have @xmath457 where the expectation of @xmath254-th row of @xmath458 yields @xmath459 . hence @xmath460 its expectation equals @xmath461 [ * proof of proposition [ prop : main2](b ) ] * note that each block @xmath301 is independent and we want to apply bernstein inequality to achieve the desired result . let @xmath462 and by definition , we have @xmath463 note that @xmath464 since implies that @xmath465 , we have @xmath466 with probability at least @xmath346 . we proceed with estimation of @xmath467 by looking at the @xmath316-th and @xmath317-th block of @xmath350 , i.e. , @xmath468 note that @xmath469 . the @xmath254-th diagonal entry of @xmath470 is @xmath471 where @xmath472 and implies @xmath473 since @xmath474 is still a unit vector ( note that @xmath475 is unitary since @xmath452 is a column of @xmath453 ) . therefore , @xmath476 by using lemma [ lem : laal ] , we have @xmath477 with @xmath478 and independence between @xmath479 and @xmath191 , we have @xmath480 where @xmath481 follows from and @xmath482 by using , , and lemma [ lem : pos ] , @xmath483 is bounded above by @xmath484 conditioned on the event @xmath485 , applying with @xmath486 gives @xmath487 with probability at least @xmath388 and it suffices to require @xmath488 recall that @xmath104 in and the only difference from is that here all @xmath36 are equal to @xmath20 . if @xmath85 , @xmath104 ( excluding the last row ) can be factorized into @xmath489 where @xmath490 is the normalized @xmath491 . before we proceed to the main result in this section we need to introduce some notation . let @xmath492 be the @xmath254-th column of @xmath493 , which is a complex gaussian random matrix ; define @xmath494 to be a matrix whose columns consist of the @xmath254-th column of each block of @xmath50 , i.e. , @xmath495 where @xmath69 " denotes kronecker product and both @xmath429 and @xmath430 are the standard orthonormal basis in @xmath67 and @xmath68 , respectively . in this way , the @xmath496 are independently from one another . by definition , @xmath497 where @xmath498 and @xmath499 with @xmath500 the expectation of @xmath501 is given by @xmath502 0.25 cm our analysis depends on the _ mutual coherence _ of @xmath172 . one can not expect to recover all @xmath172 and @xmath24 if all @xmath172 are parallel to each other . let @xmath165 be the gram matrix of @xmath172 with @xmath503 , i.e. , @xmath504 and @xmath505 and in particular , @xmath506 . its eigenvalues are denoted by @xmath507 with @xmath508 . basic linear algebra tells that @xmath509 where @xmath510 is unitary and @xmath511 let @xmath512 , then there holds @xmath513 since @xmath514 here @xmath515 and @xmath516 . in particular , if @xmath517 , then @xmath167 ; if @xmath518 for all @xmath519 , then @xmath520 and @xmath521 we are now ready to state and prove the main result in this subsection . [ prop : main3 ] there holds @xmath522 with probability at least @xmath163 if @xmath523 and each @xmath524 is i.i.d . complex gaussian , i.e. , @xmath525 . in particular , if @xmath167 and @xmath168 , becomes @xmath169 the proof of theorem [ thm : main3 ] follows exactly from that of theorem [ thm : main2 ] when proposition [ prop : main3 ] holds . hence we just give a proof of proposition [ prop : main3 ] . [ * proof of proposition [ prop : main3 ] ] * let @xmath526 , where @xmath527 and @xmath528 are defined as @xmath529 [ [ estimation - of - sum_i1mmathcalz_i1 ] ] estimation of @xmath530 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + following from , we have @xmath531 where @xmath532 is a @xmath533 matrix with orthonormal rows and hence each @xmath534 is rayleigh distributed . ( we say @xmath535 is rayleigh distributed if @xmath536 where both @xmath537 and @xmath538 are standard real gaussian variables . ) due to the simple form of @xmath527 , it is easy to see from bernstein s inequality that @xmath539 with probability @xmath540 here @xmath541 where @xmath542 . therefore , @xmath543 now we only need to find out @xmath544 and @xmath545 in order to bound @xmath546 . [ [ estimation - of - mathcalz_i2-_psi_1 ] ] estimation of @xmath547 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + denote @xmath548 and there holds , @xmath549 for @xmath550 , its @xmath551-norm is bounded by @xmath552 @xmath553 let @xmath554 and @xmath555 . the @xmath551-norm of @xmath556 yields @xmath557 where the second inequality follows from the cauchy - schwarz inequality , @xmath558 , and @xmath559 . therefore , @xmath560 and there holds @xmath561 [ [ estimation - of - sigma_02 ] ] estimation of @xmath483 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + note that @xmath562 . let @xmath563 and @xmath564 be the @xmath316-th and @xmath317-th block of @xmath565 respectively , i.e. , @xmath566_{1\leq k\leq l\leq p } + \frac{1}{m^2 } { \boldsymbol{i}}_p\otimes ( { \boldsymbol{a}}_i{\boldsymbol{a}}_i^ * - { \boldsymbol{i}}_n)^2 . \label{eq : ai22 } \ ] ] since @xmath565 is a positive semi - definite matrix , lemma [ lem : pos ] implies @xmath567 so we only need to compute @xmath568 and @xmath569 . @xmath570 where @xmath571 now we have @xmath572 for @xmath573 note that @xmath574 which follows from . by , and lemma [ lem : pos ] , there holds , @xmath575 applying to @xmath576 with and , we have @xmath577 with probability @xmath322 . by combining with and letting @xmath578 , we have @xmath579 with probability @xmath163 if @xmath580 or equivalently , @xmath581 [ lem : pos ] for any hermitian positive semi - definite matrix @xmath582 with @xmath583 , there holds , @xmath584 in other words , @xmath585 . [ thm : bern1 ] consider a finite sequence of @xmath344 of independent centered random matrices with dimension @xmath590 . we assume that @xmath591 and introduce the random matrix @xmath592 compute the variance parameter @xmath593 then for all @xmath594 @xmath595 with probability at least @xmath322 where @xmath596 is an absolute constant . [ berngaussian ] for a finite sequence of independent @xmath590 random matrices @xmath344 with @xmath601 and @xmath483 as defined in , we have the tail bound on the operator norm of @xmath600 , @xmath602 with probability at least @xmath322 where @xmath596 is an absolute constant . suppose that @xmath608 is a rademacher sequence and for any fixed @xmath605 and @xmath606 , there holds @xmath609 where @xmath610 is an @xmath62 matrix with only one nonzero entry equal to 1 and at position @xmath611 in particular , setting @xmath612 gives @xmath613 since @xmath614 is a rademacher sequence , i.e , each @xmath615 takes @xmath123 independently with equal probability , this implies @xmath616 and @xmath617 therefore , @xmath618 the @xmath619-th entry of @xmath620 is @xmath621 . [ lem : laal ] there holds @xmath629 where @xmath630 , @xmath631 and @xmath632 is a deterministic unit vector . @xmath362 is a random partial fourier / hadamard matrix with @xmath326 and @xmath117 , i.e. , the columns of @xmath119 are uniformly sampled without replacement from an @xmath121 dft / hadamard matrix ; @xmath633 is a diagonal matrix with entries random sampled from @xmath123 with equal probability ; moreover , we assume @xmath633 and @xmath119 are independent from each other . in particular , if @xmath634 , @xmath635 let @xmath637 be the @xmath254-th column of @xmath341 and the @xmath638-th entry of @xmath639 is @xmath640 where @xmath641 . the randomness of @xmath642 comes from both @xmath119 and @xmath633 and we first take the expectation with respect to @xmath633 . @xmath643 where @xmath644 follows from each entry in @xmath633 being a bernoulli random variable . hence , @xmath645 let @xmath646 be the @xmath647-th column of @xmath119 and @xmath648 and @xmath649 " denotes the hadamard ( pointwise ) product . so we have @xmath650 there holds , @xmath651 where the third equation follows from linearity of the hadamard product and from @xmath652 the last one uses the fact that @xmath653 if @xmath646 is a vector from the dft matrix or hadamard matrix . by the property of conditional expectation , we know that @xmath654 and due to the linearity of expectation , it suffices to find out for @xmath626 , @xmath655 where @xmath646 and @xmath656 , by definition , are the @xmath647-th and @xmath364-th columns of @xmath119 which are sampled uniformly without replacement from an @xmath121 dft matrix @xmath657 . note that @xmath658 is actually an _ ordered _ pair of random vectors sampled without replacement from columns of @xmath657 . hence there are in total @xmath659 different choices of @xmath660 and @xmath661 where @xmath662 is defined as the @xmath254-th column of an @xmath121 dft matrix @xmath657 . now we have , for any @xmath626 , @xmath663 where @xmath664 and @xmath665 now we return to @xmath666 . by substituting into , we end up with @xmath667 where @xmath668 follows from @xmath669 a. ahmed , a. cosse , and l. demanet . a convex approach to blind deconvolution with diverse inputs . in _ computational advances in multi - sensor adaptive processing ( camsap ) , 2015 ieee 6th international workshop on _ , pages 58 . ieee , 2015 . s. curtis , j. lim , and a. oppenheim . signal reconstruction from one bit of fourier transform phase . in _ acoustics , speech , and signal processing , ieee international conference on icassp84 . _ , volume 9 , pages 487490 . ieee , 1984 . r. gribonval , g. chardon , and l. daudet . blind calibration for compressed sensing by convex optimization . in _ acoustics , speech and signal processing ( icassp ) , 2012 ieee international conference on _ , pages 27132716 . ieee , 2012 . j. shin , p. e. larson , m. a. ohliger , m. elad , j. m. pauly , d. b. vigneron , and m. lustig . calibrationless parallel imaging reconstruction based on structured low - rank matrix completion . , 72(4):959970 , 2014 . r. vershynin . introduction to the non - asymptotic analysis of random matrices . in y. c. eldar and g. kutyniok , editors , _ compressed sensing : theory and applications _ , chapter 5 . cambridge university press , 2012 .

whenever we use devices to take measurements , calibration is indispensable . while the purpose of calibration is to reduce bias and uncertainty in the measurements , it can be quite difficult , expensive and sometimes even impossible to implement . we study a challenging problem called _ self - calibration _ , i.e. , the task of designing an algorithm for devices so that the algorithm is able to perform calibration automatically . more precisely , we consider the setup @xmath0 where only partial information about the sensing matrix @xmath1 is known and where @xmath1 linearly depends on @xmath2 . the goal is to estimate the calibration parameter @xmath2 ( resolve the uncertainty in the sensing process ) and the signal / object of interests @xmath3 simultaneously . for three different models of practical relevance we show how such a _ bilinear _ inverse problem , including blind deconvolution as an important example , can be solved via a simple _ linear least squares _ approach . as a consequence , the proposed algorithms are numerically extremely efficient , thus allowing for real - time deployment . explicit theoretical guarantees and stability theory are derived and the number of sampling complexity is nearly optimal ( up to a poly - log factor ) . applications in imaging sciences and signal processing are discussed and numerical simulations are presented to demonstrate the effectiveness and efficiency of our approach .

the realization of bec in ultracold atomic gases [ 1 - 3 ] has opened a wide area of research in atomic physics , where quantum - statistical effects are of crucial importance : upon cooling , bosonic atoms in a trap condense into a superfluid state at a rather sharply defined critical temperature @xmath0 while fermionic atoms continuously evolve into a degenerate noninteracting gas , resisting spatial compression due to their fermi pressure [ 4 ] . both features are a consequence of the purely statistical interaction between the atoms . by contrast , the actual interparticle potential plays only a comparatively minor role . to see this , we start from the standard pseudopotential description , which replaces the complicated interatomic potential by an effective contact interaction of the form @xmath1 containing the exact s - wave scattering length @xmath2 as the only parameter . for identical fermions , there is no s - wave scattering due to the pauli - principle and thus we obtain an ideal fermi gas to lowest order . for bosons , in turn , @xmath2 is finite , however at a given density @xmath3 the importance of direct interaction effects can be estimated from the ratio @xmath4 between the interaction and the kinetic energy per particle . now the average interparticle spacing @xmath5 is usually much larger than the scattering length and thus @xmath6 is very small , with typical values around @xmath7 . this puts us into the weak coupling limit where the many body ground state of @xmath8 bosons is well approximated by a simple product [ 5 ] @xmath9 in which all atoms are in the identical single particle state @xmath10 . taking ( 3 ) as a variational ansatz , the optimal macroscopic wave function @xmath10 is found to obey the well known gross - pitaevski equation . it describes - even on a quantitative level - a wealth of remarkable and nontrivial properties of trapped condensates from interference between different condensates [ 6 ] to collective modes [ 7 ] or vortices [ 8 ] . from a many body point of view , the effective single particle or hartree - description ( 3 ) is of course the simplest of all possible cases , containing no interaction induced correlations between different atoms at all . a first step to go beyond this mean field description is the well known bogoliubov theory . this is usually introduced by considering small fluctuations around the gross - pitaevski equation in a systematic expansion in the number of noncondensed particles [ 9 ] . as emphasized , for example , by leggett [ 5 ] , it is more instructive from a many body point of view to formulate bogoliubov theory in such a way that the many body boson ground state is approximated by an optimized product @xmath11 of identical , symmetric two particle wave functions @xmath12 . this allows us to build in the interaction beyond the mean field potential by suppressing configurations in which particles @xmath13 and @xmath14 are close together . the many body state thus incorporates two - particle correlations which are important e.g. to obtain the standard sound modes and the related coherent superposition of particle and hole excitations [ 10 ] . however , even the bogoliubov description is restricted to the regime @xmath15 , where interactions lead only to a small depletion of the condensate at zero temperature . the associated ground state is again characterized by a macroscopic matter wave field and continuously evolves from that of a noninteracting gas . an obvious way to go beyond the weak coupling regime is to increase the dimensionless interaction strength parameter @xmath6 by increasing the scattering length via a feshbach resonance . of course , this method is limited by the fact that the associated condensate lifetime strongly decreases due to three - body losses which occur at a rate [ 11 ] @xmath16 in spite of this problem , this method of reaching the strong coupling regime has been followed quite successfully recently , see e.g. [ 12 ] and [ 13 ] . in the following , we will discuss an alternative route to reach strong coupling even at small values of @xmath17 . it is based on confining cold atoms in the periodic potential of an optical lattice generated via the dipole force which atoms experience in a standing , off - resonant light field . depending on the sign of the polarizability , the atoms are attracted either to the nodes or the anti - nodes of the laser intensity . in this manner , one- , two- , or three - dimensional lattices can be created with a lattice constant @xmath18 which is half the laser wavelength ( typically @xmath19 is in the range between @xmath20 m and @xmath21 m ) . in the simplest case , three orthogonal , independent standing laser fields with wave vector @xmath22 produce a separable 3d lattice potential @xmath23 with a tunable amplitude @xmath24 . a convenient measure for the strength @xmath24 of the lattice potential is the recoil energy @xmath25 which is typically in the few khz range . in a deep optical lattice with @xmath26 , the energy @xmath27 of local oscillations in the well is much larger than the recoil energy and each well supports many quasi - bound states . for instance in the deepest lattices generated in the recent experiments by greiner et.al [ 14 ] with @xmath28 this number is around four while the local oscillation frequency has reached @xmath29khz . provided all the atoms are in the lowest vibrational level at each site , their motion is frozen except for the small tunneling amplitude to neighbouring sites . the atoms are then effectively confined to move in the lowest band of the lattice . with @xmath30 as the states localized at site @xmath31 , the appropriate single particle eigenstates are bloch - waves @xmath32 with quasimomentum @xmath33 and energy @xmath34 the bandwidth parameter @xmath35 is essentially the gain in kinetic energy due to nearest neighbour tunneling . in the limit @xmath26 it can be obtained from the exact result for the width of the lowest band in the 1d mathieu - equation @xmath36 obviously , in a lattice , it is @xmath35 which plays the role of the kinetic energy per particle in the homogeneous case . the effective value of @xmath37 is therefore very large in optical lattices , increasing exponentially with @xmath38 . thus it is the quenching of the kinetic energy for motion in the lowest band which drives cold atoms into the strong coupling regime , even though @xmath17 may still be much smaller than one . alternatively , one may argue that @xmath39 becomes small because atoms in a deep optical lattice have an exponentially large effective mass @xmath40 . to really obtain interesting many body effects it is of course necessary to have the interaction and the kinetic energy of the same order . this requires optical lattices in which the number of atoms per site is of order one or larger , a regime which has been possible to reach only recently with bose - einstein - condensates . in the following , i will discuss the mott - hubbard transition for bosonic atoms , as a generic example illustrating how cold atoms in optical lattices can be used to study genuine many body phenomena in dilute gases . the original idea suggesting the possibility of nontrivial many body states using cold atoms in optical lattices is due to jaksch et.al . [ their starting point is the so called bose - hubbard model , originally introduced by fisher et.al . in a rather different context [ 16 ] . it describes bosons hopping with amplitude @xmath35 to nearest neighbors on a regular lattice of sites @xmath41 . the particles interact with a zero - range , on - site repulsion @xmath42 , disfavouring configurations with more than one atom at a given site . with @xmath43 as the creation operator of a boson at site @xmath41 and @xmath44 the associated number operator , the hamiltonian reads @xmath45 here @xmath46 denotes a sum over nearest neighbour pairs , including double counting . the last term with a variable on - site energy @xmath47 is introduced to describe the effect of the trapping potential and acts like a spatially varying chemical potential . the form of the interaction term is precisely that obtained by viewing each site as a local condensate with a gross - pitaevski mean field potential . the relevant interaction parameter @xmath42 is thus given by an integral over the on - site wave function @xmath48 via @xmath49 the explicit result is obtained by taking @xmath48 as the gaussian ground state in the local oscillator potential around any of the sites . more precisely , @xmath48 is the exact wannier wave function of the lowest band . in a separable periodic potential like that in ( 6 ) , @xmath48 decays exponentially in all directions rather than in a gaussian manner [ 17 ] , however this does not seriously affect the calculation of @xmath42 in the deep lattice limit @xmath26 . from eqn.(10 ) , it is obvious that the strength of the repulsion increases with @xmath24 due to the tighter squeezing of the on - site wave function @xmath48 . with increasing @xmath38 , therefore , not only does the kinetic energy drop exponentially , but at the same time the interaction energy increases . as a result , it is possible to reach the strong coupling regime @xmath50 simply by increasing the depth of the optical lattice potential . regarding the requirements necessary for the validity of the discrete lattice model ( 9 ) , it is obvious that the atoms have to remain in the lowest vibrational state at each site even in the presence of strong interactions . we thus require that @xmath51 which is well obeyed even in deep optical lattices as long as @xmath52 . the zero temperature phase diagram of the homogeneous bose - hubbard model was first discused by fisher et.al . although there are considerable quantitative differences between the case of one- , two- or three - dimensional lattices , the qualitative structure is similar in all cases and is shown schematically in figure 1 . at large @xmath53 the kinetic energy dominates and the ground state is a delocalized superfluid ( sf ) . at small values of @xmath53 , interactions dominate and one obtains a series of so called mott - insulating ( mi ) phases with fixed integer filling @xmath54 depending on the value of the chemical potential @xmath55 . to understand the peculiar structure of these mott - lobes , consider first the case of unit filling , i.e. the number @xmath8 of atoms is precisely equal to the number @xmath56 of lattice sites . in the limit where @xmath24 is very large compared to @xmath57 , there is no hopping ( @xmath58 ) and the obvious ground state in the superfluid phase ( sf ) hit the corresponding mott - insulating ( mi ) phases at the tips of the lobes at a critical value of @xmath53 , which decreases with density @xmath59 . for @xmath60 the line of constant density stays outside the @xmath61 mi because a fraction @xmath62 of the particles remains superfluid down to the lowest values of @xmath35 . in an external trap with a @xmath63 mi phase in the center , a series of mi and sf regions appear by going towards the edge of the cloud , where the local chemical potential has dropped to zero . ] @xmath64 is a simple product of local fock - states with precisely one atom ( @xmath65 ) per site . upon lowering @xmath24 , the atoms start to hop around , which necessarily involves double occupancy , increasing the energy by @xmath42 . now as long as the gain @xmath35 in kinetic energy due to hopping is smaller than @xmath42 , the atoms remain localized although the ground state is no longer a simple product state as in ( 11 ) . once @xmath35 becomes of order or larger than @xmath42 , the gain in kinetic energy outweighs the repulsion due to double occupancies and the atoms will be delocalized over the whole lattice . in the limit @xmath66 the many body ground state becomes simply an ideal bose - einstein - condensate where all @xmath8 atoms are in the @xmath67 bloch - state of the lowest band . including the normalization factor in a lattice with a total number of @xmath56 sites , this state can be written in the form @xmath68 for large enough @xmath35 therefore , we recover a gross - pitaevski like description in terms of one , macroscopically occupied state . in two- and three - dimensional lattices , the critical value for the transition from a mi to a sf is reasonably well described by a mean - field approximation , giving @xmath69 for @xmath70 and @xmath71 for @xmath72 . here @xmath73 is the number of nearest neighbours . in one dimension there are strong deviations from a mean - field approximation and the corresponding values are @xmath74 for @xmath70 [ 18,19 ] and @xmath75 for @xmath72 . the latter result follows from mapping the bose - hubbard model to a chain of josephson junctions , for which the critical value of the transition to a mi phase is known precisely . consider now a filling with @xmath60 which is slightly larger than one . for large @xmath53 the ground state has all the atoms delocalized over the whole lattice and the situation is hardly different from the case of unit filling . upon lowering @xmath53 , however , the line of constant density remains slightly above the @xmath70 mott - lobe , and stays in the sf regime down to the lowest @xmath53 ( see fig.1 ) . for any noninteger filling , therefore , the ground state remains sf as long as the atoms can hop at all . this is a consequence of the fact , that even for @xmath76 there is a small fraction @xmath62 of atoms which remain sf on top of a frozen mi - phase with @xmath70 . indeed this fraction can still gain kinetic energy by delocalizing over the whole lattice without being blocked by the repulsive interaction @xmath42 because two of those particles will never be at the same place . the same argument applies to holes when @xmath62 is negative . in order to describe the situation in a weak harmonic trap , we use the standard appproximation that a slowly varying external potential may be accounted for by a spatially varying chemical potential @xmath77 ( we choose @xmath78 at the trap center ) . assuming that the chemical potential @xmath79 at trap center falls into the @xmath63 mott - lobe , one obtains a series of mi domains separated by a sf by moving to the boundary of the trap where @xmath80 vanishes ( see fig.1 ) [ 15 ] . in this manner , all the different phases which exist for given @xmath53 below @xmath79 are present simultaneously . since the defining property of a mi - phase is its incompressibility @xmath81 , the atomic density stays constant in the mott - phases , even though the external trapping potential is rising . an estimate for the width of the incompressible domains is obtained by noting that for @xmath76 the range in chemical potential over which the density remains constant is close to @xmath42 . in a quadratic confining potential with axial frequency @xmath82hz and with typical values @xmath83khz , the width of the incompressible mi - states is around @xmath84 m . it remains an experimental challenge to spatially resolve the sf and mi phases in a trap , thus verifying the crucial property of incompressibility . in practice , the observation of the sf to mi transition is done in the usual manner by absorption imaging the atomic cloud after a given expansion time . the corresponding series of images is shown in fig.2 for different values of @xmath24 , ranging between @xmath85 ( a ) and @xmath86 ( h ) . one observes a series of bragg - peaks around the characteristic zero - momentum peak of a condensate in the absence of an optical lattice . with increasing @xmath24 these peaks become more pronounced . beyond a critical lattice depth around @xmath87 ( e ) , this trend is suddenly reversed , however , and the bragg peaks eventually disappear completely . in order to understand , to which extent these pictures actually provide a direct evidence for the existence of a sf to mi transition predicted by the bose - hubbard model , we neglect the inhomogeneous nature of the atomic cloud and assume that the absorption images simply reflect the momentum distribution . for atoms which are confined to move in the lowest band of the lattice , it is straightforward to show that the momentum ( not quasi - momentum [ 20 ] ) distribution @xmath88 ( a ) , @xmath89 ( b ) , @xmath90 ( c ) , @xmath91 ( d ) , @xmath92 ( e ) , @xmath93 ( f ) , @xmath94 ( g ) and @xmath95 ( h ) . taken from ref . [ 14 ] with permission . ] @xmath96 can be expressed in terms of the exact one - particle density matrix @xmath97 at separation @xmath98 and the fourier transform @xmath99 of the associated wannier wave - function . the summation in ( 13 ) is over all lattice vectors @xmath98 , which are integer multiples of the three primitive vectors of the given lattice . now the sf and mi phases are distinguished quite generally by the behaviour of the one particle density matrix ( or first order coherence function in quantum optics terminology ) at large separation . in the sf , @xmath100 approaches a finite value @xmath101 which defines the condensate density @xmath102 [ 21 ] . for the mi phase , in turn , @xmath100 decays to zero exponentially . using ( 13 ) , the sf phase of cold atoms in an optical lattice can thus quite generally be characterized by the fact that at reciprocal lattice vectors @xmath103 defined by @xmath104 times an integer , the momentum distribution @xmath105 has a peak @xmath106 which scales with the total number @xmath8 of particles . this is the expected behaviour for the interference pattern from a periodic array of phase coherent sources of matter waves and is precisely analogous to the more standard bragg - peaks in the static structure factor of a solid , with the condensate fraction playing the role of the debye - waller factor [ 22 ] . the fact that the peaks in the momentum distribution at @xmath107 initially grow with increasing depth of the lattice potential is a result of the strong decrease in spatial extent of the wannier function @xmath48 , which entails a corresponding increase in its fourier transform @xmath99 at higher momenta . it is important to realize that there is no broadening of the peaks as long as @xmath108 is finite , in agreement with the experimental observations [ 14 ] . in the mi regime , where @xmath100 decays to zero , remnants of the bragg - peaks still remain ( see e.g. ( f ) in fig.2 ) as long as @xmath100 extends over several lattice spacings , because the series in ( 13 ) adds up constructively at @xmath107 . physically this reflects the fact that phase coherence is still present over distances much larger than one lattice spacing provided one is close to the transition to superfluidity . in contrast to the sf regime , however , these peaks are now broadened and do not scale with the total number @xmath8 of particles . in the extreme mi limit @xmath76 , hopping of atoms completely vanishes and @xmath100 is zero beyond @xmath109 . coherence is then completely lost and the momentum distribution is a structureless gaussian , reflecting the fourier transform of the wannier wave function ( see ( h ) in fig.2 ) . these arguments show that for a large and homogeneous system there is indeed a sharp signature of the sf to mi transition in the interference pattern . it is connected with the existence or not of ( off - diagonal ) long range order in the one particle density matrix , which effectively measures the range of phase coherence and the condensate fraction . of course the actual system is not homogeneous and a numerical computation of the interference pattern is necessary for a quantitative comparison with experiment [ 23 ] . due to the finite size and the fact that different mi phases are involved , the pattern evolves continously from the sf to the mi regime . indeed , as is evident from the phase diagram in fig.1 , the critical value of @xmath53 is different for the two different mi phases @xmath70 and @xmath110 which are present in the trap . nevertheless , a rather sharp transition is observed experimentally , because @xmath53 depends exponentially on the control parameter @xmath38 . the small change from @xmath87 in ( e ) to @xmath111 in ( f ) thus covers a range in @xmath53 wider than that which would be required to distinguish the @xmath70 from the @xmath110 transition . a second signature of the sf to mi transition is the appearance of a finite excitation gap @xmath112 in the mi . deep in the mi phase , this gap has size @xmath42 , which is just the increase in energy if an atom tunnels to an already occupied adjacent site ( note that this is much smaller than the gap @xmath113 for the excitation of the next vibrational state ) . the existence of this gap has been verified experimentally by applying a phase gradient in the mi and measuring the resulting excitations produced in the sf at smaller @xmath38 [ 14 ] . in this manner the fact that @xmath114 was verified for a range of @xmath24 , all reasonably deep in the mi phase . for reasons discussed above , however , it has not been possible to see the vanishing of the gap near the transition , which should scale like @xmath115 in the three - dimensional case [ 16 ] . in the sf regime , there is no excitation gap and instead the homogeneous system exhibits a sound like mode with frequency @xmath116 . the associated sound velocity follows from the thermodynamic relation @xmath117 and thus gives information about the superfluid density @xmath118 . the existence of a sound like excitation even in the presence of an underlying lattice which explicitely breaks translation invariance is a consequence of long range phase coherence in the sf . its observation would thus constitute an independent proof that the atoms move coherently over the whole lattice and thus phase gradients give rise to dissipationless currents . finally we discuss the change in the atom number statistics at individual sites between the sf and the mi regime . this issue has been investigated in a very recent beautiful experiment , observing collapse and revival of the matter wave due to the coherent superposition of states with different atom numbers in the sf [ 24 ] . as noted above , the ground state ( 11 ) in the extreme mi limit is a product of fock states with a definite number @xmath59 of particles at each site . at finite hopping @xmath119 , this simple picture breaks down because the atoms have a finite amplitude to be at different sites . the many body ground state can then no longer be written as a simple product state as in ( 11 ) . in the opposite limit @xmath120 , the ground state is a condensate of zero quasimomentum bloch states . it turns out , that the probability of finding precisely @xmath3 atoms at any given site in the associated state ( 12 ) is close to a poissonian distribution . more precisely , in the limit @xmath121 at fixed density @xmath122 , the state ( 12 ) becomes indistinguishable in a local measurement from a coherent state @xmath123 which factorizes into a product of local poissonian states with average @xmath124 because boson operators at different sites commute . we have thus come to the remarkable conclusion that for integer densities @xmath125 the many body ground state may be written in a local product form @xmath126 in both limits @xmath127 and @xmath120 . the associated atom number probability distribution @xmath128 is either a pure fock or a full poissonian distribution . it is now very plausible to use the factorized form ( 16 ) as an approximation for arbitrary @xmath53 , taking the coefficients @xmath129 as variational parameters which are determined by minimizing the ground state energy [ 25 ] . as first pointed out by rokhsar and kotliar [ 26 ] , this is effectively a gutzwiller ansatz for bosons . beyond being very simple computationally , this ansatz describes the sf to mi transition in a mean - field sense , becoming exact in infinite dimensions . in addition , it provides one with a very intuitive picture of the transition to a mi state , which occurs precisely at the point where the local number distribution becomes a pure fock distribution . this is consistent with a vanishing expectation value of the local matter wave field @xmath130 in the gutzwiller approximation . it is important , however , to emphasize that the ansatz ( 16 ) fails to account for the nontrivial correlations between different sites present at any finite @xmath35 . these correlations imply that the one particle density matrix @xmath100 is different from zero at finite distance @xmath131 , becoming long ranged at the transition to a sf . by contrast , in the gutzwiller approximation , the one particle density matrix has no spatial dependence at all : it is zero at any @xmath131 in the mi and is completely independent of @xmath98 in the sf . moreover , in the gutzwiller approximation the phase transition is directly reflected in the local number fluctuations , with the variance of @xmath132 vanishing throughout the mi phase . by contrast , in an exact theory local variables like the on - site number distribution will change in a smooth manner near the transition and the variance of the local particle number will only vanish in the limit @xmath127 . concerning the dynamics , one expects that the gutzwiller approximation qualitatively captures the time scales for local changes of the configuration , however it fails to correctly describe long wavelength excitations [ 26 ] . the realization of a sf to mi transition with cold atoms in optical lattices provides an essentially perfect realization of one of the most prominent models in many body physics . it allows to study a quantum phase transition by simply tuning the depth of the optical lattice . there remain , however , a number of open questions in particular concerning the detailed spatial structure in the trap and the dynamical behaviour [ 27 ] . as we have discussed above , superfluidity of cold atoms is destroyed in a deep optical lattice where the ground state looses phase coherence and - essentially - has a fixed number of atoms per site . the associated mott - hubbard transition occurs when the ratio @xmath133 is of order one . using the expressions ( 8) and ( 10 ) for @xmath35 and @xmath42 in terms of the optical lattice parameters , this translates into a condition of the form @xmath134 for the critical value of the dimensionless lattice depth @xmath135 . in the interesting regime with one or two atoms per site , the lattice constant @xmath19 is roughly equal to the the mean interparticle spacing @xmath5 . the prefactor @xmath136 in ( 18 ) thus coincides with the dimensionless interaction parameter @xmath6 introduced in ( 2 ) . for weak interactions @xmath15 therefore , the sf to mi transition requires deep optical lattices , for instance @xmath137 in the experiments by greiner et.al.[14 ] , where @xmath138 . in the following we want to adress the question what happens in a situation where the effective gas parameter @xmath6 becomes of order one or larger . the superfluid ground state is then expected to be destroyed already in a weak optical lattice , where the description in terms of a bose - hubbard model is no longer applicable . it turns out that this problem can be solved completely in the special case of one - dimensional bose gases in which the transverse motion is frozen into the lowest eigenstate of a strong confining potential [ 28 ] . as pointed out by petrov et.al . [ 29 ] , the ratio between the interaction and kinetic energy per particle in one dimension @xmath139 scales inversely with the 1d density @xmath140 . here @xmath141 is the strength of the effective delta - function interaction in 1d and @xmath142 the oscillator length for the transverse confinement [ 29 ] . in one dimension , it is thus the low density limit where interactions dominate . this somewhat counterintuitive result can be understood physically by noting that at low 1d densities , the average kinetic energy per particle @xmath143 vanishes so quickly that the atoms are perfectly reflected by the repulsive potential of the surrounding particles . for @xmath144 , therefore , the system aproaches a gas of impenetrable bosons which is called the tonks - limit [ 29 ] . in particular at @xmath145 the exact many body wave function is just the absolute value of that of a free fermi gas , as was shown a long time ago by girardeau [ 30 ] . this equivalence remains valid in the presence of an arbitrary additional one - particle potential like that of an optical lattice . for a qualitative understanding of what happens in the strongly interacting regime it is therefore useful to consider a free fermi gas in a weak periodic potential @xmath146 with @xmath24 of order @xmath57 or smaller this is an elementary problem in solid state physics , equivalent to the nearly free electron limit of a one - dimensional bandstructure . the single particle spectrum consists of a series of free particle like bands separated by energy gaps @xmath147 . the gaps become exponentially small with increasing energy , scaling like @xmath148 in the limit @xmath149 . for a commensurate density , where an integer number @xmath41 of particles fit into one unit cell , the @xmath41 lowest bands are completely filled . the groundstate of noninteracting fermions is thus a trivial band insulator . similar to the incompressible mott - insulating phase of the bose - hubbard model , the state with a fixed integer density remains locked over a finite range @xmath150 of the chemical potential . for weak optical lattices @xmath149 the lowest gap @xmath151 is much larger than the higher order ones . as a result , it is the commensurate phase with unit filling @xmath152 which has maximal stability . in the tonks limit @xmath153 we have thus found that an arbitrary weak optical lattice which is commensurate with the average density will pin the atoms into an incompressible optical crystal . the crucial question is obviously whether this peculiar feature of hard core bosons in one dimension is still present at finite and experimentally realizable values of @xmath154 . to answer that , it is convenient to use haldane s description of bose gases with arbitrary repulsive , short range interactions in terms of their long wavelength density oscillations [ 31 ] . introducing a field @xmath155 which is related to the fluctuations @xmath156 around the average density via @xmath157 , the hamiltonian in the presence of a weak , commensurate optical lattice can be shown [ 28 ] to be that of a quantum @xmath158dimensional sine - gordon model @xmath159\eqno(20)\ ] ] here @xmath160 is the actual sound velocity and @xmath161 is canonically conjugate to @xmath162 such that @xmath163=i\delta(x - x')$ ] . the coupling parameter @xmath164 is related to the dimensionless ratio @xmath165 which characterizes the power in the characteristic decay @xmath166 of the one particle density matrix in the absence of the optical lattice , typical for one - dimensional quantum liquids [ 31 ] . in haldane s description , the correlation exponent @xmath167 is a phenomenological parameter which approaches @xmath168 for hard core bosons and @xmath169 in the ideal gas limit . for given @xmath170 and strength @xmath24 of the optical lattice , the sine - gordon model ( 20 ) is an exactly soluble field theory [ 32 ] . it exhibits a transition at a critical value @xmath171 such that for @xmath172 the bose gas ground state remains gapless and superfluid in a weak optical lattice while for @xmath173 the atoms are locked even in an arbitrary weak periodic lattice as long as the deviation @xmath174 between the period @xmath19 and the average interparticle spacing @xmath175 is less than a critical value @xmath176 . the commensurate phase is characterized by a finite excitation gap [ 28 ] @xmath177 which is nonanalytic in @xmath24 . it approaches @xmath178 in the tonks gas limit @xmath168 in agreement with the ideal fermi gas picture discussed above . the size of the gap also determines the critical value @xmath179 of the deviation from exact commensurability which can still be accomodated into a locked groundstate . in order to relate @xmath167 to the microscopic and experimentally tunable parameter @xmath154 introduced in ( 19 ) , we use the exact solution by lieb and liniger [ 33 ] of the 1d bose gas with a delta - function interaction of strength @xmath141 . it turns out that @xmath180 is a monotonically decreasing function which reaches the critical value @xmath181 at @xmath182 . the transition to a commensurate , incompressible state in a weak optical lattice thus occurs long before the tonks limit is reached . as is evident from eqn . ( 22 ) , however , the gap is exponentially small in the vicinity of the critical value @xmath181 . in order to reach appreciable values of the gap , one thus needs @xmath183 where @xmath184 . in a one - dimensional bose gas we have thus found that a transition from a sf to a mi state at weak coupling @xmath15 requires a deep optical lattice , while at strong coupling @xmath185 an arbitrary weak lattice is sufficient to destroy phase coherence ( similar to the mi phase of the bose - hubbard model , the one particle density matrix @xmath186 will decay exponentially in the phase where the atoms are locked to the external lattice potential ) . for weak coupling , the criterion @xmath187 for the transition in the anisotropic , one - dimensional bose - hubbard model can be written in a form similar to eqn . ( 18 ) @xmath188 . \eqno(23)\ ] ] the solution of this transcendental equation gives a critical value @xmath189 which increases rather slowly as a function of the inverse interaction parameter @xmath190 . from the exact solution of the sine - gordon model , in turn , we know that - at least in one dimension - this critical amplitude of the optical lattice vanishes at the finite value @xmath191 . more precisely , the kosterlitz - thouless nature of the transition near @xmath181 determines the critical value of @xmath167 for small @xmath192 to behave like @xmath193 to linear order in @xmath194 [ 28 ] . the complete phase diagram for unit filling at arbitrary values of @xmath38 can then be obtained by combining these two asymptotic results in a smooth interpolation , as shown in fig.3 . the bose - hubbard transition ( bh ) for weakly interacting gases in a deep optical lattice thus continuously evolves into one of the commensurate - incommensurate ( c - ic ) type for strongly interacting gases in a weak lattice . it remains a challenge to extend these results to situations with two or more particles per lattice period and - in particular - to the case of two- or three - dimensional lattices . as a function of the inverse interaction parameter @xmath6 . the dashed line is the asymptotic behaviour near the critical point @xmath195 as determined from the sine - gordon model , while the dashed - dotted line derives from the solution of the transcendental equation ( 23 ) as obtained from the critical point of the 1d bose - hubbard model . ] regarding the prospects for an experimental observation of the c - ic transition discussed above , we have seen that realistic values of the excitation gap @xmath196 require @xmath154 to be of order @xmath91 . one - dimensional bose gases with parameters in this range may be realized with a strong optical lattice in only two directions @xmath197 which confine the atoms transversely but leave the motion along @xmath73 essentially free , except for a rather weak axial trap with frequency @xmath198 . as an example , using numbers close to those in the experiments by greiner et.al . [ 14 ] on the sf to mi transition , it is possible to generate a few thousand parallel one - dimensional gases with about @xmath199 atoms per wire . taking realistic values @xmath200 khz and @xmath201 hz , the central density for large @xmath6 is close to @xmath202m@xmath203 [ 34 ] . this is precisely commensurate with the lattice constant @xmath204 m of the optical lattice used in these experiments . adding a weak optical lattice in @xmath205direction , will thus lead to an incompressible state in the center of the trap provided @xmath154 is larger than the critical value @xmath182 . for @xmath206rb with a scattering length @xmath207 nm , the resulting @xmath154 is close to one and thus not in the required range . with a tunable scattering length as in @xmath208rb , however , it is perfectly feasible to increase @xmath2 by one order of magnitude and thus a mott - insulating state could be realized with a very weak optical lattice . since three - body losses are very strongly reduced in one dimension at large @xmath6 [ 35 ] , there is no problem with the condensate lifetime here . the transition may be observed in a similar manner than in the bose - hubbard case . more interesting however would be to measure the long range translational order present in the locked phase by bragg diffraction , as was done for cold atoms in deep optical lattices even at very low densities [ 22 ] . this method has the advantage that the signal from many parallel wires adds up constructively because they all experience the same modulation in @xmath205direction . with the recent realization of a quantum phase transition between a superfluid and a mott - insulating state by greiner et.al . [ 14 ] , the field of cold atoms has entered a regime , where strong correlation effects may be studied in an unprecedentedly clean manner . indeed , basic models in many body theory like the hubbard - model for bosons or fermions with on - site interaction , which were originally introduced in a condensed matter context as a rather schematic description of say superfluid helium in vycor or electrons in high - temperature superconductors can now be applied even on a quantitative level . moreover the crucial parameters @xmath209 and density can easily be tuned in a controlled fashion . this opens a wide area of possibilities for strong correlation physics with cold atoms , in particular if degenerate fermions may be loaded into an optical lattice . with two equivalent species of fermions the resulting version of the hubbard model displays a wealth of different phases : in the attractive case it describes the bcs- to bose - crossover for cooper - pairing , in the repulsive case antiferromagnetic or unconventional superconducting phases appear , as recently discussed by hofstetter et.al . [ 36 ] . from the perspective of atomic and molecular physics , a mott - insulating state with precisely two atoms per site is an ideal starting point for the formation of molecular condensates via local photoassociation and subsequent melting of the mott - phase as suggested by jaksch et.al . [ 37 ] . finally , controlled interactions ( collisions ) between atoms in different internal states in an optical lattice may be used to generate highly entangled states useful for quantum computation schemes [ 38,39 ] . cold atoms in optical lattices have therefore started to fascinate people not only in the field of atomic physics and quantum optics but far beyond that and it seems that the field is at the beginning of a promising area in research . < 45 > by removing the lattice potential in an _ adiabatic _ manner , it is possible also to map out the _ quasi-_momentum distribution and thus directly visualize the occupation within the brillouin zone , see greiner m 2001 _ phys . lett . _ * 87 * 160405 this property only holds in the two- or three - dimensional case , while in 1d , the behaviour is algebraic even in the ground state , as shown by eqn . ( 21 ) below . experimentally the existence of ` off - diagonal long range order ' in dilute gases was verified by i. bloch 2000 _ nature _ * 403 * 166 note that the static structure factor @xmath210 which is the fourier transform of the diagonal _ two_-particle density matrix , exhibits bragg peaks even if the atoms are localized at random sites of the optical lattice without phase coherence between them . these peaks simply reflect the externally imposed periodicity and were observed e.g. by weidemller m. 1995 _ phys . * 75 * 4583 polkovnikov a 2002 non - equilibrium gross - pitaevski dynamics of boson lattice models _ preprint _ cond - mat/0206490 and altman e and auerbach a 2002 oscillating superfluidity of bosons in optical lattices _ preprint _ cond - mat/0206157

we discuss the superfluid to mott - insulator transition of cold atoms in optical lattices recently observed by greiner et.al . ( nature 415 , 39 ( 2002 ) ) . the fundamental properties of both phases and their experimental signatures are discussed carefully , including the limitations of the standard gutzwiller - approximation . it is shown that in a one - dimensional dilute bose - gas with a strong transverse confinement ( tonks - gas ) , even an arbitrary weak optical lattice is able to induce a mott like state with crystalline order , provided the dimensionless interaction parameter is larger than a critical value of order one . the superfluid - insulator transition of the bose - hubbard model in this case continuously evolves into a transition of the commensurate- incommensurate type with decreasing strength of the external optical lattice .

zero velocity lieb - robinson bounds are a way to rigorously capture the suppression of transport . since there is no comprehensive discussion of such bounds available in the literature , we precisely discuss the relationships between different possible definitions here . depending on the precise notion of transport adapted , the formulation of the bound slightly varies . the strongest variant is that information can not be send through a system by locally changing the hamiltonian and is rigorously captured by our definition of strong dynamical localisation ( def . [ df_dynamical_localisation ] ) . ( zero velocity lieb - robinson bounds : truncated hamiltonian ) [ lr_truncation ] a hamiltonian is said to satisfy zero velocity lieb - robinson bounds in the truncated hamiltonian formulation , iff @xmath70 here @xmath71 is an observable in the heisenberg picture , @xmath72 the truncated hamiltonian including all hamiltonian terms within a distance @xmath12 of the support of @xmath4 and @xmath73 are constants . as in this whole work , @xmath74 was chosen for convenience . in constrast to the definition in the main text ( def . [ df_dynamical_localisation ] ) , we here include a factor @xmath75 , which is stronger than what we need for our proof of theorem [ main_theorem ] part * a*. however , since it seems reasonable to assume that the left hand side of eq . initially grows continuously with @xmath38 and this way of stating the bound allows for an easier connection between the different lieb - robinson versions , we decided to include the factor here . indeed the strongest standard lieb - robinson bounds in nearest - neighbour systems also include such a factor @xcite . another possible and very physical way of stating that a system has no transport , is to say that excitations can not spread through the system . mathematically , this can be captured by demanding that the action of a local unitary can not be detected far away , even for long times . ( zero velocity lieb - robinson bounds : excitations ) [ lr_excitations ] a hamiltonian is said to satisfy zero velocity lieb - robinson bounds in the excitation formulation , iff for any locally supported hermitian operator @xmath45 with @xmath76 @xmath77 where @xmath78 is an arbitrary state vector and @xmath79 are @xmath38 independent constants . ( zero velocity lieb - robinson bounds : commutator ) [ lr_commutator ] a hamiltonian is said to satisfy zero velocity lieb - robinson bounds in the commutator formulation , iff @xmath80 \| & \leq \min(t,1 ) c^ { '' } e^{- \mu { \mathrm{d}(a , b ) } } , \end{aligned}\ ] ] with suitable @xmath38 independent constants @xmath81 . going from the truncation formulation ( def . [ lr_truncation ] ) to the commutator formulation ( def . [ lr_commutator ] ) is straightforward @xmath80 \| & \leq \| [ a(t)-e^{i t h_l } a e^{- i t h_l } , b ] \| \\ & + \| [ e^{i t h_l } a e^{- i t h_l } , b ] \|\\ & \leq \min(t,1 ) c^ { '' } e^{- \mu { \mathrm{d}(a , b ) } } , \end{aligned}\ ] ] where @xmath84 has been chosen to make the second commutator zero . to show the one implication of the second part of the lemma , going from def . [ lr_excitations ] to def . [ lr_commutator ] , we rewrite the excitations formulation as @xmath85 where @xmath86 $ ] is the commutator and @xmath87 denotes multiple applications of it . dividing by @xmath88 and taking the limit @xmath89 yields @xmath90 { \left \vert \psi \right \rangle}| \leq \min(t,1 ) c^ { ' } e^{- \mu d(a , g ) } \;. \end{aligned}\ ] ] since the bound holds for arbitrary hermitian observables @xmath45 and arbitrary state vectors @xmath78 , this concludes the proof . to show the converse direction , going from def . [ lr_commutator ] to def . [ lr_excitations ] , we express the left hand side of def . [ lr_excitations ] as an integral @xmath91 e^{- i r g } { \left \vert \psi \right \rangle}|\\ & \leq \int_0^s { \mathrm d}r \| [ g , a(t ) ] \|\\ & \leq s \min(t,1 ) c^ { '' } e^{- \mu { \mathrm{d}(a , b ) } } , \end{aligned}\ ] ] where we used def . [ lr_commutator ] in the final step . using common tools @xcite , one can also use the commutator formulation to arrive at estimates similar to the truncation formulation , but with a slightly weaker pre - factor , depending on the time @xmath38 . a tool that plays an important role in this work is energy - filtering with respect to a suitable filter function . energy - filtering of a local observable is defined as follows @xmath92 here @xmath57 is a so - called filter function , usually taken as a @xmath93-function . here , @xmath94 refers to time evolution under the full hamiltonian @xmath10 , but we will later also consider filters with respect to truncated hamiltonians , such as @xmath95 . energy - filtering allows to alter the matrix elements of a local observable in the eigenbasis of a hamiltonian , while still keeping some form of locality @xcite . we will work with two types of filter functions . gaussian filter functions in particular provide a good compromise between locality in fourier space and decay behaviour in real time and hence allow us to pick out narrow energy windows , while still preserving the approximate locality of the observable . [ filter : gaussian ] a gaussian filter is defined as @xmath96 where @xmath97 defines the sharpness of the filter . the matrix elements in the eigenbasis of the hamiltonian fulfil @xmath98 for strongly localizing systems ( def . [ df_dynamical_localisation ] ) , local observables filtered with a gaussian filter still remain approximately local in the sense that @xmath99 for systems with a mobility edge ( def . [ def : subsp_loc ] ) , we have @xmath100 { \left \vert k \right \rangle}\right| } } \leq c_\mathrm{mob } e^{-d(a , b)}. \end{aligned}\ ] ] the gaussian suppression of off - diagonal elements readily follows from the definition @xmath101 and the fact that the fourier transform of a gaussian is again a gaussian . assuming strong dynamical localisation , deriving locality is straightforward , @xmath102 where have used that the gaussian filter is normalised . the case with a mobility edge can be shown in the same way . [ filter : high - pass ] a high - pass filter is defined by @xmath103 here @xmath97 describes the sharpness of the filter . the matrix elements can be bounded for any @xmath104 by @xmath105 local observables remain approximately local under a high - pass filter , in the sense that @xmath106 { \left \vert k \right \rangle}| \leq \frac{e^{-\mu { \mathrm{d}(a , b)}}}{2 \pi } \left(4 + \ln \frac{\pi}{4 \alpha } \right ) . \end{aligned}\ ] ] calculating the off - diagonal elements of such a high - pass filter relies on the subsequent bound on the error function of a gaussian random variable proven in ref . @xcite , stated here as lemma [ lem : gauss_tails ] . making use of the identity proven in ref . @xcite @xmath107 the bounds for the matrix elements of @xmath108 follow from lemma [ lem : gauss_tails ] . the locality statement can be shown by splitting the integral into three parts @xcite and using our assumption of dynamical localisation @xmath109 { \left \vert k \right \rangle } \leq \frac{e^{-\mu { \mathrm{d}(a , b)}}}{2\pi}\times\\ \left(\int_{{\ensuremath{\left| t\right|}}\leq 1 } e^{-\alpha t^2 } { \mathrm d}t + \int_{1\leq{\ensuremath{\left| t\right|}}\leq \lambda } \frac{e^{-\alpha t^2}}{t } { \mathrm d}t + \int_{{\ensuremath{\left| t\right|}}\geq\lambda } \frac{e^{-\alpha t^2}}{t } { \mathrm d}t\right)\\ \leq \frac{e^{-\mu { \mathrm{d}(a , b)}}}{\pi } \left(1 + \ln\lambda + \frac{1}{2\lambda } \sqrt{\frac{\pi}{\alpha } } \right ) . \end{gathered}\ ] ] here estimating the first term used the @xmath110 factor included in the definition of a mobility edge ( def . [ def : subsp_loc ] ) . choosing @xmath111 concludes the proof . for later use , we give the gaussian and high pass filters of sharpness @xmath61 the symbols @xmath59 and @xmath118 , respectively . in case the local observable is filtered with the full system hamiltonian , we will often omit the @xmath10 . it is an interesting insight that the locality structure of a gaussian filter is independent of its sharpness @xmath61 for dynamically localising systems , while it still depends on @xmath61 for a high - pass filter . it is precisely this property of gaussian filters that will allow us to prove part @xmath119 of theorem [ main_theorem ] . in this appendix , we briefly comment on how local constants of motion could be constructed using energy filtering . a natural candidate for local constants of motion are the energy - filtered versions of the local hamiltonian terms @xmath120 since they act non - trivially on most eigenstates , it is clear that they are retained by the energy - filtering without being averaged to zero and thus indeed provide useful quantum numbers . what is more , strong dynamical localisation implies that they will stay approximately local . this construction can be optimised by choosing suitable filter functions , allowing for an interpolation between the locality of the observables and making them completely constant in time . in particular , taking the time - average is equivalent to choosing a constant filter function @xcite leading to quantities that are completely preserved in time , while they will usually not be strictly local anymore . naturally , this approach is not limited to taking the local hamiltonian terms and in principle any local observable that does not vanish after applying the energy filter will provide useful quantum numbers . specifically , polynomials of the local hamiltonian terms can equally well be used . in systems with a mobility edge , where not all transport is suppressed , this construction is still possible , but the resulting constants of motion might no longer be approximately local . since local constants of motion by definition commute with the hamiltonian , their local eigenbasis is compatible with the global energy eigenbasis of the hamiltonian . thus , they could be used to directly construct local states in a matrix - product state form , as long as enough constants of motion exist in order to use their quantum numbers to divide the full hilbert space into suitable small fractions . in this appendix , we will formulate the details of the proof of theorem [ main_theorem ] part * a * , which shows that strong dynamical localisation implies exponential clustering of all eigenvectors . for this , we will rely on the following lemma . [ lm_energy_nondegenerate ] under the assumption of locally independent gaps ( * aii * ) , energy - filtering of two observables can be factorized into local energy filters @xmath121 with @xmath122 . here @xmath20 and @xmath21 are chosen to include all hamiltonian terms within distance @xmath123 of the support of @xmath4 and @xmath5 respectively ( see fig . [ fig : hamiltonian_decomposition ] ) . to show that instead of applying an energy filter to @xmath60 , we can also apply it to the observables individually , we need to use that the eigenvalues of @xmath124 are disconnected on the two regions and we can thus label them by two different quantum numbers @xmath125 . we hence get @xmath126 we will use the triangle inequality and proceed to show that both energy filters give only the diagonal entires up to a small error . starting with the first term gives the following estimate , @xmath127 where we assumed locally independent gaps ( * aii * ) . the second term yields the following estimate , @xmath128 here , @xmath129 is a diagonal matrix with entries @xmath130 for @xmath131 and 0 otherwise and we assumed locally independent gaps ( * aii * ) . finally let us briefly comment that the local independence of gaps can alternatively be ensured by demanding that the joint hamiltonian @xmath132 supported on two disjoint rectangular regions has non - degenerate energies , since these energies are all possible sums of the local energies @xmath133 . the proof uses the local hamiltonians @xmath20 and @xmath21 ( see fig . [ fig : hamiltonian_decomposition ] ) and the following estimates @xmath135 in each step , an error term of the form @xmath136 with @xmath137 to @xmath138 will be introduced . the constant @xmath139 appearing in the main theorem will simply be the sum of them . constants @xmath140 directly follow from the properties of the applied gaussian filter ( def . [ filter : gaussian ] ) and constant @xmath141 is derived in lemma [ lm_energy_nondegenerate ] @xmath142 the constants @xmath141 and @xmath138 can be chosen arbitrarily small , by picking a sharp enough filter function , meaning a sufficiently small @xmath61 . in particular , we can choose @xmath143 , which yields @xmath144 choosing @xmath145 concludes the proof . [ mobility edge ] a hamiltonian @xmath10 is said to have a mobility edge at energy @xmath37 iff its time evolution satisfies for all times @xmath38 , @xmath39 @xmath146\right)\vert \leq \min(t,1 ) c_\mathrm{mob } e^{-\mu { \mathrm{d}(a , b)}}\ ; , \end{aligned}\ ] ] where @xmath147 is a @xmath38 independent constant . that is , all transport is suppressed for states supported only on the low - energy sector below energy @xmath37 . the first step for our proof will be to show that each eigenvectors only contributes to the correlation function by a term that is exponentially suppressed with the distance of the two observables . for this , we need the localisation assumption only for the eigenvector @xmath48 . [ lem : exp_dec_matrix_elem ] let @xmath148 be local observables and @xmath48 a weakly localised eigenvector of @xmath10 , i.e. satisfying @xmath149 { \left \vert k \right \rangle}\right|}}\leq c_{\rm mob } e^{-\mu { \mathrm{d}(a , b ) } } . \end{aligned}\ ] ] then the contribution of any eigenvector @xmath150 , to the correlation function will be exponentially suppressed , in the sense that @xmath151 the proof of this lemma again uses an energy filter with a gaussian filter function ( def . [ filter : gaussian ] ) and works with @xmath152 , which can always be achieved by using a shifted observable @xmath153 . for this proof the filter function will be multiplied by a complex factor @xmath154 such that its fourier transform approximately suppresses all transitions except the one from level @xmath155 to level @xmath12 . in a mild variant of the above filter function , we define @xmath57 and the corresponding filter as @xmath156 with this energy filter , we can proceed to prove the lemma . @xmath157 { \left \vert k \right \rangle}\right| } } + { \ensuremath{\left| { \left \langle k \right \vert } b i_f(a ) { \left \vert k \right \rangle}\right| } } + 2^n e^{-{\gamma^2}/({4 \alpha})}\\ \leq & c_\mathrm{mob } e^{-d(a , b ) } + 2^n \left ( e^{-{\gamma^2}/({4 \alpha } ) } + e^{-{\zeta^2}/({4 \alpha } ) } \right ) . \end{aligned}\ ] ] here , @xmath113 refers to the smallest gap and @xmath158 is the smallest degeneracy of the gaps for fixed @xmath29 as defined in the main text in * ai * and * aiii * and in the last step we used weak dynamical localisation of the eigenvector @xmath48 . we can now conclude the proof by choosing @xmath61 small enough such that @xmath159 which means that we pick a filter function that is narrow enough in fourier space . setting @xmath160 it would even be enough to choose @xmath161 independently of @xmath6 . * if the hamiltonian has a mobility edge at energy @xmath50 and its spectrum fulfils assumptions * ai * and * aiii * , then all eigenvectors up to that energy @xmath50 cluster exponentially @xmath162 where @xmath53 is the number of states at energy @xmath31 and @xmath54 is a constant that can be chosen arbitrarily to optimise the bound . the proof runs along and builds upon the lines of thought of both ref . @xcite and of @xcite , and generalises both . the basic idea is again to start from the correlation function and to transform it into an expression depending on the commutator . for this , we fix a constant @xmath51 and define @xmath163 to be the projector onto the subspace of energies smaller than @xmath55 and write @xmath164 for @xmath165 . decomposing the identity with respect to @xmath166 and @xmath164 and using a high - pass filter @xmath167 for a suitable @xmath97 ( def . [ filter : high - pass ] ) , we separate the correlator into terms @xmath168 { \left \vert k \right \rangle}\;. \end{aligned}\ ] ] lets consider the first term on the right - hand side of eq . that contains @xmath166 and expand the projector in the eigenbasis of @xmath10 , which gives @xmath169 here , we have used that all matrix elements decrease under a high - pass filter ( lemma [ filter : high - pass ] ) for all @xmath97 . as before , we will set @xmath152 w.l.o.g . and then proceed by bounding all the terms in the sum individually . using lemma [ lem : exp_dec_matrix_elem ] yields @xmath170 where @xmath171 simply is the number of eigenvectors contained in @xmath166 . the other term in eq . ( [ eq : subsp_decomp ] ) containing @xmath166 can be bounded analogously , yielding @xmath172 the terms containing @xmath164 in eq . ( [ eq : subsp_decomp ] ) are bounded using the explicit form of the matrix elements of a high - pass filter ( lemma [ filter : high - pass ] ) , which delivers @xmath173 where @xmath63 is a diagonal matrix whose entries are bounded by @xmath174 . a simple norm estimate concludes the estimation of this term @xmath175 the other term containing @xmath164 can be estimated in the same way @xmath176 the commutator term in eq . ( [ eq : subsp_decomp ] ) is bounded by using locality of a high - pass filter ( lemma [ filter : high - pass ] ) . we are still free to choose a value for @xmath61 in the high - pass filter . taking @xmath177 gives @xmath178 { \left \vert k \right \rangle } & \leq & \bigg ( 12 \pi { \theta}(e_k + \kappa ) c_\mathrm{mob } \\ & + & 2 \pi + 4 + \ln \frac{\pi \mu { \mathrm{d}(a , b)}}{\kappa^2 } \bigg ) \frac{e^{-\mu { \mathrm{d}(a , b)}}}{2 \pi } \;. \end{aligned}\ ] ] this concludes the proof . it follows from the above results that in the case of 1d systems , the corresponding eigenstates satisfy an area law for a suitable entanglement entropy , which in turn implies that they can be well approximated by a matrix - product state of a low bond dimension . we will bound the bond dimension by using theorem 1 from ref . @xcite . for convenience , we repeat our corollary [ mps_theorem ] from the main text . * if the hamiltonian shows strong dynamical localisation and its spectrum fulfils assumptions * ai * and * aii * , then the statement holds for all eigenvectors @xmath48 and the approximation has a bond dimension @xmath179 for some constant @xmath67 . * if the hamiltonian has a mobility edge at energy @xmath37 , and its spectrum fulfils assumptions * ai * and * aiii * , then the statement holds for all eigenvectors below this energy @xmath37 and the bond dimension is given by @xmath180 for any fixed @xmath54 which enters in the precise form of the polynomial . . theorem 1 in ref . @xcite allows to bound the bond dimension across cuts by providing an upper bound to the smooth max entropy . the reduced state of the suitably normalised matrix product state is @xmath181 . ] . theorem 1 in ref . @xcite allows to bound the bond dimension across cuts by providing an upper bound to the smooth max entropy . the reduced state of the suitably normalised matrix product state is @xmath181 . ] as a first step of the proof , we reformulate our clustering of correlation results ( theorem [ main_theorem ] ) in terms of a correlation length according to ref . @xcite . for this , we need a decay of the correlator of the form @xmath182 and start by rewriting our bound as @xmath183 next we split the decaying term and use one part to get rid of the constant prefactor @xmath139 and the other to preserve an exponential decay with a correlation length @xmath184 . this yields @xmath185 where @xmath186 . following ref . @xcite , we will use the exponential clustering to obtain a description of the eigenvector @xmath48 in terms of mps ( see also fig . [ fig_mps ] ) . this is based on the so called smooth max entropy , which is defined as @xmath187 with @xmath188 being a ball around @xmath189 ( see appendix a of ref . @xcite for details ) . here @xmath190 denotes the trace distance . according to theorem 1 in ref . @xcite , we can obtain an upper bound for the smooth max entropy for any bipartite cut , as long as the system size is larger than @xmath191 with a constant @xmath192 . picking the approximation parameter in the smooth max entropy to be @xmath193 , then ref . @xcite provides a bound of the form @xmath194 for all @xmath195 with constants @xmath196 . in the following , we will make corollary 3 in ref . @xcite explicit , by deriving concrete bounds on the bond dimension of the matrix - product state chosen to approximate @xmath48 . naturally , the first step for this is to express @xmath48 as a matrix - product state vector with , a priori , exponentially large bond dimension ( fig . [ fig_mps ] ) . for simplicity , we restrict the proof here to a linear system with open boundary conditions , but it can equivalently be reformulated for a periodic system on a ring @xcite . the bond dimension of the matrix - product state will be bounded by truncating at each cut explicitly . for this , we start at one end of the chain and look at each cut separately . in each step , we apply a singular value decomposition ( see also fig . [ fig_mps ] ) . since we are truncating spectral values in each step , the positive operators will no longer be normalised to unit trace and are states only up to normalisation . this results in a reduction on the left side of the cut of @xmath181 , with @xmath43 being unitary and @xmath197 diagonal . the goal is now to truncate the diagonal matrix @xmath198 to a fixed bond dimension @xmath63 while creating only a small discarded weight @xcite . following lemma 14 in appendix b of ref . @xcite , for any @xmath199 , we can choose the bond dimension as @xmath200 and create a discarded weight @xmath201 where @xmath202 are the eigenvalues of @xmath203 or equivalently of @xmath204 . each time we create discarded weight , the fidelity with our original pure quantum state potentially is reduced by the discarded weight @xcite , upper bounded by @xmath205 . the truncation will result in a subnormalised state . renormalising will , however , only increase the fidelity , allowing us to obtain a normalised mps approximation . thus , in total , the fidelity of our mps approximation with bond dimension @xmath206 is bounded by @xmath207 where @xmath3 is the size of the linear 1d system . for a fixed global error @xmath208 , we therefore obtain an allowed local error @xmath209 we now fix @xmath210 to take the role of @xmath211 in . plugging this bound into eq . and using @xmath193 , we obtain @xmath212 and thus obtain an approximation with @xmath213 coming back to our original clustering assumption in eq . , this yields @xmath214 with @xmath215 . thus , we finally have a polynomial scaling in the constant @xmath67 as well as the in the system size and in the inverse error of the approximation @xmath216 . the rest of the proof follows directly by inserting the constants from theorem [ main_theorem ] . in this subsection we make the link between dynamical localisation and the absence of thermalisation more explicit . on intuitive grounds , such a connection is much expected : if no transport happens , then the system will retain too much memory of the initial state for all times . this initial state dependence , then necessarily leads to an absence of thermalisation . here , we will formulate the link between strong dynamical localisation in the sense of definition [ df_dynamical_localisation ] and the absence of thermalisation along the lines of ref . @xcite and consistent with ref . @xcite . to state this concisely , denote with @xmath203 a gibbs state of some inverse temperature @xmath217 . we denote with @xmath4 both an observable as well as the support of @xmath4 on the lattice . @xmath5 is the region of distance no more than @xmath12 from @xmath4 . the following lemma shows that under the condition of strong localisation , there is an initial state @xmath218 different from @xmath203 on @xmath4 only , such that @xmath219 will still be locally statistically distinguishable from @xmath203 for all times @xmath220 on the region @xmath5 , in stark contrast to a presumed thermalisation . that is to say , in the sense of ref . @xcite the system retains too much of a memory of the initial condition to thermalise . hence , the system must also violate the eth . let @xmath203 be a gibbs state . then there exists a state @xmath218 , different from @xmath203 on @xmath4 only , such that the trace distance of the time evolved state @xmath219 and the gibbs state @xmath203 reduced to @xmath5 is lower bounded by @xmath221 with @xmath222 , where @xmath223 and @xmath224 are the reductions to @xmath4 and @xmath5 , respectively . for the distinguished region of the lattice , refer to @xmath223 as its reduced state and @xmath225 its complement . take as initial state @xmath226 , where @xmath227 is chosen such that @xmath228 using that the trace norm is a unitarily invariant norm , it is clear that such a state @xmath227 can always be found . denote with @xmath4 the observable satisfying @xmath229 that saturates the trace distance . then @xmath230 where we used that the gibbs state is invariant under the time evolution of its hamiltonian . applying the triangle inequality , the condition of strong localisation , and the fact that @xmath231 is an observable supported on @xmath5 with operator norm upper bounded by unity , we get @xmath232 for @xmath12 suitably chosen . from this the statement follows .

the phenomenon of many - body localisation received a lot of attention recently , both for its implications in condensed - matter physics of allowing systems to be an insulator even at non - zero temperature as well as in the context of the foundations of quantum statistical mechanics , providing examples of systems showing the absence of thermalisation following out - of - equilibrium dynamics . in this work , we establish a novel link between dynamical properties a vanishing group velocity and the absence of transport with entanglement properties of individual eigenvectors . using lieb - robinson bounds and filter functions , we prove rigorously under simple assumptions on the spectrum that if a system shows strong dynamical localisation , all of its many - body eigenvectors have clustering correlations . in one dimension this implies directly an entanglement area law , hence the eigenvectors can be approximated by matrix - product states . we also show this statement for parts of the spectrum , allowing for the existence of a mobility edge above which transport is possible . the concept of disorder induced localisation has been introduced in the seminal work by anderson @xcite who captured the mechanism responsible for the absence of diffusion of waves in disordered media . this mechanism is specifically well understood in the single - particle case , where one can show that in the presence of a suitable random potential , all eigenfunctions are exponentially localised @xcite . in addition to this spectral characterisation of localisation there is a notion of dynamical localisation , which requires that the transition amplitudes between lattice sites decay exponentially @xcite . naturally , there is a great interest in extending these results to the many - body setting @xcite . in the case of integrable systems that can be mapped to free fermions , such as the xy chain , results on single particle localisation can be applied directly @xcite . a far more intricate situation arises in interacting systems . such _ many - body localisation _ @xcite has received an enormous attention recently . in terms of condensed - matter physics , this phenomenon allows for systems to remain an insulator even at non - zero temperature @xcite , in principle even at infinite temperature @xcite . in the _ foundations of statistical mechanics _ , such many - body localised systems provide examples of systems that fail to thermalise . when pushed out of equilibrium , signatures of the initial condition will locally be measurable even after long times , in contradiction to what one might expect from quantum statistical mechanics @xcite . despite great efforts to approach the phenomenon of many - body localisation , many aspects are not fully understood and a comprehensive definition is still lacking . similar to the case of single - particle anderson localisation , there are two complementary approaches to capture the phenomenon . on the one hand probes involving real - time dynamics @xcite have been discussed , showing excitations `` getting stuck '' , or seeing suitable signatures in density - auto - correlation functions @xcite , leading to a dynamic reading of the phenomena . on the other hand , statistics of energy levels has been considered as an indicator @xcite as well as a lack of entanglement in the eigenbasis @xcite and a resulting violation of the eigenstate thermalisation hypothesis ( eth ) @xcite . in such a static description of many - body localisation , it has for example been suggested to define many - body localised states as those states that can be prepared from product states or slater determinants of single - particle localised states with a finite - depth local unitary transformation . accordingly , a many - body localised phase is one where most eigenvectors have this property @xcite . it seems fair to say that the connection between these static and dynamic readings is still unclear . in this work , we make a substantial first step in relating these approaches : if the evolution of an interacting many - body system is dynamically localising ( and its spectrum is generic ) , then all eigenvectors have exponentially clustering correlations in general and , in one dimension , satisfy an area law and can be efficiently written as a matrix - product state ( mps ) @xcite with a low bond dimension . our proof uses advanced mathematical tools developed for the analysis of interacting many - body systems , such as energy filtering and does not rely on the precise details of the considered model . we show that short ranged correlations in individual energy eigenstates necessarily follow from absence of transport in localising models . _ setting . _ for simplicity , we will state all our results in the language of spin systems of local dimension @xmath0 , but they can equivalently be derived for fermionic systems . we consider local hamiltonians @xmath1 , where each interaction term @xmath2 is supported on finitely many sites of a lattice with @xmath3 vertices , and the physical system associated with each site is finite - dimensional . we make for most of the argument no assumptions concerning the dimensionality or the kind of the lattice . in what follows , we will consider observables @xmath4 and @xmath5 that are supported on finitely many sites on the lattice . there is a natural distance @xmath6 in the lattice related to the length of the shortest path of interactions connecting the supports of @xmath4 and @xmath5 . for convenience of notation , we will choose @xmath7 from here on , which does not restrict generality @xcite . for any local hamiltonian of this sort there is an upper bound @xmath8 to the largest group velocity , referred to as lieb - robinson velocity . there are several possible formulations of such a bound ( see appendix [ lr ] for a detailed discussion ) . one convenient and common way is to compare the time - evolution of a local observable @xmath9 under the full hamiltonian @xmath10 with its time - evolution under a truncated hamiltonian @xmath11 that only includes interactions @xmath2 contained in a region of distance no more than @xmath12 from the support of @xmath4 . the lieb - robinson theorem then states that there exists a constant @xmath13 and a velocity @xmath8 such that @xmath14 is true for all times @xmath15 . such a bound limits the _ speed of information propagation _ : in any such lattice with short - ranged interactions , all interactions are causal in this sense . a related , but in the case of @xmath16 weaker estimate is given by the following commutator bound ( see also [ lr ] ) @xmath17 \| & \leq c e^{- \mu ( { \mathrm{d}(a , b)}- v t ) } . \label{eq : lrcomm}\end{aligned}\ ] ] our results will rely on two generic and natural assumptions on the spectrum of the hamiltonian . to state those , we express the many - body hamiltonian @xmath10 in its eigenbasis as @xmath18 * _ non - degenerate energies ( * ai * ) : _ the energies of the full hamiltonian are assumed to be non - degenerate . the smallest gap between these energies will be called @xmath19 . * _ locally independent gaps ( * aii * ) : _ the energies of reduced hamiltonians @xmath20 and @xmath21 which include all interactions inside rectangular regions @xmath4 and @xmath5 respectively , are assumed to be non - degenerate when viewing them as operators on their respective truncated hilbert spaces @xmath22 and @xmath23 . the smallest gap will be called @xmath24 . moreover , the gaps of these hamiltonians need to be locally independent with respect to each other , in the sense that @xmath25 where @xmath26 and @xmath27 label the eigenvalues of @xmath20 and @xmath21 respectively . the smallest difference of these gaps will be called @xmath28 . * _ non - symmetric gaps ( * aiii * ) : _ for all eigenvalues @xmath29 of the full hamiltonian the spectrum is no - where symmetric in the sense that @xmath30 similar reasonable conditions have already been considered in the context of equilibration of closed systems and are assumed to hold when a small amount of random noise is added to a local hamiltonian @xcite . especially in the context of localising systems which are typically based on large random terms in the hamiltonian , these assumptions are very natural indeed . our assumptions are , however , considerably weaker than assuming fully non - degenerate energy gaps @xcite . another quantity that will be important later on is the number of states up to energy @xmath31 given by @xmath32 . _ notions of dynamical localisation . _ one defining feature of localisation is that the system shows no transport of information . there are many possible readings of this . the first and strongest is that the time evolution can be truncated to a finite region independent of time and is directly connected to an absence of thermalisation in the model ( see appendix [ absence ] ) . [ df_dynamical_localisation ] a hamiltonian @xmath10 exhibits strong dynamical localisation iff its time evolution satisfies @xmath33 for a suitable @xmath34 , where @xmath4 is an arbitrary local observable , @xmath35 a constant independent of the system size and @xmath11 denotes a hamiltonian which includes all interactions contained in a region of distance no more than @xmath12 from the support of @xmath4 . this corresponds to setting the lieb - robinson velocity @xmath36 to zero in eq . . a potential candidate for such a model is given by free fermionic systems with strong local disorder . there , it is known that transport is strongly suppressed and it is still being debated to what extent they satisfy zero velocity lieb - robinson bounds @xcite . this definition also implies that information can not be transported by encoding it locally in the chosen hamiltonian ( see appendix [ lr ] for details ) . as such , it does not allow for a growth of entanglement entropies logarithmically in time , a phenomenon that has been numerically observed in spin chains with random onsite noise @xcite . in a second step , we relax our dynamical assumption to a weaker variant of zero velocity lieb - robinson bounds and restricting it to a suitable subspace . it is natural to assume that the transport properties of an electronic system are closely connected to the energy available in the system . in fact , often the notion of a _ mobility edge _ is introduced @xcite . the intuition is that for all energies below this mobility edge , transport is fully blocked , while it might be possible for higher energies . the following definition is making this precise and allows for transport in highly excited sectors . [ def : subsp_loc ] a hamiltonian @xmath10 is said to have a mobility edge at energy @xmath37 iff its time evolution satisfies for all times @xmath38 , @xmath39 @xmath40\right)\vert \leq \min(t,1 ) c_\mathrm{mob } e^{-\mu { \mathrm{d}(a , b)}}\ ; , \end{aligned}\ ] ] with constants @xmath41 independent of @xmath38 . that is , all transport is suppressed for states supported only on the low - energy sector below energy @xmath37 . this definition relies on a rather weak version of zero velocity lieb - robinson bounds ( see appendix [ lr ] ) . while our results will be applicable to all eigenvectors below the mobility edge , they can naturally also be used to capture the ground state , yielding an extension of the previous results @xcite . in experiments , a natural setting for dynamical localisation is one where a system is excited locally . for example , in case of a ground state vector @xmath42 , this corresponds to the action of a unitary operator @xmath43 with local support @xmath44 where @xmath45 denotes its generator . in this setting , suppression of transport means that the excitation can not be measured at far distances , even for long times , in the sense that @xmath46 for arbitrary observables @xmath4 . here @xmath47 are constants independent of the time @xmath38 and the system size . if excitations are stuck in the above sense in an entire certain energy subspace , then this implies our zero velocity lieb - robinson estimate in eq . , thus relating definition [ def : subsp_loc ] to a quantity that can in principle be experimentally measured ( see appendix [ lr ] ) . _ main result . _ we are now in the position to state our main result . it provides a clear link between the dynamical properties of the system and the static correlation behaviour of the corresponding eigenvectors and can be applied to quantum systems of arbitrary dimension . in one dimension it directly implies an area law and the existence of an approximating mps ( see corollary [ mps_theorem ] ) . [ main_theorem ] the dynamical properties of a local hamiltonian imply clustering correlations of its eigenvectors in the following way . * if the hamiltonian shows strong dynamical localisation and its spectrum fulfils assumptions * ai * and * aii * then all its eigenvectors @xmath48 have exponentially clustering correlations , i.e. @xmath49 * if the hamiltonian has a mobility edge at energy @xmath50 and its spectrum fulfils assumptions * ai * and * aiii * , then all eigenvectors @xmath48 up to that energy @xmath50 cluster exponentially , in the sense that for all @xmath51 @xmath52 where @xmath53 is the number of eigenstates up to energy @xmath31 and @xmath54 can be chosen arbitrarily to optimise the bound . part * b * of the theorem only requires lieb - robinson bounds on a subspace and is thus perfectly compatible with a growth of entanglement entropies following quenches . in fact , for the proof it is sufficient to assume eq . on the level of the individual eigenstates ( see appendix [ proof : part_b ] ) . it further allows to freely choose an energy cut - off @xmath54 that serves as an artificial energy gap and can be used to optimise the bound . for typical local models , we expect the density of states to behave like a gaussian , an intuition that can be numerically tested for small systems ( see fig . [ fig : gaussian ] ) , and rigorously proved in a weak sense @xcite . in this case the number of states will behave like a low - order polynomial in the system size at energies close to the ground state energy @xmath55 . thus fixing a @xmath51 independent of the system size will lead to a pre - factor that scales like a low - order polynomial in the system size where the order of the polynomial will increase as one moves to higher energies . in the bulk of the spectrum , the number of states will grow exponentially with the system size , thus rendering our bounds useless as one moves to high energies . interestingly , this feature of stronger correlations , associated with a larger entanglement in the state , at higher energies seems to be shared by the heisenberg chain with random on - site magnetic field ( see fig . [ fig : gaussian ] ) . naturally our theorem can also be applied to the ground state where it extends previous results @xcite and for example for the case of an almost degenerate ground state will be a substantial improvement . naturally , our results can also be applied to the highly excited states at the other end of the spectrum . the proofs for both parts of the theorem rely on energy filtering techniques @xcite . this is a versatile tool especially in the study of perturbation bounds for hamiltonian systems and allows to partly diagonalise a local observable in the energy eigenbasis , while still keeping some locality structure . energy filtering of an observable is defined as @xmath56 here @xmath57 is a _ filter function _ that can be used to interpolate between locality of the resulting observable and the strength of the off - diagonal elements in the hamiltonian basis ( see appendix [ appendix : filtering ] ) . in order to show part * a * of the theorem , we will make use of energy filterings of the local observables @xmath4 and @xmath5 with respect to the full hamiltonian @xmath10 as well as with respect to local restrictions @xmath20 and @xmath21 . those restrictions contain the support of the corresponding observable and are chosen as large as possible , while still satisfying @xmath58=0 $ ] . the main idea is to choose a suitable ( gaussian ) filter function @xmath59 and use the gap assumptions * ai * and * aii * to show that the joint filter of the observables @xmath60 decouples into energy filters for @xmath4 and @xmath5 separately . then strong dynamical localisation can be used to make these filters local . finally choosing the width of the energy filter @xmath61 small enough , and in particular exponentially small in the system size , allows to conclude the proof ( for details see appendix [ part_a ] ) . the proof of part * b * of the theorem is more involved , relies on a high - pass filter ( see appendix [ appendix : filtering ] ) and is contained in appendix [ proof : part_b ] . _ implications on area laws and matrix - product states . _ in one dimension , the conclusions of our main theorem about the correlation behaviour of eigenvectors can be turned into a statement about their entanglement structure . it has been noted before that many - body localisation should be connected to eigenvectors fulfilling an area law ( see conjecture 1 in ref . @xcite ) , and eigenvectors being well approximated by matrix - product state vectors of the form @xmath62 where @xmath63 is the bond dimension and @xmath64 for all @xmath65 . our main theorem allows to rigorously prove this connection . [ mps_theorem ] an eigenvector @xmath48 of a localising hamiltonian can be approximated by an mps with fidelity @xmath66 , where the bond dimension for a sufficiently large system is given as follows . * if the hamiltonian shows strong dynamical localisation and its spectrum fulfils assumptions * ai * and * aii * , then the statement holds for all eigenvectors @xmath48 and , for some constant @xmath67 , the approximation has a bond dimension @xmath68 * if the hamiltonian has a mobility edge at energy @xmath37 , and its spectrum fulfils assumptions * ai * and * aiii * , then the statement holds for all eigenvectors below this energy @xmath37 and the bond dimension is given by @xmath69 for any fixed @xmath54 which enters in the precise form of the polynomial . the proof is a direct consequence of our main theorem together with the fact that exponential clustering in one dimension implies strong bounds on entanglement entropies between any bipartite cut of the chain @xcite . using techniques from @xcite this leads to an efficient mps approximation with the above bounds on the bond dimension ( see appendix [ mps_proof ] for details ) . _ summary & outlook . _ despite considerable progress in understanding the effects of random potentials on quantum many - body systems , a precise definition of the phenomenon of many - body localisation continues to be elusive . attempts to capture the phenomenology can largely be classified into two complementary approaches : one of them puts characteristic properties of the eigenfunctions into the focus of attention and asks for a lack of entanglement and a violation of the eth . the other one takes the suppression of transport as the basis , which seems closer to being experimentally testable . in this work , we have established a clear link between these two approaches , by showing that dynamical localisation implies that eigenvectors cluster exponentially . this result , together with the existence of an approximating mps description in one dimension , reminds of the definition of ref . @xcite , that defines many - body localisation in terms of matrix - product state approximations of eigenstates . in contrast , in our work , this feature is shown to follow from an absence of transport . for future research , it would be interesting to further explore this connection and to address the converse direction , namely to establish that an area - law for all eigenvectors implies that excitations can not travel through the system . a different approach towards approximating the individual eigenstates with a matrix - product state could potentially be provided by constructing meaningful local constants of motion that give a set of local quantum numbers @xcite . it seems likely that tools using energy filtering will again prove useful in this context , a prospect that we briefly discuss in appendix [ constants ] . on the practical side , further numerical effort will be needed to understand the behaviour of individual models and to fully understand the transport properties for different energy scales . our work could well provide a first stepping stone for further endeavours in this direction : showing that eigenstates are well approximated by matrix - product states implies that not only ground states , but in fact also excited states can efficiently be described in terms of tensor networks . our result as such does not yet provide an efficient algorithm to find the respective matrix - product states : this reminds of the situation of the existence of lattice models for which the ground states are exact matrix - product states , but it amounts to a computationally difficult problem to find them @xcite . still , this appears to be a major step in the direction of formulating such numerical prescriptions of describing the low - energy sector of many - body localizing systems . eventually , the leading vision in any of these endeavours appears to be a rigorous proof of many - body localisation in the spirit of the original results by anderson . for this , creating a unifying framework and linking the possible definitions seems a key first step . _ acknowledgements . _ we thank h. wilming , c. gogolin , t. osborne , f. verstraete and f. pollmann for insightful discussions and m. goihl for collaboration on the numerical code . we would like to thank the eu ( siqs , aqus , raquel , cost ) , the erc ( taq ) , the bmbf , and the studenstiftung des deutschen volkes for support . vbs is supported by the swiss national science foundation through the national centre of competence in research ` quantum science and technology ' and by an eth postdoctoral fellowship . wb is supported by epsrc . 10 url # 1`#1`urlprefix[2]#2 [ 2][]#2 . _ _ * * , ( ) . ( ) . . . in _ _ , vol . of _ _ ( , ) . & . . & . _ _ ( ) . & . _ _ * * , ( ) . , & . _ _ * * , ( ) . , & . _ _ * * , ( ) . , & . . , & . _ _ * * , ( ) . & . _ _ * * , ( ) . & . _ _ * * , ( ) . , & . _ _ * * , ( ) . , & . . , & . _ _ * * , ( ) . , & _ _ * * , ( ) . . _ _ * * , ( ) . . _ _ * * , ( ) . , & . _ _ * * , ( ) . , & . _ _ * * , ( ) . , , & . _ _ * * , ( ) . & . _ _ * * , ( ) . & . _ _ * * , ( ) . & . _ _ * * , ( ) . , & . _ _ * * , ( ) . , & . _ _ * * , ( ) . , , & . _ _ * * , ( ) . , & . _ _ * * , ( ) . . . , & . . , , & . _ _ * * , ( ) . & . _ _ * * , ( ) . & . _ _ * * , ( ) . , , & . . , & . _ _ * * , ( ) . . _ _ * * , ( ) . & . _ _ * * , ( ) . , & . . & . in & ( eds . ) _ _ , vol . of _ _ ( , ) . . _ _ * * , ( ) .

in the local universe , large scale gas outflows are observed to arise in galaxies exhibiting high surface densities of star formation . while the precise roles of such outflows , including galactic `` superwinds '' , in galaxy evolution are still being determined , simulations suggest that the balance between outflows and the accretion of cool gas is one of the primary mechanisms by which star formation is regulated in individual halos ( e.g. , oppenheimer et al . , 2009 ; brooks et al . , 2009 ) . at the current epoch , the highest star formation rate ( sfr ) surface densities and therefore galactic winds are preferentially found in relatively low - mass halos , such as those hosting dwarf starburst galaxies . however , the mass of halos containing the highest specific star formation rates ( ssfrs ) are thought to increase with increasing look - back time , consistent with `` top - down '' galaxy formation scenarios ( cowie et al . , 1996 ; neistein et al . , 2006 ) . tracking the occurrence of galactic winds across cosmic time , particularly in manners unbiased by luminosity or halo mass , would provide a powerful way to study such models . furthermore , outflows are believed to be required to explain a wide variety of astrophysical observations , from the shape of the galaxy luminosity function ( benson et al . , 2003 ; khochfar et al . , 2007 ) , to the stellar mass - metallicity relation ( tremonti , 2004 ; erb et al . , 2006 ; brooks et al . , 2007 ; finlator & dav , 2008 ) , the large extent of dust and metals in galactic halos and in the intergalactic medium ( scannapieco , ferrara , & madau , 2002 ; oppenheimer & dav , 2006 ; kobayashi et al . , 2007 ) , and many other related phenomena . despite their clear importance in the galaxy formation process , outflows have generally been overlooked theoretically and , until recently , have proved difficult to study observationally at redshifts @xmath3 , the epoch when the universe formed most of its stars and superwinds were ubiquitous . surveys have identified outflowing gas from galaxies at redshift @xmath4 through strong , blue - shifted resonance - line absorption in their spectra arising in low - ion gas entrained in the flows ( e.g. , pettini et al . , 2001 ; shapley et al . , 2003 ; tremonti , moustakas , & diamond - stanic , 2007 ; wiener et al . , 2009 ) . however , there are two important limitations inherent in methods which rely only on spectra of the outflow hosts . first , they provide no information on the location of the outflowing gas ; it is only presumed that the material reaches the igm . secondly , these surveys searched for evidence of outflows in spectra of either the brightest galaxies at the relevant redshift , known star - forming galaxies , or known post - starburst galaxies . what is needed is the reverse experiment : a survey of the galaxies from which known large - scale outflows originate . this begs the question : how does one identify a galactic wind without _ a priori _ knowledge of the galaxy itself ? quasar absorption lines may offer such an opportunity , as they select galaxies based on the gas absorption cross section , with no direct dependence on emission from the galaxy . the physical processes that determine the properties of intervening low - ion quasar absorption line systems are not well understood . while it has long been known that such absorbers can in general be identified with individual galaxies ( e.g. , bergeron & boiss , 1991 ; steidel , dickinson , & persson , 1994 ) correlations between emission ( i.e. , of the galaxy ) and absorption ( i.e. , strength and velocity structure of the absorbing gas ) properties have been elusive ( kacprzak et al . , 2007 ) and/or inconclusive ( e.g. , steidel et al . , 2002 ; kacprzak et al . , 2010 ) . the strongest absorbers , which have until recently been neglected due to their relative scarcity , may hold important clues . for example , bond et al . ( 2001 ) considered the velocity profiles of the strongest absorbers known at the time ( rest equivalent widths @xmath5@xmath6 ) measured with high - resolution spectroscopy , and proposed that such systems may arise in galactic superwinds . detections of outflows through broad low - ion absorption in the spectra of starbursting galaxies suggest that galactic winds _ can _ result in very strong absorption along a sightline past a galaxy . however , this does not necessarily imply that all or even any of the strongest intervening absorbers detected in quasar spectra actually _ do _ arise in the winds of foreground galaxies . indeed , models have been proposed that account for the observed distribution of absorption strength without relying on outflows ( e.g. , tinker & chen 2008 ) . alternatively , the huge kinematic spreads that define the strongest systems may be due simply to the chance intersection of the sightline with multiple `` normal '' absorbing galaxies in a rich group or cluster , as first suggested by pettini et al . ( 1983 ) . very large absorber surveys ( e.g. , nestor et al . , 2005 ; prochter , prochaska , & burles , 2006 ; quider et al . , 2010 ) , which are now becoming available , have uncovered large numbers ( e.g. , @xmath7 in quider et al . ) of `` ultra - strong '' ( ) systems with @xmath5@xmath8 . catalogs of such systems afford the opportunity to explore in depth the proposed absorber - galactic wind connection . recent work has already given support for a connection between the strongest systems and star forming galaxies , which is usually considered as support of an outflow scenario . for example , by stacking thousands of relatively - shallow sloan digital sky survey ( sdss ) images of the fields of strong absorption systems zibetti et al . ( 2007 ) demonstrated the strongest systems are associated with bluer galaxies closer to the sightline to the background quasar compared to weaker systems . similarly , bouch et al . ( 2007 ) have detected strong h@xmath9 emission at the absorption redshift towards strong absorbers and rubin et al . ( 2009 ) have identified an absorber in the spectrum of a background galaxy that they identify with a wind from a foreground galaxy . perhaps the most compelling evidence suggesting a connection between absorbers and star formation is the relation between @xmath5 and emission discussed by mnard et al . ( 2009 ) , wherein they demonstrate that the strongest absorbers are on average associated with the highest luminosity densities and are therefore likely to be hosted by vigorously star - forming galaxies . nestor et al . ( 2007 ; hereafter ntrq ) published the first imaging survey aimed specifically at the strongest absorption systems , including images of the fields of thirteen moderate - redshift ( @xmath10 ) systems . these revealed bright galaxies at relatively low impact parameter to the absorption sightline ( compared to the fields of most absorbers ) . while consistent with the outflow model in general and , e.g. , the results of zibetti et al . , in particular , detailed study of the galaxies associated with absorbers is needed to test this putative connection . if the `` ultra - strong '' nature of these absorbers is indeed linked to galactic winds , we expect to find evidence of recent , high mass fraction starbursts in one or more of the low impact parameter ( low-@xmath11 ) galaxies . in this paper , we present the results of an imaging and spectroscopic study of the galaxies detected in two absorber fields from the ntrq sample , conducted to test both the absorber / outflow connection and alternative models . in 2 we describe the selection of targets , observations and reductions of our data , and our basic observational results . in 3 we discuss the properties of the galaxies determined to be at similar redshift to the systems . we present further discussion in 4 and 5 , and summarize the paper in 6 . throughout the paper we assume a cosmology with @xmath12 , @xmath13 and @xmath14 and state magnitudes in the ab system . the targets for imaging in ntrq were chosen from the now publicly - available pittsburgh absorber catalog ( quider et al . , 2010 ) based solely on @xmath5 , absorption redshift and observability , and thus should represent an otherwise unbiased sample of absorber environments at those redshifts . many of the images from that study reveal a fairly bright ( @xmath15 ) isolated galaxy or clump of emission at small impact parameter ( @xmath16 ) to the sightline , representing clear candidates for the object(s ) associated with the absorption . we were awarded 23 hours of the gemini - north queue in semester 2008a to obtain spectra of these candidates using the gemini multi - object spectrograph ( gmos ) . however , as the time was awarded in `` band 3 '' , it was likely that the observations would take place in sub - optimal conditions . thus , it would be exceedingly difficult to obtain observations of targets with particularly faint apparent magnitudes or those with small angular separation from the background quasar . therefore , we limited the observations to long exposures of three fields from ntrq : those towards sdss j074707.62 + 305415 ( hereafter q0747 + 305 ; @xmath17 , @xmath5@xmath18 ) , sdss j101142.01 + 445155.4 ( q1011 + 445 ; @xmath19 , @xmath5@xmath20 ) and sdss j141751.84 + 011556.1 ( q1417 + 011 ; @xmath21 , @xmath5@xmath22 ) . the primary absorber - galaxy candidates in these fields are all relatively bright ( @xmath23 ) although the long exposures allow the determination of redshifts for fainter targets , as well and at sufficiently large impact parameter ( @xmath24 ) to place slitlets that avoid the point spread function of the quasar even in poor seeing . these are the three fields that ntrq categorize as `` bright '' in their descriptions of the absorber environments . it is important to emphasize that , as these fields differ in appearance from the majority of those imaged in ntrq , they might not be representative of systems in general . we discuss this further in [ sect : discusion ] . we show the absorption region of the sdss spectrum of the quasar in the q0747 + 305 and q1417 + 011 fields in figures [ fig:0747 ] and [ fig:1417 ] . the absorber towards q0747 + 305 has an observed profile consistent with the minimum possible velocity spread given its @xmath5 , @xmath25 . for the absorber towards q1417 + 011 , the doublet members are blended but the profiles appear to span @xmath26 . the vertical lines in each figure correspond to relative velocity offsets of galaxies in the field ( see below ) . pre - imaging was obtained for the three fields with the @xmath27-band filter on gmos early in semester 2008a . as the wiyn photometry from ntrq utilized the @xmath28-band for these three fields , the relatively deep pre - imaging also provided @xmath29 ( @xmath30 rest - frame @xmath31 at @xmath32 ) colours for sources detected in both data sets . reduced and combined images were provided by gemini . sextractor ( bertin & arnouts , 1996 ) was used for deblending of sources and obtaining photometric measurements . photometric zero - points were determined using bright unsaturated sources in our images having sdss magnitudes available from the sdss website . slit masks were designed with 2-wide slitlets and lengths which varied to optimize placement in the crowded regions near the quasar while still allowing for individual sky measurements . spectra were obtained in the q0747 + 305 field on 4 and 13 march 2008 with total integration time of 22800s over twelve individual exposures , and in the q1417 + 011 field on 1 , 2 , 10 and 12 may and 11 june 2008 with a total integration time of 28500s over fifteen exposures . a single exposure of 1900s in the q1011 + 445 field was obtained on 12 april 2008 , but with an insufficient signal to noise ratio for scientific purposes ; we thus limit further discussion to the q0747 + 305 and q1417 + 001 fields . all spectra were obtained using the r400 grating and og515 filter with three different grating tilts to obtain continuous coverage across the detector gaps . spectra were reduced and extracted in standard fashion with exposures of quartz halogen and cuar lamps used for flat - fielding and wavelength calibration . individual sky - subtracted extracted spectra were combined with weighting by signal to noise ratio . slitlets were placed over stars in each field , which aided in the removal of telluric features in the spectra . a single observation of hz44 on 13 march 2008 was provided to facilitate relative spectro - photometric calibration . absolute spectro - photometric calibration was achieved by comparing synthetic magnitudes derived from the spectra to our photometry . the pre - imaging was accomplished under better than expected conditions , with low atmospheric attenuation and seeing of @xmath33 for each field . the images are deeper than the @xmath28-band images from ntrq , with @xmath34 down to @xmath35 for compact sources . the signal to noise ratios of the resulting spectra varied with both wavelength and the brightness of the source , and for our primary targets ranged from @xmath36 to @xmath37 . subtraction of sky emission lines was problematic , particularly in the red wavelengths and for the fainter sources . similarly , correcting for telluric absorption was only moderately successful . we tested our relative spectrophotometric calibration by comparing synthetic colours derived from the calibrated spectra of our brighter sources to colours from the sdss , as well as a direct comparison of the spectrum of one source that was spectroscopically observed by both us and the sdss . various issues complicated such comparisons , such as slit losses and photometric uncertainties . thus , we are only able to put a limit on our relative spectrophotometric accuracy of better than @xmath38 twenty per cent . emission lines were fit with single gaussian profiles ( no emission multiplets were resolved ) and line fluxes and flux uncertainties determined using an optimal extraction procedure . the region of the gmos @xmath27-band image surrounding the sightline for this field is shown with slitlet positions overlaid in figure [ fig:0747 ] . we were able to place slitlets over all resolved sources brighter than @xmath42 within an impact parameter of @xmath43kpc at @xmath44 , and brighter than @xmath45 within @xmath46kpc . comparing to the characteristic magnitude @xmath47 at redshifts @xmath48 ( gabasch et al . , 2006 ) , these limits correspond to approximately 0.08 @xmath49 and 0.23 @xmath49 , respectively , for a sc - like @xmath50-correction . in table 1 , we present the photometry for all galaxies detected with @xmath51 within @xmath52kpc of the sightline at @xmath44 and/or with a slit spectrum from our observations , together with the determined redshift if spectroscopically observed . values of @xmath53 are from ntrq . .observed galaxies towards q0747 + 305 ( @xmath40 ) [ cols="^,^,^,^,^,^,^ " , ] @xmath54 may be inaccurate due to poorly - measure flux ; see text . the @xmath55 galaxies are bright , with rest - frame @xmath56-band luminosities 2.4 @xmath57 and 0.9 @xmath57 for g07 - 1 and g07 - 2 respectively , and 1.8 @xmath57 and 0.3 @xmath57 for g14 - 1 and g14 - 2 , where @xmath57 is given at similar redshift by gabasch et al . ( 2004 ) . comparing the rest - frame @xmath58 colours to local galaxies in the sdss from blanton et al . ( 2003)-corrected to @xmath59 , the effect on the colours is negligible . ] , we find that g07 - 1 and g07 - 2 are typical of bright galaxies , while g14 - 1 and g14 - 2 are considerably bluer than most bright galaxies , suggesting seds dominated by young ob stars . while we lack accurate dust - corrected sfrs estimates , all four of the galaxies clearly exhibit significant ongoing star formation . in order to compare our estimated sfrs with those of other galaxies at similar redshifts , we consider the sfrs corresponding to the characteristic uv and h@xmath9 luminosities ( i.e. , @xmath60 and @xmath61 ) using luminosity functions at @xmath2 from the compilation of hopkins et al . we find values for `` sfr@xmath62 '' between 1.6 - 9.3 m@xmath63 yr@xmath64 . thus , galaxies with comparable or greater sfrs are relatively rare by number but account for a significant amount of the star formation density in the universe at @xmath2 . the results of our template - fitting suggest that g07 - 1 is quite massive , sitting above the knee in the galaxy stellar mass function at @xmath65 ( e.g. , drory et al . , 2009 ) , that g07 - 2 and g14 - 1 lie in the relatively flat part of the stellar mass function , and that g14 - 2 is a notably low - mass galaxy . combining the mass estimates with the photometry , we compute b - band mass - to - light ratios which we compare to models for bursty spiral galaxies by bell & de jong ( 2001 ) . g07 - 1 and g07 - 2 have m / l@xmath66 values as expected of galaxies with neutral b@xmath67r colours , while g14 - 1 and g14 - 2 have particularly low m / l@xmath66 values even for very blue galaxies . again , this implies that g14 - 1 and g14 - 2 are undergoing notably high mass - fraction starbursts . we can also obtain estimates of the rotation velocities , @xmath68 , through the tully - fisher relation using either the absolute b magnitudes or stellar masses ( e.g. , fernndez lorenzo et al . 2009 ; kassin et al . 2007 ) , which , in turn , allow us to approximate the respective escape velocities , @xmath69 ( veilleux , cecil , bland - hawthorn , 2005 ) . we find @xmath70 - 800 and @xmath71 - 600 for g07 - 1 and g07 - 2 , respectively , and @xmath72 - 800 and @xmath73 - 500 for g14 - 1 and g14 - 2 , respectively . the mass estimates allow us to compute ssfrs for each galaxy , which are shown in table 4 . these can be compared to galaxies at similar mass and redshift ( feulner et al . , 2005 ) . the ssfrs for all four galaxies are among the highest at their mass / redshift ( see feulner et al . , figure 1 ) . g14 - 1 and g14 - 2 are well above and g07 - 1 and g07 - 2 near the `` doubling '' line in ssfr , which is often used to distinguish between passively star forming and starbursting galaxies . the metallicities of each of the four galaxies are typical for their mass at @xmath32 ( savaglio et al . , 2005 ; maiolino et al . , it is not clear how to simultaneously interpret gas - phase metallicities in region determined from galactic emission lines together with those determined from absorbing regions at significant galactocentric distance . it is noteworthy , however , that while measurements reported in the literature for `` typical '' low - ion absorption systems are largely metal poor ( e.g. , pettini et al . , 1999 ; meiring et al . , 2008 ; nestor et al . , 2008 ) , it has been demonstrated that gas - phase metallicity is correlated with rew ( e.g. , nestor et al . , 2003 ; turnshek et al . , 2005 ; murphy et al . , 2007 ) , and those in are expected to approach those measured in our galaxies . finally , the pc1 and pc2 amplitudes for each galaxy can be compared to the categories defined by these indices in figure 9 in wild et al . the points in that figure correspond to spectra of local ( @xmath74 ) sdss galaxies , which are dominated by light from the inner @xmath75kpc as the sdss fibers are generally significantly smaller than the sizes of galaxies at those redshifts . nonetheless , the physical interpretation of the principal component amplitudes ( i.e. , for categorizing stellar populations as quiescent , starburst , etc . ) is robust . and have pc1 and pc2 amplitudes placing them firmly in the post - starburst region of the pc1-pc2 plane , while and are firmly in the starburst region . notably , these results are consistent with the findings from each of our other methods described above . comparing these pc1 and pc2 values to the models ( 3.3 ) , we find estimates for the ages for the starbursts of @xmath76 gyr for and , @xmath77 myr for and @xmath78 myr for . it is interesting that the star formation outbursts for the two galaxies in each field appear to be coeval . this strongly suggests that the starbursts are related to their proximity ; i.e. , triggered by an interaction . to summarise our observational results : ( i ) each field contains a pair of emission - line galaxies with @xmath79 , at impact parameters @xmath80kpc and @xmath81kpc to the absorption sightlines ; ( ii ) and appear to be fairly massive , bright galaxies ; ( iii ) and are also bright but less massive and have very blue rest - frame @xmath82 colours ; ( iv ) all four galaxies have high ssfrs and have metallicities typical for their mass ; ( v ) and are roughly 1 gyr removed from a starburst phase , though both exhibit ongoing star formation ; and ( vi ) and appear to currently be undergoing a starburst episode . we undertook the observations discussed above with the goal of uncovering the physical mechanism driving the extreme absorption velocity spreads that define systems . in this section , we test the star formation - driven galactic wind model in light of the results of these observations , and find abundant circumstantial evidence in its favour . we conclude the section by considering other popular models , and find tidally - stripped gas from interacting galaxies is also consistent with the observational results for some systems . first , we consider the possibility that the presence of starburst galaxies in the vicinities of ultra - strong absorption are coincidental . indeed , the existence of a strong absorber requires a galaxy at @xmath83 . however , galaxies with such properties as we find for , , and are uncommon ( e.g. , bell & de jong 2001 ; feulner et al . , 2005 ; wild et al . , 2009 ) . similarly , the presence of a galaxy produces a likelihood of detecting a absorber with @xmath84 , but systems with @xmath5@xmath85 account for only 1 per cent of strong ( @xmath5@xmath86 ) systems , and those with @xmath5@xmath87 only 0.05 per cent . thus it is exceedingly unlikely that the ( post-)starburst nature of the galaxies and the ultra - strong nature of the absorption are unrelated . this does not necessarily imply the starbursts are driving a wind that is responsible for the absorption . for example , it is conceivable that they share a common cause , such as a major interaction stripping gas out to large galactic radii while simultaneously triggering a starburst , or that such an interaction channeled gas to the galaxy centers and fed galactic nuclear activity , which in turn drove the outflow . the minimum rest - frame velocity width of a completely saturated , opaque @xmath88 feature is given by @xmath89(@xmath5@xmath901)@xmath91 . for systems ( i.e. , 3 @xmath92 @xmath5 @xmath93 6 ) , this corresponds to 320 @xmath94 640 . actual kinematic spreads are typically larger due to partially non - opaque profiles . as discussed in [ sec : targets ] , the absorber towards q0747 + 305 ( figure [ fig:0747 ] ) and q1417 + 011 ( figure [ fig:1417 ] ) have observed profiles consistent with @xmath95 , and @xmath26 , respectively . thus , the physical mechanism behind systems must be one that is able to produce cool gas that , along a single line - of - sight , continuously spans many hundreds of with a total dynamic spread of up to @xmath96 1000 . such a dynamic range can naturally be obtained along a sightline passing though a galactic wind at some angle to the outflow orientation . the impact parameters of the four @xmath79 galaxies in the present study are comparable to those of the general population of absorbers ( see , e.g. , kacprzak et al . , 2007 ) . they are , however , at the extreme end of the @xmath11-distribution for systems ( ntrq ) . it is therefore worthwhile to consider the inferred velocities and timescales of the putative outflows in light of the distances needed to be traversed by outflowing material to cover the sightline to the qso . the observed absorption velocity spread arises from projections of the outflow velocity ( of sufficient columns of low - ion gas ) onto the sightline , and thus depends on the unknown geometry . however , order of magnitude estimates of the outflow speeds can be made by considering the red- and blue - most velocity extents relative to the galaxy systemic velocity ( e.g. , figures [ fig:0747 ] and [ fig:1417 ] ) together with a conic outflow geometry featuring opening angles between @xmath97 and @xmath98 ( veilleux , cecil , bland - hawthorn , 2005 ) . while such a calculation is overly simplistic , it should give an order of magnitude estimate for the outflow speeds which can be interpreted together with our estimations of the age of the most recent starburst . for the q0747 + 305 field , we estimate an outflow velocity of @xmath99 - 600 . this range is comparable to our approximation of the escape velocities for these galaxies ( [ sec : props ] ) . if outflows from both galaxies contribute to the observed profile , then the velocity could be as low as @xmath100 . at these velocities , the gas would have to have flowed for a minimum of @xmath101 - 500 myrs to have reached the distance to the sightline . these timescale estimates are well below our estimate of a @xmath102 gyr age for the bursts in and . thus , the relatively large impact parameters in this field are completely consistent with a scenario involving outflows driven by the starburst event . the starbursts in the q1417 + 011 field are likely much younger . as can be seen in figure [ fig:1417 ] , however , much larger velocities are also necessary to explain the absorption . for an outflow to account for the absorption , we estimate @xmath103 - 1000 , which exceeds our approximations of the escape velocities . the velocity estimate changes little if both galaxies contribute , as they are at similar redshift and both @xmath104 . though rather large , such velocities are indeed seen in some galactic winds , especially at high redshift ( e.g. , pettini et al . , 2002 ; quider et al . , 2009 ; dessauges - zavadsky et al . , this translates into minimum travel times of @xmath105 - 200 myrs , which is roughly consistent with our age estimates for the bursts in this field . unfortunately we do not cover or other strong low - ion absorption features in our spectra of the @xmath83 galaxies . thus , we are not able to directly identify cold gas entrained in outflows via profiles blue - shifted with respect to the emission lines . however , we reiterate that strong , broad , blue - shifted low - ion absorption is ubiquitous in the spectra of star - forming galaxies at all redshift ( heckman , 2000 ; tremonti , moustakas , & diamond - stanic , 2007 ; weiner et al . , 2009 ; vanzella et al . , 2009 ; steidel et al . , an example of blue shifted _ ultra_-strong low - ion absorption in a galaxy spectrum is the well - known @xmath106 starbursting galaxy 1512-cb58 , with rews in and lines of 4.4 ( pettini et al . , 2000 ) . these features are generally considered signs of large - scale outflows . finally , we note that the incidence of systems ( nestor et al . , 2005 ; nestor et al . , in preparation ) and the global sfr density ( hopkins et al . , 2006 ) show a remarkably similar fractional decrease from @xmath107 to @xmath2 . the cause and consequences of the equivalence of these trends are unclear ; however it suggests that absorbers have some strong dependence on star formation . for the systems studied here , the associated galaxies have properties indicating significant recent / ongoing star formation episodes , suggesting a connection between the absorbers and star formation - driven galactic outflows . this is in contrast to the galaxies associated with the majority of weaker absorption , which are found to span a range of colours and morphologies , with most considered `` normal '' galaxies . the physical nature of these absorbers is still an open question . traditionally , the absorption kinematics have been thought to arise from a variety of causes including rotating gaseous disks , virialised clouds , and accreting gas . as we have not _ directly _ observed a galactic wind in our systems , we must consider if the phenomena thought to be responsible for the absorption kinematics in weaker systems , or perhaps some other uncommon phenomenon , can lead to absorption alone , without recourse to a galactic wind . detailed studies of disk - galaxy/-absorber pairs have shown that extended rotating gas disks likely account for part of the absorption kinematics in some systems ( steidel et al . , 2002 ; kacprzak et al . , 2010 ) . however , the projected spread in velocity of a sightline through a gas disk should be less than the disk rotation velocity . spreads in systems are much larger than those expected for galaxies of the luminosities of our galaxies ( catinella , giovanelli , & haynes , 2006 ) . comparison of the neutral hydrogen gas density of galaxies with that of the cosmic star formation rate density reveals the necessity for continual replenishment of ( hopkins mcclure - griffiths & gaensler , 2008 ) . could the observed velocity spreads in absorbers be due to sightlines through such condensing gas , e.g. , in streams or clouds of accreting cool gas ? again , it is difficult to reconcile the huge range in absorption velocity along a single sightline with accreting gas alone . a combination of a contribution from a rotating disk and accretion is more palatable for less - ultra strong systems such as q0747 + 035 . the accretion scenario , however , may favor finding the host galaxy in a pre - starburst or early starburst phase , whereas and appear to be in a post - starburst phase . the large spread in absorption velocity may be due to the chance intersection of the sightline with multiple halos hosting less - strong `` typical '' absorbers . this scenario has additional appeal for the two sightlines in the present study since they both contain a pair of galaxies at @xmath108 as opposed to the majority of the ntrq sample which have only a single absorber - galaxy candidate . this is particularly true for q0747 + 035 , in which the two galaxies bracket the absorption redshift . this scenario would be supported if the kinematics split into two or more distinct groupings of velocity components when observed with higher resolution . unfortunately , we do not have such observations for our systems . notwithstanding , insight can be gained by considering high - spectral resolution observations of other absorbers , including systems . in figure [ fig : uvesspecs ] we show the @xmath88 or @xmath109 ( as the red - most components of the @xmath88 lines overlap the saturated regions of the @xmath109 line in two systems ) transition for five @xmath110 systems observed with uves on the vlt . these data will be discussed in detail in an forthcoming paper . the rew of systems ( c ) , ( d ) and ( e ) in figure [ fig : uvesspecs ] are dominated by a single , wide , opaque component , while the dominant component in systems ( a ) and ( b ) are strong enough to qualify as absorbers on their own . while many intermediate strength ( i.e. , 1@xmath93 @xmath5 @xmath111 ) systems break up into multiple weaker components not resembling those shown in figure [ fig : uvesspecs ] , some are dominated by a single broad opaque component . for example , c.f . figure 2 of mshar et al . ( 2007 ) , which shows the kinematics of a sample of absorbers ranging from @xmath5@xmath112 to @xmath5@xmath113 , as well as a pair of systems . returning to q0747 + 035 , could two separate weaker systems be the cause of the apparent absorber ? the width of the absorption in the sdss spectrum implies a system dominated by a single opaque component ( [ sec : kin ] ) . thus , this scenario would require two @xmath5@xmath114 opaque single - component systems at just the right velocity separation ( @xmath115 200 ) . while possible , we consider this scenario to be less likely than the outflow ( starburst - driven or tidally - stripped ) scenarios . the multiple overlapping - absorber scenario is less appealing as an explanation for the velocity spread in the q1417 + 011 system due to the extreme rew and that both galaxies have velocities @xmath116 below the red - edge of the observed absorption . absorbers ( nestor et al . , in preparation ) . panels ( a ) , ( c ) and ( e ) show the @xmath88 transition , while panels ( b ) and ( d ) show @xmath109 . blends from other lines are pixelated . the rew of systems ( c ) , ( d ) and ( e ) are dominated by a single , wide , opaque component , while the dominant component in systems ( a ) and ( b ) are strong enough to qualify as absorbers on their own.,width=302 ] gaseous disks do in some cases contribute to absorption kinematics , the accretion of cool gas must occur at these redshifts and may lead to absorption in , and some close pairs of systems must exits . the magnitude of the velocity extent of the _ strongest _ absorbers , however , favours outflows . this conclusion is strengthened by the presence of galaxies hosting recent starbursts in both of the fields in this study . as we have mentioned above , however , it is not as clear that the starbursts are actually driving the outflows . tidally - stripped gas would likely also span a relatively large range in velocity and the interaction would be expected to trigger star formation episodes , making such a scenario difficult to distinguish observationally from a star formation - driven wind . the presence of two @xmath83 galaxies in both of the fields in this study is consistent with this scenario . typical velocity spreads in stripped gas might be expected to be of the order of the velocity dispersion of groups , where gas - rich major interactions are most common . although the velocity spreads of ( relatively ) modest systems are comparable to that of typical groups , the strongest systems , including the system toward q1417 + 011 , exceed typical group dispersions . furthermore , we note that many systems appear isolated in the ntrq imaging data . while those data are of insufficient quality to rule out minor interactions , which may in principle trigger starbursts , no evidence of the major interactions likely needed to strip large amounts of gas across such huge velocity spreads , such as distorted morphologies or tidal tails , is seen in most fields . regarding agn , we lack the necessary emission - line diagnostics to discriminate between starbursts and liner / seyfert galaxies . however , the 4000 break and balmer absorption strengths clearly indicate recent / ongoing starbursts . nonetheless , while the data and kinematics favour star formation - driven outflows over gas stripped from galaxies undergoing a high mass - fraction interaction or agn - driven winds , the latter scenarios should not be dismissed , particularly as an explanation for some fraction of the less - extreme systems . either way , each of these scenarios involve outflowing gas associated with star - forming galaxies ; the primary difference being the energy source responsible for the kinematics . this work presents the first two system host galaxies studied in detail selected from a statistically - understood quasar absorption line sample . we have shown that outflows driven by galactic star formation appear to be the source of the strongest absorbers ( see also rubin et al . 2009 ; steidel et al . significantly , these systems are identified _ without direct dependence on galaxy luminosity . _ thus , systems provide a method of tracing galactic winds and thus high densities of star formation in the universe selected in a manner complementary to , and without the biases of , emission - based methods . we have now collected imaging data for the majority of relatively low - redshift systems in the sdss spectra through the fourth data release and intend to investigate the nature of the host galaxies using the entire sample . such larger studies are needed to accurately map the connection between @xmath5 and the detailed properties of the associated galaxies , which will provide important constraints on our understanding of the global evolution of star - forming galaxies . in this section , we discuss some of the implications of this connection . we have argued that the presence of the _ strongest _ absorbers is related to highly enhanced levels of star formation in galaxies . however , the connection between star formation and @xmath5 discussed by mnard et al . ( 2009 ) is significant for systems ranging from absorbers down to at least @xmath5@xmath117 . usmgii absorber galaxies almost certainly have significant cross section for absorption at all @xmath5@xmath118 , and therefore are only observed as usmgii systems , as opposed to weaker systems , by the chance alignment of the sightline through the region of larger @xmath5 . ( for the same reason , some fraction of systems observed as weaker absorbers must also have significant usmgii absorption cross section . ) furthermore , the intrinsic @xmath5-distribution exhibits the form of a single exponential above @xmath5@xmath119 that is featureless despite excellent sampling ( nestor et al . , 2005 ; nestor et al . , in prep ) , suggesting a common underlying cause for absorption at these strengths . thus , it may be that _ all _ @xmath5@xmath120 systems are associated with galaxies that have experienced enhanced star formation within their past few gyr . strong absorbers ( @xmath5@xmath120 ) are known to trace high - column densities of neutral hydrogen ( @xmath121 @xmath122 ) . damped ly@xmath9 absorbers ( dlas , defined as having @xmath123 @xmath122 ) , which have near - unity neutral fractions of atomic hydrogen and are generally metal - poor , have long been considered the reservoirs of cool gas for star formation . below @xmath124 @xmath122 , subdlas exhibit a significant ionisation fraction and higher metallicities . rao , turnshek , & nestor ( 2006 ) investigated the relationship between strong @xmath5 and @xmath125 over @xmath126 , finding : ( i ) at a given @xmath5 absorbers span @xmath127 to 4 orders of magnitude in @xmath125 ; ( ii ) @xmath125 is not a good predictor of @xmath5 , although the lowest @xmath125 subdlas are predominantly @xmath5@xmath128 ; and ( iii ) the likelihood of a absorber having @xmath123 @xmath122 correlates with @xmath5 . these trends fit the outflow scenario very well . the bulk of the in dlas is almost certainly associated with only a portion of the velocity profile in the corresponding absorption . thus , absorbers and dlas are largely not arising in the same gas . as high ssfrs galaxies should also be gas - rich , a sightline passing through a star formation - driven wind may also pass through a gas - rich portion of the galaxy . furthermore , that most dlas are not absorbers is consistent with the fact that not all gas - rich galaxies having sufficiently high ssfrs to drive outflows . the relative contribution of absorber galaxies to the global star formation density at any redshift can be used to infer the proportion of star formation that takes place at high enough star formation surface density to drive winds at that epoch . we can write this fraction as @xmath129 where @xmath130 is the average sfr in galaxies selected by absorption , @xmath131 is the line - of - sight number density of absorbers , @xmath132 is the differential proper distance , @xmath133 is the average cross section for absorption , and @xmath134 is the global sfr density . at @xmath135 , @xmath136 m@xmath63 yr@xmath64 mpc@xmath137 ( hopkins & beacom , 2006 ) . we have an accurate measurement of @xmath131 from our sdss catalogs ( nestor et al . , 2005 ; nestor et al . , in prep . ) , log(@xmath131 ) @xmath138 . thus , for given values of the average sfr for galaxies selected by absorption and the average absorption cross section , we can determine @xmath139 . figure [ fig : rho ] shows the values of @xmath130 and @xmath133 that result in various values for this fraction . we note that this fraction represents all galaxies having non - zero cross - section for usmgii absorption , regardless of whether it is actually observed as an usmgii system . , for values of the average sfr in galaxies associated with absorbers and the average absorption cross section for absorption . the ellipse represents the range preferred by the data in conjunction with our model , though the actual uncertainties are unclear ( see text).,width=302 ] the results of mnard et al . allow us to accurately compute the average luminosity from galaxies emitted at low - enough @xmath11 to the quasar sightline to fall within the sdss spectroscopic fiber used to obtain the quasar spectrum . this leads to a very firm lower - limit of @xmath140 m@xmath63yr@xmath64 . using the images presented in ntrq , we determine the fraction of galactic emission , on average , falling within the fiber to be @xmath141 per cent , or @xmath142 per cent when weighting by luminosity . finally , we determine the average reddening corrections for from moustakas , kennicutt , & tremonti ( 2006 ) considering the typical luminosity of galaxies in ntrq to be a factor of @xmath143 . thus , we estimate @xmath144 to 8 m@xmath63yr@xmath64 . this value is consistent with the uncorrected sfrs presented in this work . while it is lower than our inferred dust - corrected sfrs , we again note that the two systems in this study correspond to two of the brightest , highest-@xmath11 host galaxies from ntrq , and therefore may be expected to have sfrs exceeding the average for systems in general . to approximate @xmath133 , we consider the distribution of impact parameters in ntrq and a bi - conic model geometry with a range of opening angles and base - widths , and account for non - uniformity of the absorption covering factor .- distribution with @xmath11 to @xmath30 15 or 20kpc in the ntrq sample is consistent with the average absorption covering factor being @xmath30 constant with radius out to this distance . at larger radius , the covering factor must drop but remain non - zero out to @xmath30 60kpc , to account for the @xmath30 50 or 30 per cent of absorbers spread over larger values of @xmath11 . ] doing so , we favour values for @xmath133 of @xmath145 - 1350kpc@xmath146 ( c.f . the approximation of @xmath147kpc@xmath148 in nestor et al . , 2005 ) . we show these ranges of @xmath130 and @xmath133 as the ellipse in figure [ fig : rho ] . the preferred range for @xmath139 is then 0.25 to 0.75 , indicating a significant contribution to the global sfr at @xmath149 from selected galaxies . we caution that the above calculation for the preferred range of @xmath139 relies on untested assumptions about the absorption geometry and informal estimation of the uncertainty in the average sfr . thus , it should be considered suggestive until future work is able to better constrain these quantities . nonetheless , the results have interesting implications on the nature of star formation as a function of redshift . if @xmath139 is indeed close to the global value at a given redshift , it implies that the bulk of the star formation at that epoch takes place at high enough star formation surface density to drive winds . alternatively , epochs where @xmath139 is found to be only a small fraction of the global star formation density , @xmath134 must be dominated by relatively diffuse star formation . finally , if it were to be found that @xmath150 at any epoch , it would suggest a significant contribution from faint dwarf galaxies which are being unaccounted for in the current or uv surveys yet have high enough sfr surface densities to drive winds we have presented a deep imaging and spectroscopic study of the fields of two systems from the ntrq sample of systems . in each field we find that there are two galaxies at the absorption redshift having strong emission lines of , , and . the emission line fluxes indicate the galaxies are metal rich and have significant ongoing star formation . we employed sed template fitting to estimate stellar masses , which indicates relatively high . analysis of the 4000 break and balmer absorption strengths indicate the @xmath55 galaxies toward q0747 + 035 underwent a starburst @xmath30 1 gyr in the past , while those towards q1417 + 001 are currently in a starburst phase . it is extremely unlikely to find galaxies with such properties at the same location as the absorption unless they are related in some way to the low - ion velocity spreads that define systems . we consider various popular models for the nature of absorption systems . given the ( post-)starburst natures of the galaxies , their velocities relative to the observed absorption kinematics , the burst ages , together with the impact parameters of the sightlines to the galaxies , we conclude that starburst - driven galactic winds are the most likely causes of the absorption . however , a scenario in which gas is tidally stripped by galaxy - galaxy interactions which simultaneously trigger starbursts is also consistent with the data . while past studies have found blueshifted absorption in the spectra of galaxies at cosmological distances , identifying outflows in this manner unambiguously demonstrates that the material reaches the igm . finally , we estimate that the star formation density traced by absorbers is , roughly , within at least an order of magnitude of the total global density , indicating that , though rare , absorbers are a powerful tracer of star formation in the universe . dbn and bdj acknowledge support from the stfc - funded galaxy formation and evolution programme at the institute of astronomy . vw acknowledges european union support from a marie curie intra - european fellowship . based on observations obtained at the gemini observatory , 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 science and technology facilities council ( united kingdom ) , the national research council ( canada ) , conicyt ( chile ) , the australian research council ( australia ) , ministrio da cincia e tecnologia ( brazil ) and ministerio de ciencia , tecnologa e innovacin productiva ( argentina ) .

star formation - driven outflows are a critically important phenomenon in theoretical treatments of galaxy evolution , despite the limited ability of observational studies to trace galactic winds across cosmological timescales . it has been suggested that the strongest absorption - line systems detected in the spectra of background quasars might arise in outflows from foreground galaxies . if confirmed , such `` ultra - strong '' ( ) absorbers would represent a method to identify significant numbers of galactic winds over a huge baseline in cosmic time , in a manner independent of the luminous properties of the galaxy . to this end , we present the first detailed imaging and spectroscopic study of the fields of two absorber systems culled from a statistical absorber catalog , with the goal of understanding the physical processes leading to the large velocity spreads that define such systems . each field contains two bright emission - line galaxies at similar redshift ( @xmath0 ) to that of the absorption . lower - limits on their instantaneous star formation rates ( sfr ) from the observed and line fluxes , and stellar masses from spectral template fitting indicate specific sfrs among the highest for their masses at these redshifts . additionally , their 4000 break and balmer absorption strengths imply they have undergone recent ( @xmath1 - 1 gyr ) starbursts . the concomitant presence of two rare phenomena starbursts and absorbers strongly implies a causal connection . we consider these data and absorbers in general in the context of various popular models , and conclude that galactic outflows are generally necessary to account for the velocity extent of the absorption . we favour starburst driven outflows over tidally - stripped gas from a major interaction which triggered the starburst as the energy source for the majority of systems . finally , we discuss the implications of these results and speculate on the overall contribution of such systems to the global sfr density at @xmath2 . [ firstpage ] intergalactic medium quasars : absorption lines ism : jets and outflows galaxies : starburst .

in the present analysis , we estimate a reference value of @xmath96 defined by eq . in the heavy hadron limit by using the following naive scaling property : @xmath97 where the coupling constants are defined in the relativistic form of the interaction lagrangian among a vector meson @xmath98 and two pseudoscalar mesons @xmath99 and @xmath100 espressed as @xmath101 the coupling @xmath102 appears in the decay width of @xmath103 as @xmath104 which with @xmath105mev and @xmath63gev leads to @xmath106 and @xmath107 are related to the vector meson masses as @xcite @xmath108 where @xmath56 and @xmath109 are the masses of @xmath14 and @xmath110 mesons , @xmath33 and @xmath111 are the pion and kaon decay constants , and @xmath112 is the gauge coupling constant of the hidden local symmetry @xcite . using @xmath65mev , @xmath113mev , @xmath114mev and @xmath115mev , the ratio of two couplings in eq . ( [ eq : k couplings ] ) is estimated as @xmath116 which with eq . ( [ couplingdstdpi ] ) leads to @xmath117 from the heavy quark lagrangian in eq . ( [ pionlagrangian2 ] ) , the @xmath14-@xmath5-@xmath15 interaction is written as @xmath118 with a scaling factor of the mass of heavy meson @xmath119 . comparing this with eq . ( [ vpp lagrangian ] ) , we estimate @xmath95 as @xmath120 which is the value used in the present work . the work of m. h. was supported in part by the jsps grant - in - aid for scientific research ( c ) no . m. was supported in part by national science foundation of china ( nsfc ) under grant no . 11475071 , 11547308 and the seeds funding of jilin university . c. e. detar and t. kunihiro , phys . d * 39 * , 2805 ( 1989 ) . w. g. paeng , h. k. lee , m. rho and c. sasaki , phys . d * 88 * , 105019 ( 2013 ) . y. l. ma , m. harada , h. k. lee , y. oh , b. y. park and m. rho , phys . d * 88 * , no . 1 , 014016 ( 2013 ) erratum : [ phys . d * 88 * , no . 7 , 079904 ( 2013 ) ] . m. a. nowak , m. rho and i. zahed , phys . d * 48 * , 4370 ( 1993 ) . w. a. bardeen and c. t. hill , phys . d * 49 * , 409 ( 1994 ) . a. hosaka , t. hyodo , k. sudoh , y. yamaguchi and s. yasui , arxiv:1606.08685 [ hep - ph ] . m. harada , m. rho , and c. sasaki , phys . d * 70 * , 074002 ( 2004 ) . a. mishra , e. l. bratkovskaya , j. schaffner - bielich , s. schramm and h. stoecker , phys . c * 69 * , 015202 ( 2004 ) . b. friman , j. phys . g * 30 * , s895 ( 2004 ) . s. yasui and k. sudoh , phys . c * 87 * , no . 1 , 015202 ( 2013 ) [ arxiv:1207.3134 [ hep - ph ] ] . d. suenaga , b. r. he , y. l. ma and m. harada , phys . c * 89 * , no . 6 , 068201 ( 2014 ) .

we study the omega meson effect as well as the sigma meson effect at mean field level on the density dependence of the masses of heavy - light mesons with chiral partner structure . it is found that the omega meson affects the masses of the heavy - light mesons and their antiparticles in the opposite way , while it affects the masses of chiral partners in the same way . as a result , the mass difference between chiral partners is proportional to the mean field of sigma , reflecting the partial restoration of chiral symmetry in the nuclear matter . in addition to the general illustration of the density dependence of the heavy - light meson masses , we also consider two concrete models for nuclear matter , the parity doublet model and skyrmion crystal model in the sense of mean field approximation . spontaneous chiral symmetry breaking is one of the most important properties of low energy qcd . it is expected that the spontaneous chiral symmetry breaking characterized by non - zero value of the quark condensate generates a part of hadron masses and causes the mass splitting between chiral partners . then , schematically , hadron masses can be expressed as a sum of the chiral invariant mass and the chiral non - invariant mass coming from the spontaneous chiral symmetry breaking . for example , for the nucleon mass , one has @xcite @xmath0 where @xmath1 is the chiral invariant mass and @xmath2 is the part of the mass that vanishes in the chiral symmetric phase . naturally , it is interesting to ask how much amount of a hadron mass is generated by the chiral symmetry breaking . an ideal environment to estimate the magnitude of the hadron mass coming from the spontaneous chiral symmetry breaking is qcd at extreme condition in which the chiral symmetry is believed to be partially restored . in such an environment this can be accessed by studying the temperature or / and , which will be done in this work , the density dependence of hadron mass . in the nucleon sector , by using an effective model with parity doublet structure of baryons , it was found that @xmath3 of nucleon mass comes from chiral symmetry breaking @xcite . however , when the baryon as a topological soliton in the hidden local symmetry lagrangian is immersed in the dense matter which is treated as skyrmion matter , people found that the chiral invariant mass composes @xmath4 of nucleon mass @xcite which roughly the same as that obtained based on the renormalization group analysis of hidden local symmetry lagrangian with baryons @xcite . in this paper , we study the medium modified mass splitting of heavy - light mesons with chiral partner structure in which , it is widely accepted that the mass splitting arises from the spontaneous breaking of chiral symmetry @xcite . such a kind of study can be tested in the planned experiments in j - parc , fair and so on . studying the properties of heavy - light mesons in medium is also expected to give clues for understanding the chiral symmetry structure ( see , e.g. , ref . @xcite for a review ) . the medium modified heavy - light meson spectrum has been studied by several groups in the literature @xcite . in ref . @xcite , it was shown that the @xmath5 meson ( @xmath6 ) is mixed with @xmath7 meson ( @xmath8 ) in the spin - isospin correlated matter , in which the mixing strength reflects the strength of the correlation . in refs . @xcite , by regarding the @xmath9 ( @xmath10 ) and @xmath11 ( @xmath12 ) mesons as the chiral partners to @xmath5 and @xmath7 mesons , it was shown that the mass splitting of the chiral partner is reduced at high density and temperature . in particular in ref . @xcite , by replacing the chiral field for pions interacting with the heavy mesons with its mean field value obtained in the nuclear matter created by the skyrmion crystal approach @xcite , it was shown that the masses of @xmath5 and @xmath7 increase with density while the masses of @xmath13 and @xmath11 decrease , and that their masses approach the average value . in other word , the degenerated mass ( actually , the difference between the degenerated mass and the heavy quark mass ) agrees with the chiral invariant mass , which is given by the average at vacuum . however , in the analyses of refs @xcite , only the pion is included in the light hadron sector , and effects of other mesons are not included . in particular , the analysis in ref . @xcite shows that the @xmath14 meson increases the mass of @xmath5 meson , while it decreases the @xmath15 meson . in this paper , we study the effects of @xmath14 meson as well as the @xmath16 meson on the density dependence of effective masses of heavy - light mesons . we show that the effect of the @xmath16 meson increases the masses of ( @xmath5 , @xmath7 ) heavy quark doublet , while it decreases the masses of the chiral partners , i.e. ( @xmath13 , @xmath11 ) doublet , similarly to the analysis in refs . @xcite . on the other hand , the effect of the @xmath14 meson increases the masses of both doublets nevertheless , the difference between the masses of chiral partners decreases proportional to the mean field value of the @xmath16 meson , which reflects the partial chiral symmetry restoration . as a result , the masses of ( @xmath5 , @xmath7 ) doublet and ( @xmath13 , @xmath11 ) doublet approach a certain degenerate value . differently from the previous analysis , the degenerate value does not agree with the average value at vacuum which is the chiral invariant mass of those doublets . in the following analysis , after a general consideration , we consider two concrete models , the parity doublet model @xcite and skyrmion crystal model based on hidden local symmetry @xcite to give quantitative results . for explaining the main point explicitly , we work in the heavy quark limit and consider a simple chiral effective model for a heavy meson multiplet of charmed mesons with @xmath17 , @xmath18 , @xmath19 and @xmath20 based on the chiral doubling structure @xcite . let @xmath21 and @xmath22 denote the heavy - quark doublets of heavy - light mesons with the expression @xmath23 , \nonumber\\ g & = & \frac{1+v^\mu \gamma_\mu } { 2}\left[d_0^\ast - i \gamma^\mu { d}_{1\mu}'\gamma_5 \right ] , \label{hg doublet}\end{aligned}\ ] ] where @xmath24 is the velocity of the heavy - light mesons , and @xmath5 , @xmath25 , @xmath9 and @xmath26 are corresponding meson fields . we introduce the chiral fields @xmath27 as @xmath28 \ , \ \ \mathcal{h}_l = \frac{1}{\sqrt{2}}\left[g - i h\gamma_5\right ] , \label{eq : hlrgh}\end{aligned}\ ] ] which transform linearly under the chiral symmetry : @xmath29 with @xmath30 . the relevant lagrangian used in the present calculation is expressed as @xcite @xmath31 + { \rm tr}[\mathcal{h}_r(iv\cdot\partial)\bar{\mathcal{h}}_r\right ] \notag\\ & & { } - g_{\omega dd } \,\mbox{tr } \left [ { \mathcal h}_l v^\mu \omega_\mu \bar{\mathcal h}_l + { \mathcal h}_r v^\mu \omega_\mu \bar{\mathcal h}_r \right ] \notag\\ & & { } + \frac{\delta_m}{2f_\pi } \mbox{tr } \left[\mathcal{h}_l m \bar{\mathcal{h}}_r+\mathcal{h}_r m^{\dagger}\bar{\mathcal{h}}_l\right ] \notag\\ & & { } - i\frac{g_{a}}{2f_\pi}{\rm tr}\left[\mathcal{h}_r\gamma_5\gamma^{\mu}\partial_{\mu}m^{\dagger}\bar{\mathcal{h}}_l - \mathcal{h}_l\gamma_5\gamma^{\mu}\partial_{\mu}m\bar{\mathcal{h}}_r\right ] , \nonumber\\ \label{pionlagrangian}\end{aligned}\ ] ] where @xmath32 is the mass difference between @xmath22 and @xmath21 doublets , @xmath33 is the pion decay constant , @xmath34 is a dimensionless real parameter . in the above lagrangian , the omega meson field @xmath35 is introduced as a chiral singlet and the field @xmath36 is parametrized as @xmath37 with the pauli matrix @xmath38 , which transforms as @xmath39 . we rewrite the effective lagrangian in terms of @xmath21 and @xmath22 fields as @xmath40\notag\\ & & { } + \frac{\delta_m}{4 f_\pi } \mbox{tr } \left[g\left(m+m^{\dagger}\right)\bar{g } + h\left(m+m^{\dagger}\right)\bar{h } - ig\left(m - m^{\dagger}\right)\gamma_5\bar{h } + ih\left(m - m^{\dagger}\right)\gamma_5\bar{g}\right ] \notag\\ & & { } - \frac{ig_{a}}{4f_\pi } \mbox{tr } \left [ g\gamma_5\left({{\ooalign{\hfil/\hfil\crcr$\partial$}}}m^{\dagger } - { { \ooalign{\hfil/\hfil\crcr$\partial$}}}m\right)\bar{g } -h\gamma_5\left({{\ooalign{\hfil/\hfil\crcr$\partial$}}}m^{\dagger}-{{\ooalign{\hfil/\hfil\crcr$\partial$}}}m\right)\bar{h } + ig\left({{\ooalign{\hfil/\hfil\crcr$\partial$}}}m^{\dagger}+{{\ooalign{\hfil/\hfil\crcr$\partial$}}}m\right)\bar{h}-ih\left({{\ooalign{\hfil/\hfil\crcr$\partial$}}}m^{\dagger}+{{\ooalign{\hfil/\hfil\crcr$\partial$}}}m\right)\bar{g } \right ] \ . \label{pionlagrangian2}\end{aligned}\ ] ] now , we replace the light meson fields by their mean field values in medium . here we consider the symmetric matter only and assume no pion condensation , so that @xmath41 and @xmath42 . note that the mean field value of @xmath16 at vacuum agrees with the pion decay constant , @xmath43 . from the above form , we obtain @xmath44 \notag\\ & & { } - { \rm tr}\left [ h \left(i\partial_0 + g_{\omega dd } \langle \omega_0 \rangle \right ) \bar{h } \right ] \notag\\ & & { } + \frac{\delta_m}{2f_\pi } \left\langle \sigma \right\rangle \mbox{tr } \left[g\bar{g}+h\bar{h}\right ] , \label{eq : ga1ga20}\end{aligned}\ ] ] where we used @xmath45 . so that , the effective masses of @xmath21 and @xmath22 doublets are obtained as @xmath46 where @xmath47 is the average mass of the @xmath21 and @xmath22 doublets with @xmath48 . the masses of @xmath21 and @xmath22 doublets are determined by the spin average of the physical masses as @xmath49 we should note that , for the anti - charmed mesons @xmath15 , @xmath50 , @xmath51 and @xmath52 , the sign in front of the coupling to the omega meson is flipped , so that the effective masses are written as @xmath53 now , let us study the density dependence of masses in eqs . ( [ eff mass ] ) and ( [ eff mass bar ] ) . as for the mean field value of @xmath14 , we simply take @xmath54 where @xmath55 is the omega meson coupling to the nucleon , @xmath56 is the mass of omega meson and @xmath57 is the baryon number density . as for the mean field of @xmath16 we adopt the linear densty approximation as @xmath58 where @xmath59 is the coefficient of the @xmath60-@xmath61 sigma term . for making a numerical estimation , we use @xmath62gev , @xmath63 gev , and @xmath64 for masses , in addition to @xmath65 mev , @xmath66mev and @xmath67mev . as for other parameters , we use @xmath68mev , @xmath69 estimated in appendix [ app : a ] and @xmath70 , is the corrected value obtained for the chiral invariant mass @xmath71mev . ] which lead to @xmath72 , as reference values . for the concreteness of the discussion , we take @xmath73 below meson carries the anti - light - quark number , we expect that the @xmath14 meson provides an attractive interaction to the nucleon . in this sense , we think that @xmath74 is a natural choice . ] . when we study the case with @xmath75 , we just exchange @xmath21 with @xmath76 and @xmath22 with @xmath77 in the following discussion . we plot the density dependence of the masses in fig . [ fig : mass ] . this shows that the masses of @xmath21 and @xmath76 doublets as well as those of @xmath22 and @xmath77 doublets are split by @xmath14 contribution . from the @xmath14 contribution combined with the @xmath16 contribution , the mass of @xmath22 doublet ( indicated by red - dashed curve ) decreases with increasing density , and the @xmath76 mass ( by blue - dotted curve ) increases . on the other hand , @xmath21 mass ( by black - solid curve ) and @xmath77 mass ( by green - dotdashed curve ) are rather stable . aa a result , the @xmath22 mass tends to degenerate with the mass of @xmath21 doublet at certain high density . if one measures the mass of @xmath21 only , one might think that the chiral invariant mass would be almost same as the mass of @xmath21 doublet . however , the actual chiral invariant mass is larger than the @xmath21 mass at vacuum , which can be obtained by averaging the masses of the particles ( @xmath22 and @xmath21 ) and anti - particles ( @xmath76 and @xmath77 ) , as shown in fig . [ fig : sum of mass ] . we should note that the sums of masses of particle and anti - particle are actually independent of the sign of @xmath78 . mass difference between the chiral partners , i.e. , the @xmath21 doublet and the @xmath22 doublet , is caused by the spontaneous chiral symmetry breaking . this structure is seen by subtracting the mass of @xmath21 doublet from that of @xmath22 doublet with eq . ( [ eff mass ] ) as @xmath79 so the mass difference is expected to give a clue for the chiral condensate . in the mean field approximation , it is actually proportional to the mean field @xmath80 as shown in fig . [ fig : diff ] . this figure clearly shows that , with the increasing of the nuclear matter density , chiral symmetry is ( partially ) restored . for checking the @xmath60-@xmath61 sigma term dependence of the effective masses , we vary the value of @xmath59 as @xmath81 and @xmath82mev , which are plotted in fig . [ fig : mass pin ] . + this shows that the difference between the masses of @xmath21 and @xmath22 as well as that between @xmath76 and @xmath77 decreases more rapidly for larger value of @xmath59 . as a result , the chiral symmetry restores more rapidly for the larger @xmath59 . we next check the dependence on the value of @xmath83 in fig . [ fig : mass gg ] , by taking 30% deviation from the estimated value . + this shows that the masses change more rapidly for larger value of @xmath83 . after the above general discussion , let us study the density dependences of the effective masses based on some specific models . here we use the the nuclear matter described by the parity doublet model @xcite and by the skyrmion crystal model based on the hidden local symmetry @xcite . we first show the density dependence obtained from the parity doublet model in fig . [ fig : masspdm ] . here we use @xmath69 estimated in appendix [ app : a ] as a typical value and substitute the mean field values of @xmath80 and @xmath84 calculated in ref . @xcite into eqs . ( [ eff mass ] ) and ( [ eff mass bar ] ) . this shows that , the same as the previous plots , the masses of @xmath21 and @xmath76 doublets as well as those of @xmath22 and @xmath77 doublets are split by @xmath14 contribution . compared to fig . [ fig : mass ] , the density dependence of @xmath85 and @xmath86 is weaker for @xmath87 . this is because the sigma contribution for @xmath88 is smaller in the parity doublet model . we plot in fig . [ fig : massskyr ] the density dependence of the effective masses of charmed mesons by using @xmath89 and @xmath90 with @xmath91 calculated by the skyrmion crystal model based on the hidden local symmetry @xcite and other parameters are the same as that used in the plot of fig . [ fig : mass ] . in this model , we find that both @xmath22 and @xmath77 masses decrease with density while both @xmath21 and @xmath76 increase with density . because the density dependence of @xmath76 mass and @xmath22 mass is stronger than that of @xmath77 mass and @xmath21 mass , @xmath21 and @xmath22 as well as @xmath76 and @xmath77 become degenerate at density @xmath92 at which the skyrmion phase transits to half - skyrmion phase . in this work , by regarding the ( @xmath13 , @xmath11 ) heavy quark doublet as the chiral partner of the ( @xmath5 , @xmath7 ) doublet , we explicitly showed that the effect of the @xmath14 meson decreases the masses of both doublet , while ( @xmath93 , @xmath52 ) and ( @xmath15 , @xmath50 ) meson masses are increased . we explicitly point out that the @xmath14 meson effect is significant for understanding the density dependence of effective hadron masses in medium . even though the qualitative dependence is model dependent , the tendency that the masses of the heavy - light mesons and their antiparticles are split due to the @xmath14 meson effect is robust . we would like to note that the result of the omega meson effect to @xmath5 and @xmath15 mesons are consistent with the result obtained in ref . @xcite . in our analysis , we further introduce @xmath94-wave excited @xmath5 and @xmath15 mesons by using the chiral doubling model , and we found that the difference between the masses of chiral partners decreases in proportion to the mean field value of the @xmath16 meson , which reflects the partial chiral symmetry restoration even if the @xmath14 meson contribution enters . in our calculation , we simply take the mean field approach . an extension of the present work to include some loop contributions will be reported in @xcite . in the present work , we only discussed the medium modified charmed mesons . the results presented here are intact for their bottom cousins except the average mass @xmath47 should be taken the value of bottom mesons .

one of the key problems at the interface between fundamental physics and cosmology is to understand the physical mechanism behind the late - time acceleration of the universe . in principle , this phenomenon may be the result of unknown physical processes involving either modifications of gravitation theory or the existence of new fields in high energy physics . although the latter route is most commonly used , which gives rise to the idea of a dark energy component ( see , e.g. , @xcite ) , following the former , at least two other attractive approaches to this problem can be explored . the first one is related to the possible existence of extra dimensions , an idea that links cosmic acceleration with the hierarchy problem in high energy physics , and gives rise to the so - called brane - world cosmology @xcite . the second one , known as @xmath0 gravity , examine the possibility of modifying einstein s general relativity ( gr ) by adding terms proportional to powers of the ricci scalar @xmath6 to the einstein - hilbert lagrangian @xcite . the cosmological interest in @xmath0 gravity comes from the fact that these theories can exhibit naturally an accelerating expansion without introducing dark energy . however , the freedom in the choice of different functional forms of @xmath0 gives rise to the problem of how to constrain on theoretical and/or observational grounds , the many possible @xmath0 gravity theories . much efforts within the realm , mainly from a theoretical viewpoint , have been developed so far @xcite ( see also refs . @xcite for recent reviews ) , while only recently observational constraints from several cosmological data sets have been explored for testing the viability of these theories@xcite . an important aspect worth emphasizing concerns the two different variational approaches that may be followed when one works with @xmath0 gravity theories , namely , the metric and the palatini formalisms ( see , e.g. , @xcite ) . in the metric formalism the connections are assumed to be the christoffel symbols and variation of the action is taken with respect to the metric , whereas in the palatini variational approach the metric and the affine connections are treated as independent fields and the variation is taken with respect to both . in fact , these approaches are equivalents only in the context of gr , that is , in the case of linear hilbert action ; for a general @xmath0 term in the action they give different equations of motion . in the present paper we will restrict ourselves to the palatini formalism for gravitation and will focus on its application to the flat friedmann - robertson - walker ( frw ) cosmological model . we will derive constraints on the two parameters @xmath2 and @xmath4 of the @xmath5 gravity theory from the most recent compilations of type ia supernovae ( sne ia ) observations , which includes the recent large samples from supernova legacy survey ( snls ) , the essence survey , distant sne ia observed with hst , and others , giving a sample of 307 sne ia events @xcite . we also combine the sne ia data with information from the baryon acoustic oscillation ( bao ) @xcite and the cmb shift parameter @xcite in order to improve the sne ia bounds on the free parameters of the theory . the action that defines an @xmath0 gravity is given by @xmath7 where @xmath8 , @xmath9 is the determinant of the metric tensor and @xmath10 is the standard action for the matter fields . treating the metric and the connection as completely independent fields , variation of this action gives the field equations @xmath11 where @xmath12 is the matter energy - momentum tensor which , for a perfect - fluid , is given by @xmath13 , where @xmath14 is the energy density , @xmath15 is the fluid pressure and @xmath16 is the fluid four - velocity . here , we adopt the notation @xmath17 , @xmath18 and so on . in ( [ field_eq ] ) @xmath19 is given in the usual way in terms of the independent connection @xmath20 , and its derivatives , which is related with the christoffel symbol @xmath21 of the metric @xmath22 by @xmath23 and @xmath24 . we assume a homogeneous and isotropic frw universe whose metric is @xmath25 , where @xmath26 is the cosmological scale factor . the generalized friedmann equation can be written in terms of redshift parameter @xmath27 and the density parameter @xmath28 as ( see ref . @xcite for details ) @xmath29 where @xmath30 and @xmath31 is the matter density today . the trace of eq . ( [ field_eq ] ) gives another relation @xmath32 and , as can be easily checked , for the einstein - hilbert lagrangean ( @xmath33 ) eq . ( [ fe3 ] ) reduces to the known form of friedmann equation . by assuming a functional form of the type @xmath5 , one may easily show that eq . ( [ trace2 ] ) evaluated at @xmath34 imposes the following relation among @xmath2 , @xmath3 and @xmath4 @xmath35 where @xmath36 , the value of the ricci scalar today , is determined from the algebraic equation resulting from equating ( [ trace2 ] ) and ( [ fe3 ] ) for @xmath34 . hence , specifying the values of two of these parameters the third is automatically fixed . in other words , in the palatini approach , the two parameter of @xmath37 can be thought as the pair ( @xmath38 ) or ( @xmath39 ) . 0.1 in 0.1 in since the very first results showing direct evidence for a present cosmic acceleration ( using a small number of sne ia events ) @xcite , the number and quality of sne ia data available for cosmological studies have increased considerably due to several observational programs . the most up to date set of sne ia has been compiled by kowalski _ _ @xcite and includes recent large samples from snls @xcite and essence @xcite surveys , older data sets and the recently extended data set of distant supernovae observed with hst . the total compilation , the so - called _ union _ sample , amounts to 414 sne ia events , which was reduced to 307 data points after selection cuts . in this section , we will use this sne ia sample to place limits on the @xmath40 ( or , equivalently , @xmath41 ) parametric space . this analysis , therefore , updates the results of refs . we also perform a joint analysis involving the _ union _ sne ia sample and measurements of the baryonic acoustic oscillations ( bao ) from sdss @xcite and the cmb shift parameter as given by the wmap team @xcite to break possible degeneracies in the @xmath40 plane ( for more details on the statistical analyses discussed below we refer the reader to ref . @xcite ) . the predicted distance modulus for a supernova at redshift @xmath42 , given a set of parameters @xmath43 , is @xmath44 where @xmath45 and @xmath46 are , respectively , the apparent and absolute magnitudes , and @xmath47 stands for the luminosity distance ( in units of megaparsecs ) , @xmath48 where @xmath49 is given by eqs . ( [ fe3 ] ) - ( [ trace2 ] ) . we estimate the best fit to the set of parameters @xmath50 by using a @xmath51 statistics , with @xmath52^{2}}{\sigma_i^{2}}},\ ] ] where @xmath53 is given by eq . ( [ dm ] ) , @xmath54 is the extinction corrected distance modulus for a given sne ia at @xmath55 , and @xmath56 is the uncertainty in the individual distance moduli . since we use in our analysis the _ union _ sample ( see @xcite for details ) , @xmath57 . figure ( 1 ) shows the hubble diagram for the 307 sne ia events of the _ union _ sample . the curves stand for the best - fit @xmath0 models obtained from sne ia and sne ia + bao + cmb analysis . for the sake of comparison , the standard @xmath58cdm model with @xmath59 is also shown . note that all models seem to be able to reproduce fairly well the sne ia measurements . in fig . ( 2a ) we show the first results of our statistical analyses . contour plots ( 68.3% , 95.4% and 99.7% c.l . ) in the @xmath60 plane are shown for the @xmath61 given by eq . ( [ chi2307 ] ) . we clearly see that sne ia measurements alone do not tightly constrain the values of @xmath2 and @xmath3 , allowing for a large interval of values for these parameters , with @xmath2 ranging from -1 to even beyond 1 , and @xmath3 consistent with both vacuum solutions ( @xmath62 ) , as well with universes with up to 90% of its energy density in the form of non - relativistic matter . the best - fit values for this analysis are @xmath63 and @xmath64 , with the reduced @xmath65 ( @xmath66 is defined as degrees of freedom ) . 0.4 in 0.1 in the acoustic oscillations of baryons in the primordial plasma leave a signature on the correlation function of galaxies as observed by eisenstein _ this signature furnishes a standard rule which can be used to constrain the following quantity : @xmath67 where the observed value is @xmath68 , @xmath69 is the typical redshift of the sdss sample and @xmath70 is the dilation scale , defined as @xmath71^{1/3}$]with the comoving distance @xmath72 given by @xmath73 . in fig . ( 2b ) we show the confidence contours ( 68.3% , 95.4% and 99.7% c.l . ) in the @xmath40 plane arising from this measurement of @xmath74 . as expected , since this quantity has been measured at a specific redshift ( @xmath75 ) , it forms bands on this parametric space , instead of ellipsoids as in the case of sne ia data . the shift parameter @xmath76 , which determines the whole shift of the cmb angular power spectrum , is given by @xcite @xmath77 where the @xmath78 is the redshift of the last scattering surface , and the current estimated value for this quantity is @xmath79 @xcite . note that , to include the cmb shift parameter into the analysis , the equations of motion must be integrated up to the matter / radiation decoupling , @xmath80 . since radiation is no longer negligible at this redshift , a radiation component with an energy density today of @xmath81 has been included in our analysis . figure ( 2c ) shows the constraints on the @xmath40 plane from the current wmap estimate of @xmath76 . in fig . ( 3a ) we show the results of our joint sne ia + bao + cmb analysis . given the complementarity of these measurements in the @xmath40 plane [ see figure ( 2 ) ] , we obtain a considerable enhancement of the constraining power over @xmath2 and @xmath3 from this combined fit . note also that the best - fit value for the matter density parameter , i.e. , @xmath82 , is consistent with current estimates of the contribution of non - relativistic matter to the total energy density in the universe ( see , e.g. , @xcite ) . the joint fit also constrains the parameters @xmath2 , @xmath3 , and @xmath4 to lie in the following intervals ( at 99.7% c.l . ) @xmath83 , \quad \omega_{mo}\in [ 0.22 , 0.32 ] \quad \mbox{and } \quad \beta\in [ 1.3 , 5.5],\ ] ] which is consistent with the results obtained in refs . @xcite using the supernova _ gold _ and the snls data sets , respectively . lcrl + test & ref . & @xmath2 & @xmath4 + + sne ia ( _ gold _ ) & @xcite & 0.51 & 10 + sne ia ( _ gold _ ) + bao + cmb & @xcite & -0.09 & 3.60 + sne ia ( snls ) & @xcite & 0.6 & 12.5 + sne ia ( snls ) + bao + cmb & @xcite & 0.027 & 4.63 + h(z ) & @xcite & -0.90 & 1.11 + h(z ) + bao + cmb & @xcite & 0.03 & 4.70 + lss & @xcite & 2.6 & - + sne ia ( _ union _ ) & this paper & 0.99 & - + cmb & this paper & -0.75 & 0.48 + bao & this paper & 1.56 & - + sne ia ( _ union _ ) + bao + cmb & this paper & -0.12 & 3.45 + + recently , amendola _ et al . _ @xcite showed that @xmath0 derived cosmologies in the metric formalism can not produce a standard matter - dominated era followed by an accelerating expansion . to verify if the same undesirable behavior also happens in the palatini formalism adopted in this paper , we first derive the effective equation of state ( eos ) @xmath84 as a function of the redshift . figure ( 3b ) shows the effective eos as a function of @xmath85 for the best - fit solution of our joint sne ia + bao + cmb analysis . note that , for this particular combination of parameters , the universe goes through the last three phases of cosmological evolution , i.e. , radiation - dominated ( @xmath86 ) , matter - dominated ( @xmath87 ) and the late time acceleration phase ( in this case with @xmath88 ) . therefore , the arguments of ref . @xcite about the @xmath89 in the metric approach seem not to apply to the palatini formalism , at least for the interval of parameters @xmath2 , @xmath3 and @xmath4 given by our statistical analysis . in table i we summarize the main results of this paper compare them with recent determinations of the parameters @xmath2 and @xmath4 from independent analyses . @xmath0-gravity based cosmology has presently been thought of as a realistic alternative to general relativistic dark energy models . in this paper , we have worked in the context of a @xmath90 gravity with equations of motion derived according to the palatini approach . we have performed consistency checks and tested the observational viability of these scenarios by using the latest sample of sne ia data , the so - called _ union _ sample of 307 events . although the current sne ia measurements alone can not constrain significantly the model parameters @xmath2 , @xmath3 and @xmath4 , when combined with information from bao and cmb shift parameters , the fit leads to very restrictive constraints on the @xmath40 ( or , equivalently , @xmath41 ) parametric space . at 99.7% c.l . , e.g. , we have found the intervals @xmath91 $ ] and @xmath92 $ ] ( @xmath93 $ ] ) . we note that , differently from results in the metric formalism @xcite , the universe corresponding to the best - fit solution for a combined sne ia+bao+cmb @xmath61 minimization ( @xmath59 and @xmath94 ) shows all three last phases of the cosmological evolution : radiation era , matter era and a late time cosmic acceleration . the authors are very grateful to edivaldo moura santos for helpful conversations and a critical reading of the manuscript . js and np thank financial support from pronex ( cnpq / fapern ) . fcc acknowledge financial support from fapesp . jsa s work is supported by cnpq . v. sahni and a. a. starobinsky , int . j. mod d * 9 * ( 2000 ) 373 ; 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a class of modified gravity , known as @xmath0-gravity , has presently been applied to cosmology as a realistic alternative to dark energy . in this paper we use the most recent type - ia supernova ( sne ia ) data , the so - called _ union _ sample of 307 sne ia , to place bounds on a theory of the form @xmath1 within the palatini approach . given the complementarity of sne ia data with other cosmological observables , a joint analysis with measurements of baryon acoustic oscillation peak and estimates of the cmb shift parameter is also performed . we show that , for the allowed intervals of @xmath2 , @xmath3 , and @xmath4 , models based on @xmath5 gravity in the palatini approach can produce the sequence of radiation - dominated , matter - dominated , and accelerating periods without need of dark energy .

michael lifshits ( unpublished ) , exploiting a type of construction attributed to r. penrose ( see , e.g. , @xcite ) , constructed a class of pairs of planar domains that , while not isometric , have periodic geodesics of exactly the same lengths ( including multiplicities ) . at least when the boundaries are smooth ( @xmath0 ) , it follows that the two billiards have the same wave invariants , in the sense that the traces of their wave groups , @xmath1 , differ at most by a smooth function @xcite . in a recent review of the inverse spectral problem @xcite s. zelditch asked whether the dirichlet laplacians , @xmath2 , for the two domains are necessarily isospectral , judging that proposition `` dubious '' but not yet refuted . given the refutation , such billiards provide a kind of converse to the famous examples of `` drums that sound the same '' @xcite , being drums that sound different but are very similar geometrically in fact , in the geometrical features deemed most relevant to spectrum . in this paper we show how to construct smooth penrose lifshits mushroom pairs that are not isospectral , and we argue that inequality of the dirichlet spectra is , in fact , quite generic . since the domains are smooth ( but not convex ) , the spectral difference is not attributable to diffraction from corners , which would muddy the definition or the relevance of `` periodic geodesics '' . the construction of a mushroom starts from a half - ellipse @xmath3 with foci @xmath4 and @xmath5 : we use the tilde , whether applied to regions , curves , or points , to indicate the operation of reflection through the minor axis of the ellipse . if two entities are interchanged by that reflection , we call them _ dual_. next , add two bumps , @xmath6 on the left and @xmath7 on the right , with @xmath8 , to form a smooth domain @xmath9 : finally , add another bump ( not self - dual ) between the foci in two dual ways ( @xmath10 and @xmath11 ) to get two domains @xmath12 : we call the domains @xmath13 and @xmath14 constructed in this manner _ a penrose lifshits mushroom pair_. we repeat that the bumps can be added in such a way that the boundaries remain smooth . that assumption , however , is needed mainly to draw a clean conclusion about equality of the length spectra . the conclusions about the dirichlet spectra hold even if the domain has corners ( in which case bump @xmath7 is superfluous ) . [ t : non - isospectral ] if @xmath6 and @xmath7 are given and not dual , then there exist dual bumps @xmath10 and @xmath11 such that the resulting penrose lifshits mushrooms @xmath15 have the same length spectra and wave invariants but are not isospectral . first we review the proof that the length spectra coincide @xcite . the geodesics in an ellipse fall into two disjoint categories @xcite : those that intersect the major axis between the foci , and those that do so at or beyond the foci . ( the only exception is the major axis itself . the smoothness assumption guarantees that the major axis will not bifurcate in @xmath15 by diffraction . ) it follows that a similar division holds for the domains @xmath15 we have just described : any geodesic originating in a bump @xmath6 or @xmath7 can never reach a bump @xmath10 or @xmath11 , and vice versa . now , the geodesics that do not intersect the focal segment @xmath16 are exactly the same for the two domains . on the other hand , those for @xmath13 that do intersect this segment are identified one - to - one with their duals in @xmath14 by the reflection operation . this shows length isospectrality . equality of the wave traces modulo smooth functions follows from the hyperbolic propagation of singularities along geodesics see @xcite and references therein . our main task is to show nonisospectrality for some choice of @xmath10 . consider the spectrum of @xmath13 assuming that the bump @xmath10 is small and has support on the left half of the focal segment ; i.e. , to construct @xmath13 the ( open ) segment @xmath17 in the boundary of @xmath9 is perturbed by the graph of a smooth , compactly supported ( and nonpositive ) function @xmath18 , where @xmath19 is a small parameter . let @xmath20 be the ground state of the dirichlet laplacian on @xmath9 and @xmath21 be the corresponding lowest eigenvalue . the known rayleigh hadamard formula for change of the spectrum under domain perturbations ( e.g. , @xcite or ( * ? ? ? * section 15.1 , exercise 9 ) ) shows that if @xmath22 where @xmath23 is the normal derivative of the eigenfunction on the boundary , then the lowest eigenvalue @xmath21 changes under the perturbation . in fact , this integral gives the derivative at @xmath24 of the lowest eigenvalue with respect to @xmath19 . thus , if we could guarantee that the values of this integral are different for the two small perturbation domains @xmath12 , this would imply their non - isospectrality : the lowest dirichlet eigenvalues would change with different velocities . since the choice of the perturbation shape @xmath25 is in our hands , in order to make these integrals different , and thus domains nonisospectral , it is sufficient to have two mutually dual segments inside the focal segment such that the square of the normal derivative of the ground state is not an even function on their union , @xmath26 . indeed , in this case we could find an even perturbation @xmath25 that would provide nonequal integrals ( in fact , almost any perturbation would do ) . [ l : lemma ] if the bumps @xmath27 are not dual with respect to the minor axis of the ellipse , there is no self - dual union @xmath26 of two segments inside @xmath16 such that the square of the normal derivative @xmath28 of the ground state @xmath20 for @xmath9 is even on @xmath26 . * proof of the lemma . * suppose that @xmath29 is even on @xmath26 . since the normal derivative is continuous , by shrinking @xmath26 if necessary , we may assume that @xmath30 itself is either even or odd on @xmath26 . suppose first that the normal derivative is even . introduce the orthogonal cartesian coordinates centered at @xmath31 and with @xmath32-axis going along the major axis . consider the function @xmath33 . both @xmath20 and @xmath34 satisfy the same eigenfunction equation inside the half - ellipse @xmath3 and have the same cauchy data on @xmath26 . therefore , according to holmgren s uniqueness theorem , they agree on their common domain . in particular , @xmath20 must satisfy zero dirichlet boundary conditions not only on @xmath35 , but also on its mirror reflection with respect to the minor axis of the ellipse . since the bumps @xmath27 are assumed not dual to each other , we conclude that @xmath20 vanishes somewhere inside @xmath9 ( or @xmath34 somewhere inside @xmath36 ) . that is , @xmath20 has a nodal curve , which is well known to be impossible for a ground state ( e.g. , @xcite ) . if @xmath28 is odd on @xmath26 , one only needs to define @xmath34 as @xmath37 to obtain an analogous contradiction . this concludes the proof of the lemma , and hence of theorem [ t : non - isospectral ] . in fact , a closer look at the proof of the theorem shows that the nonisospectrality holds for smooth penrose lifshits mushrooms @xmath15 for any nondual bumps @xmath27 and for `` generic '' dual bumps @xmath38 : [ t : generic ] for any fixed choice of nondual bumps @xmath39 , nonisospectrality holds for an open and dense ( in @xmath0-topology ) set of penrose lifshits pairs @xmath12 . indeed , the set of nonisospectral pairs @xmath15 is obviously open . the previous theorem states that the closure of this set contains the domain @xmath9 ( i.e. , the one where the bumps @xmath40 are absent ) . to show density , one can apply a similar proof by small perturbation to any pair of mushroom domains @xmath15 of the type constructed above . indeed , if the pair is already non - isospectral , there is nothing to prove . if it is isospectral , let @xmath41 be the ground state in @xmath15 . as in theorem [ t : non - isospectral ] , the perturbation method described above works if one can show absence of a dual pair @xmath42 of pieces of the boundaries @xmath43 such that @xmath44 , @xmath45 and that @xmath46 is equal after reflection to @xmath47 . now , the same consideration as in the proof of lemma [ l : lemma ] applies to justify this claim . * a different proof of generic non - isospectrality claimed in theorem [ t : generic ] follows from existence of non - isospectral mushroom domains ( theorem [ t : non - isospectral ] ) , analytic dependence of the ground state on the domain @xcite , and connectedness of the manifold of these domains . * as it is not hard to establish , the set of non - isospectral mushroom pairs is open in a much weaker topology than @xmath0 . indeed , if the domains @xmath15 are distorted by a pair of dual ( in the sense used in this text ) @xmath48-diffeomorphisms that are @xmath48-close to identity , the non - isospectrality is preserved . * one can find discussion of the effects of domain variation for general elliptic boundary value problems in the nice little book @xcite , which regrettably is available only in russian . some of its results can be found in preceding publications of the authors of that book . this small note is the result of discussion at a working seminar of the recent survey @xcite by steve zelditch . the authors would like to thank the seminar participants g. berkolaiko , j. harrison and b. winn , as well as s. zelditch and j. zhou , for discussion and useful comments . the work of p. kuchment was partially supported by the nsf grant dms 0406022 . p. kuchment expresses his gratitude to nsf for this support . the content of this paper does not necessarily reflect the position or the policy of the federal government of the usa , and no official endorsement should be inferred . r. b. melrose , the inverse spectral problem for planar domains . _ instructional workshop on analysis and geometry _ , part i ( canberra , 1995 ) , 137160 , proc . centre math . univ . , * 34 * , austral . univ . , canberra , 1996 ( ` http://www-math.mit.edu/~rbm/papers/anulec/anulec.pdf ` ) .

penrose lifshits mushrooms are planar domains coming in nonisometric pairs with the same geodesic length spectrum . recently s. zelditch raised the question whether such billiards also have the same eigenvalue spectrum for the dirichlet laplacian ( conjecturing `` no '' ) . here we show that generically ( in the class of smooth domains ) the two members of a mushroom pair have different spectra .

the cosmological principle is one of the cornerstones of modern cosmology . roughly speaking , the principle states that the universe is homogeneous and isotropic on large scales . although large - scale homogeneity and isotropy were initially postulated , in recent decades the principle has received mounting experimental support , and today there is little doubt about its validity . the cosmological principle has a slightly preciser formulation , which states that a perturbed friedman - robertson - walker metric provides an accurate description of the universe . thus , according to the cosmological principle , the universe is well - described by the perturbed spacetime metric @xmath1,\ ] ] with sufficiently small ( scalar ) perturbations at long wavelengths @xmath2 but apart from that , the principle has nothing to say about the properties of these perturbations . many of the advances in modern cosmology consist in the characterization of the metric perturbations @xmath3 and @xmath4 . though not often explicitly emphasized , one of the key assumptions is that these perturbations are just a particular realization of a random process in a statistical ensemble . hence , we do not really try to describe the actual perturbations @xmath5 and @xmath6 ; our goal is to characterize the statistical properties of the random fields @xmath7 and @xmath8 . let @xmath9 denote any random field in the universe , such as the metric perturbations considered above , or the energy density of any of the components of our universe . the statistical properties of the random field are uniquely specified by its probability distribution functional . it turns to be simpler however to study the moments of the field @xmath10 where @xmath11 denotes ensemble average , and all the fields are evaluated at a common but arbitrary time @xmath0 , which we suppress for simplicity . the cosmological principle has formal counterparts in the properties of the perturbations , though , as we emphasized above , the cosmological principle itself only requires the actual perturbations in our universe to be small . we say that a random field @xmath12 is statistically homogeneous ( or stationary ) , if all its moments are invariant under translations , @xmath13 in some cases , statistical homogeneity may apply only to some field moments . the random field is _ stationary in the mean _ if @xmath14 and it is _ stationary in the variance _ if @xmath15 where we have defined @xmath16 . if the random field is gaussian , the one- and two - point functions uniquely determine all the remaining moments of the field . a gaussian random field stationary in the mean and in the variance is hence automatically fully stationary . parallel definitions apply to the properties of the perturbations under rotations . in particular , we say that a random field @xmath12 is _ isotropic in the mean _ if @xmath17 analogously , a random field is _ isotropic in the variance _ if @xmath18 since there is always a rotation that maps @xmath19 to @xmath20 , and because any two points related by a rotation always differ by a translation , equations ( [ eq : stationary mean ] ) and ( [ eq : isotropic mean ] ) imply that homogeneity and isotropy in the mean are equivalent . but homogeneity in the variance _ does not _ imply isotropy in the variance , though the converse is true @xcite , @xmath21 homogeneity and isotropy in the mean have an important consequence : equations ( [ eq : stationary mean ] ) or ( [ eq : isotropic mean ] ) immediately imply that the expectation of a stationary field is constant , @xmath22 and , conversely , any random field with constant mean is homogeneous and isotropic in the mean . because , by definition , cosmological perturbations always represent deviations from a homogeneous and isotropic background , it is then always possible to assume that the constant value of their mean is zero , if they happen to be stationary . for example , in perturbation theory we write the total energy density @xmath23 as a background value @xmath24 plus a perturbation @xmath25 , @xmath26 this split into a background value and a perturbation is essentially ambiguous , unless we specify what the background actually is . in cosmology , what sets the background apart from the perturbations is symmetry . because of the cosmological principle , the background energy density @xmath24 is _ defined _ to be homogeneous . hence , if the constant mean of the stationary random field @xmath25 is not zero , we may redefine our background and perturbations by @xmath27 without affecting the overall value of the energy density , @xmath28 . in this case the redefined perturbation @xmath29 has zero mean , while the redefined background @xmath30 is still space - independent . it is important to recognize that cosmological perturbations can be assumed to have zero mean if and only if their mean is a constant . consider again the example of the energy density ( [ eq : split ] ) , but now assume that @xmath25 is not stationary . although the redefinitions ( [ eq : redefinition ] ) allow us to set the mean of the perturbations @xmath31 to zero , the redefined background @xmath30 is inhomogeneous in this case , in contradiction with our definition of the background density @xmath24 in equation ( [ eq : split ] ) . therefore , we conclude that homogeneity in the mean , isotropy in the mean and zero mean are all equivalent , @xmath32 homogeneity and isotropy in the variance also have important implications @xcite . if a random field is stationary in the variance , its two point function in momentum space has to be proportional to a delta function , @xmath33 and if the variance is isotropic , the power spectrum @xmath34 can only depend on the magnitude of @xmath35 , @xmath36 based on the equivalences ( [ eq : equivalencea ] ) and ( [ eq : equivalenceb ] ) , there are hence six possible different combinations of the statistical properties of the primordial perturbations , which we list in table [ tab : cases ] . hypothesis @xmath37 describes the standard assumption that underlies most analyses of cosmological perturbations , and case @xmath38 describes what is usually known as a violation of statistical isotropy . in this article we focus on violations of the zero mean hypothesis , cases @xmath39 through @xmath40 . our goal is to test the standard assumption @xmath37 against one of its non - zero mean alternatives . .the six possible different combinations of statistical properties of the primordial perturbations . we are concerned here with the mean and variance alone . [ tab : cases ] [ cols="^,^,^,^ " , ] at present , the arguably cleanest and widest window on the primordial perturbations in our universe stems from the temperature anisotropies observed in the cosmic microwave background radiation ( cmb ) . hence , if we want to test whether cosmological perturbations have zero mean , we need to explore how these temperature anisotropies are related to the random fields that we have considered in the introduction . in a homogeneous and isotropic universe , different fourier modes of cosmological perturbations evolve independently in linear perturbation theory . hence , we may always assume that the temperature anisotropies ( of primordial origin ) observed at point @xmath19 in the direction @xmath41 are given by @xmath42 where @xmath43 are the fourier components of a random field at a sufficiently early time , and @xmath44 is a transfer function whose explicit form we shall not need . say , in a standard @xmath45cdm cosmology we have @xmath46 , where @xmath47 is the primordial newtonian potential , which , because of the absence of anisotropic stress , also equals @xmath48 . due to the linear relation between temperature anisotropies and primordial perturbations , it immediately follows that zero mean in the primordial perturbations implies zero mean of the temperature anisotropies . for many purposes , it is more convenient to represent functions on a sphere , like the temperature fluctuations , by their spherical harmonic coefficients @xmath49 throughout this article we work with _ real _ spherical harmonics @xmath50 , whose properties are summarized in appendix a. to calculate the spherical harmonic coefficients of the temperature anisotropies @xmath51 we note that because linear perturbations evolve in an isotropic background ( by definition ) , the transfer function @xmath52 can only depend on the two scalars @xmath53 and @xmath54 . hence , we may expand the latter in legendre polynomials @xmath55 , @xmath56 with real functions @xmath57 . substituting then equation ( [ eq : legendre expansion ] ) into ( [ eq : t anisotropies ] ) , setting @xmath58 , and using the addition theorem for ( real ) spherical harmonics in equation ( [ eq : addition theorem ] ) we get @xmath59 clearly , if primordial perturbations have zero mean , so do the spherical harmonic coefficients of the temperature anisotropies : @xmath60 in particular , a violation of the condition @xmath61 would thus imply a violation of statistical homogeneity . later we shall also need to know the covariance of the temperature multipoles , which follows from equation ( [ eq : a ] ) . if the random field @xmath12 is homogeneous and isotropic in the variance , the covariance matrix of the multipoles has elements @xmath62 recall that for arbitrary @xmath63 we define @xmath64 thus , @xmath65 denotes departures from a homogeneous and isotropic background , whereas @xmath66 denotes deviations from the ensemble mean . in the following , we drop the hat from random variables and fields . unfortunately , the temperature anisotropies we actually observe in the sky are not entirely of primordial origin . they are a superposition of primordial anisotropies @xmath67 and foreground contributions , such as dust emission and synchrotron radiation from our very own galaxy . appropriate foreground templates allow the wmap team to eliminate foregrounds in some regions of the sky @xcite , but the cleaning procedure does not completely remove foreground contamination along the galactic disc . it is thus necessary to subject these maps to additional processing . since the actual temperature measurements involve a convolution with the detector beam @xmath68 , and also include detector noise @xmath69 , we model the temperature anisotropies in a smoothed , foreground - reduced map by @xmath70,\ ] ] where @xmath71 is the smoothing kernel , the star denotes convolution , and @xmath72 represents the residual foreground contamination . we assume that the smoothing kernel and the detector beam are rotationally symmetric . this is actually not the case for the wmap beam , but it should be a good approximation at the scales we are going to consider . in that case , in harmonic space , the convolution acts on the spherical harmonic coefficients simply by multiplication , say , @xmath73 in order to remove the residual foregrounds @xmath72 , the contaminated sky regions have to be masked out . let @xmath74 be the corresponding mask , which is defined by @xmath75(\hat{n})=0}$ ] . then , by construction , the masked sky @xmath76 does not contain foregrounds , @xmath77(\hat{n}).\ ] ] it proves then useful to define an hypothetical unmasked " smoothed sky map which is free of foregrounds , but contains the effects of noise and detector beam , and whose multipoles are hence given by @xmath78 the multipole coefficients of the masked sky @xmath76 are obtained by multiplication with an appropriate convolution matrix . if @xmath79 denotes the ( real ) spherical harmonic coefficients of the mask , it is easy to show that the elements of the convolution matrix are given by @xmath80 where @xmath81 is defined in equation ( [ eq : d ] ) . in particular , the spherical harmonic coefficients of the masked sky are given by @xmath82 in practice , we need to work with finite matrices , so we restrict our attention to a finite range of multipole values , @xmath83 . in particular we assume that @xmath84 is a square matrix . the wmap team has found that detector pixel noise is well described by a gaussian distribution with zero mean @xcite , which implies that @xmath85 therefore , the masked temperature anisotropies have zero mean if the primordial anisotropies do . according to the wmap team @xcite , the noise variance is inversely proportional to the number of times point each pixel is observed . this is not the same for all pixels , but it is is fairly isotropic ( see figure [ fig : noise ] ) . hence , assuming a constant @xmath86 , and that the noise in different pixels ( of area @xmath87 ) is uncorrelated , with variance @xmath88 , we find @xmath89 where @xmath90 . in any case , at the scales we are interested in , the contribution of the noise to the variance of the masked temperature anisotropies is subdominant . this is however not crucial for our analysis , which simply assumes that the noise satisfies equation ( [ eq : noise covariance ] ) , without reference to the actual magnitude of @xmath91 . . the figure shows a plot of the power spectrum " @xmath92 in units of the monopole @xmath93 . weighted by @xmath94 , this captures the degree of anisotropy of the noise variance on angular scales @xmath95 . , height=302 ] in order to test whether the primordial temperature anisotropies have zero mean ( which follows from @xmath96 ) we need additional information about the distribution of the harmonic coefficients @xmath97 . at this point , there is no evidence for non - gaussian primordial perturbations @xcite , so we assume that the latter are normally distributed . in order to uniquely characterize their distribution , it suffices then to consider their variance . among the hypothesis with zero mean in table [ tab : cases ] , @xmath37 is the one that underlies most of our analyses of structure . we shall therefore adopt @xmath37 as our null hypothesis . if @xmath37 is true , then , according to equation ( [ eq : hom and iso variance ] ) , the variates @xmath98 are independent and have a common variance for the same values of @xmath99 . therefore , the standard assumption @xmath37 can be cast as a precise form of the distribution of the temperature multipoles @xmath97 , which follows from equations ( [ eq : zero mean ] ) and ( [ eq : hom and iso variance ] ) , * primordial perturbations are normally distributed with zero mean , homogeneous and isotropic variance @xmath100 @xmath101 da_{\ell m}.\ ] ] note that equations ( [ eq : hom and iso variance ] ) and ( [ eq : noise covariance ] ) , together with the definition ( [ eq : b ] ) , imply that the covariance matrix of the unmasked , foreground - reduced temperature anisotropies is also diagonal , @xmath102 clearly , if the null hypothesis @xmath37 does not appropriately fit the data , we wo nt be able to determine whether this is because temperature fluctuations are non - gaussian , non - isotropic , non - homogeneous , or simply because we used the wrong power spectrum . we need to analyze the data in the face of an alternative hypothesis , namely , that primordial perturbations do not have zero mean . among all the cases with non - zero mean in table [ tab : cases ] , the minimal deviation from @xmath37 is hypothesis @xmath39 , which also leads to the covariances ( [ eq : hom and iso variance ] ) . therefore , we choose as alternative * primordial perturbations are normally distributed with non - zero mean and homogeneous and isotropic variance @xmath100 @xmath103 da_{\ell m}.\ ] ] in this case , the covariance of the unmasked , foreground - reduced temperature anisotropies is again given by equation ( [ eq : b covariance ] ) . we test @xmath37 against @xmath39 . what singles out our test is the ability to examine the zero mean hypothesis against the alternative hypothesis of non - zero means . indeed , the only difference between @xmath37 and @xmath39 lies in the assumptions about the mean of the perturbations . without the alternative hypothesis , we would be simply conducting a goodness - of - fit test . mathematically it is certainly sensible to postulate hypothesis @xmath39 , but the reader may wonder whether @xmath39 is also physically reasonable . in fact , we think it is . suppose for instance that primordial perturbations are created during an inflationary period in a slightly inhomogeneous universe ( after all , if inflation is supposed to explain cosmic homogeneity , it should start with an inhomogeneous universe . ) if we regard these small inhomogeneities as first order perturbations , in linear perturbation theory the properties of the created perturbations vacuum fluctuations of an appropriate field only depend on the homogeneous and isotropic background . hence , the resulting seeded perturbations turn out to be homogeneous and isotropic in the mean and the variance , as in the conventional case , but they have to be added on top of the already existing initial inhomogeneities . in fact , similar ideas have been already discussed in the literature @xcite . if we just happened to know the temperature multipoles @xmath97 ( or their foreground - cleaned counterparts @xmath104 ) , a test of hypothesis @xmath37 against its alternative @xmath39 would be straight - forward . under the null hypothesis , for fixed @xmath99 , the variables @xmath97 form a set of independent , normally distributed variates with zero mean and common variance @xmath105 . the standard and time - honored way to test the latter involves student s @xmath0 statistic , @xmath106 where @xmath107 are , respectively , unbiased estimators of the mean and variance of the distribution . under the null hypothesis , @xmath0 follows student s distribution with @xmath108 degrees of freedom , while under the alternative hypothesis , its square is distributed like a ratio of non - central chi - squares ( more about this below . ) note that we do not need to make any assumption about the actual values of the @xmath105 in order to know how @xmath0 is distributed under the null hypothesis . as we mentioned above , though , it is not possible to subtract part of the galactic contamination , so we are forced to work with the masked sky in equation ( [ eq : masked sky ] ) . while masking preserves the property of zero mean , equation ( [ eq : masked mean ] ) , it does not preserve the diagonal form of the covariance matrix hence , one of the key assumptions of student s test is lost . to bring the problem back to the realm of student s @xmath0 , we shall impose additional symmetries on the problem . on large angular scales , the main source of foreground contamination stems from the galactic disc , which can be covered by a mask that spans galactic latitudes in the range @xmath109 . let us hence assume that the mask is symmetric under rotations around the galactic @xmath110-axis . under such rotations , the real spherical harmonics transform according to equation ( [ eq : m rotation ] ) . hence , rotational invariance implies @xmath111 for @xmath112 . when we substitute the last relation into equation ( [ eq : mask matrix ] ) we find , using the results in the appendix , @xmath113 in other words , the mask matrix is diagonal in @xmath114 space , and we can basically restrict our attention to @xmath115 . of course , a mask with these symmetry properties can not eliminate all sources of foreground contamination ( primarily point sources ) . we shall address this problem by restricting our analysis to large angular scales , for which the contribution of point sources is small . for notational simplicity let us call @xmath116 the ( symmetric ) matrix with elements @xmath117 , and let us collect all the multipoles @xmath118 for fixed @xmath114 into a single vector @xmath119 in this notation then , @xmath120 , and equation ( [ eq : c ] ) reads @xmath121 suppose now that we find a positive integer @xmath122 and a vector @xmath123 such that @xmath124 in other words , suppose that we find a sky @xmath125 with no components along the contaminated region , such that @xmath126 for _ all _ values of @xmath114 between @xmath127 and @xmath128 . if such a vector exists , it must clearly have vanishing components for @xmath129 , since @xmath130 can not exceed @xmath99 . under these conditions then , the @xmath131 variables @xmath132 ( with @xmath133 satisfy @xmath134 where we have used equation ( [ eq : s ] ) and @xmath135 . in particular , these variates do not contain galactic residual backgrounds , because they can be constructed from a masked sky , they have zero mean under the null hypothesis , @xmath136 and , because of equation ( [ eq : b covariance ] ) , they have covariances @xmath137 where @xmath138 the off - shot of this construction is not only that the variables @xmath139 are uncorrelated ( and hence independent ) , but also that their variance , equation ( [ eq : sigma ] ) , is the same for all of them . note that , by construction , the variables @xmath140 only contain temperature multipoles @xmath104 in the range @xmath141 , where @xmath142 . equation ( [ eq : covariances ] ) states that the @xmath131 variates @xmath139 ( @xmath143 ) form a set of normally distributed _ independent _ variables , with common variance @xmath144 . they are thus statistically analogous to the primordial temperature multipoles @xmath97 , which are independent and have the same variance for fixed @xmath99 . under the null hypothesis @xmath37 the @xmath139 have zero mean , and under the alternative hypothesis @xmath39 their mean is generically non - zero . because the @xmath139 are linear combinations of the normally distributed @xmath97 , the distribution of the former is also gaussian . it is hence natural to choose student s @xmath0 as test statistic , although , for later convenience , we shall actually work with its square , @xmath145 where @xmath146 intuitively , the nature of the test statistic is clear : up to factors that involve @xmath128 , @xmath147 is just the square of the ratio of sample mean to sample standard deviation . we would expect this ratio to be small if the variables indeed have zero mean , and large if they do nt . one of the keys of our test statistic is that we know its distribution both under the null and the alternative hypothesis . the identity @xmath148 implies by an extension of cochran s theorem ( section 15.20 in and section 35.7 in ) that the numerator and denominator of equation ( [ eq : t squared ] ) are independent variables , both under @xmath37 and @xmath39 . again by the same extension of cochran s theorem , the numerator ( divided by @xmath144 ) follows a non - central chi - square distribution with @xmath149 degrees of freedom , and non - central parameter @xmath150 whereas @xmath151 ( divided by @xmath144 ) follows a non - central chi - square distribution with @xmath152 degrees of freedom , and non - central parameter @xmath153 the statistic @xmath147 is thus a ratio of non - central chi - squares divided by their respective number of degrees of freedom . the latter follows a doubly non - central @xmath72 distribution , with probability density given by ( section 24.30 in ) @xmath154 where @xmath68 is the beta function . if all the ensemble means are equal , @xmath155 , @xmath156 vanishes , and the distribution of @xmath147 simplifies to a non - central @xmath72-distribution with degrees of freedom @xmath149 and @xmath152 , and non - central parameter @xmath157 . under the null hypothesis @xmath37 the non - central parameter is @xmath158 , and the distribution reduces to a central @xmath72 with @xmath159 and @xmath160 , which is just the square of student s @xmath0 distribution . we carry out a one - sided test of the null - hypothesis @xmath37 at significance @xmath161 ( say , @xmath162 ) by rejecting the null hypothesis if @xmath147 is larger than @xmath163 , where @xmath163 is the @xmath161-point of the central @xmath164 distribution , @xmath165 this amounts to a two - sided test of @xmath37 using student s @xmath0 , in which we reject the null hypothesis for sufficiently large deviations ( of either sign ) of @xmath0 from zero . in order to evaluate the power of this test , @xmath166 , we need to determine the probability of rejecting the null hypothesis when the alternative hypothesis @xmath39 is true , @xmath167 the power is a function of the hypothetical standardized non - zero ensemble means @xmath168 . in the absence of any particular model , and for the purpose of illustration , we shall consider the simple ( though perhaps somewhat unrealistic ) case of equals means : @xmath169 . using equations ( [ eq : e and b ] ) and ( [ eq : b ] ) this translates into @xmath170 in general , we may regard @xmath171 as a measure of the order of magnitude of the mean of the primordial temperature anisotropies in the corresponding multipole range , even if the means are not common . we plot the power of the test as a function of @xmath172 , for @xmath173 , in figure [ fig : power ] . we see for instance that in a test with @xmath173 , if the mean of our variables is about @xmath174 times their standard deviation ( or larger ) , the @xmath147 statistic will fall in the critical region almost with certainty . significance and @xmath173 as a function of a standardized common mean . the solid curve shows the power of a test based on student s @xmath0 , equation ( [ eq : t squared ] ) . the dashed curves show the power of a test based on the statistic ( [ eq : tk ] ) for @xmath175 and @xmath176 . clearly , student s @xmath0-test is more powerful than any of the @xmath177 tests . ( note that we can restrict our attention to @xmath178 , since for higher @xmath53 values the statistic @xmath177 has the same distribution as @xmath179 , with @xmath180 . ) for different choices of parameters , the power curves have the same qualitative form . evidently , as the number of degrees of freedom @xmath160 increases , so does the power of any of these tests.,height=302 ] the value of the @xmath147 statistic tells us not only whether the null hypothesis holds , but also how far from zero the mean may be if the alternative hypothesis @xmath39 is the correct one . let @xmath181 denote the @xmath161-point of student s @xmath147 distribution , and , for simplicity , let us assume again that all the means of the primordial anisotropies have a common value @xmath171 . then , since the statistic @xmath182 is distributed like @xmath147 under the alternative hypothesis @xmath39 , we can write @xmath183 . following the conventional approach to classical interval estimation , we can cast the last relation as a confidence interval that formally involves @xmath184 , @xmath185 this is a frequentist interval : if we repeat the same procedure to derive the confidence interval @xmath69 times , as @xmath69 approaches infinity our interval will contain the true mean @xmath171 in @xmath186 cases . we can also extend this analysis to a set of several independent @xmath147 statistics , designed to probe the temperature anisotropies at different angular scales . suppose for instance that we find not one , but a set of @xmath187 different vectors @xmath188 ( @xmath189 ) that satisfy the set of equations ( [ eq : s ] ) , with @xmath190 and @xmath191 . then , for fixed @xmath192 , the variates @xmath193 still have a common variance , as in equation ( [ eq : covariances ] ) . we can therefore define a set of @xmath187 different statistics @xmath147 simply by replacing @xmath194 by @xmath195 and @xmath128 by @xmath196 in equations ( [ eq : t squared ] ) and ( [ eq : t expl ] ) . but if the ranges @xmath197 are disjoint , the vectors @xmath198 do not have any common element , and the variables @xmath195 and @xmath199 are also uncorrelated for @xmath200 . in that case , the different @xmath147 statistics are mutually independent , and probe the temperature anisotropies in the disjoint multipole ranges @xmath201 a perhaps undesirable property of our test statistic is that @xmath147 is not a scalar under rotations . since the mask breaks rotational symmetry anyway , this is not a problem by itself . nevertheless , the mask does preserve the symmetry under azimuthal rotations , so it would be natural to demand at least invariance of our statistic under this unbroken subgroup . under an azimuthal rotation by an angle @xmath202 , the variates @xmath203 transform as in equation ( [ eq : m rotation ] ) , with @xmath97 replaced by @xmath203 . clearly , the statistic @xmath147 in equation ( [ eq : t squared ] ) does not remain invariant under such rotations . but the transformation law ( [ eq : m rotation ] ) immediately suggests how to address the problem . indeed , the variables @xmath204 are invariant under azimuthal rotations by construction , and they also share the same variance . hence , any ratio of the sum of two disjoint subsets of these squared variables is distributed like a ratio of independent ( eventually non - central ) @xmath205 distributions . consider for instance the set of statistics @xmath206 where @xmath53 is an arbitrary parameter . as before , under the alternative hypothesis @xmath39 the probability density of @xmath207 is given by equation ( [ eq : dp ] ) , with degrees of freedom and non - central parameters given by , respectively , @xmath208 in particular , under the null hypothesis @xmath37 , @xmath209 follows a central @xmath72 distribution with @xmath210 and @xmath211 degrees of freedom . using equation ( [ eq : dp ] ) we calculate the power of the set of alternative tests based on the statistic ( [ eq : tk ] ) . assuming that all the means are common , as above , we find the power curves in figure [ fig : power ] . inspection of the figure quickly reveals that the test based of student s @xmath147 statistic is uniformly more powerful than any test based on a @xmath207 statistic . this is no coincidence at all ; student s @xmath0 test is widely employed because of its optimal properties ( see example 23.14 in . ) therefore , in this article we just focus on student s @xmath147 . as long as we stick to a single ( random ) orientation of the sky , our results have a straight - forward statistical interpretation , since all we need to know is how the test statistic is distributed for an arbitrary ( but fixed ) sky orientation . in addition , the non - scalar nature of the statistic may help us to identify that area of the sky eventually responsible for a violation of the zero mean hypothesis . to conclude , we should also point out that student s @xmath0 test is known to be robust to departures from normality , at least for _ independent _ variables drawn from the same distribution ( section 31.3 in ) . our data analysis pipeline consists of four main steps . first , we construct an appropriate mask to eliminate residual galactic foregrounds . then , we identify a vector @xmath123 that belongs to the range of all the mask matrices @xmath212 for @xmath143 . we degrade the cosmic microwave maps to lower resolution , and apply our test statistic to these maps . finally , we check for an eventual residual contamination in our results . this section lists the details of each of these steps . the reader not interested in technical details is welcome to skip this part and jump to the next section for the actual results . as mentioned above , in order to eliminate galactic contamination , and preserve azimuthal symmetry at the same time , we construct a mask invariant under rotations along the galactic @xmath110-axis . our starting point is a healpix pixelization of the sphere with @xmath213 . we set all pixels in the mask with galactic latitude @xmath214 to zero , and all the remaining pixels to one . the effective area covered by the mask is @xmath215 of the full sky . we label the components of the pixelized mask by @xmath216 , where @xmath192 runs over all the @xmath217 pixels of the mask . this particular mask is in fact also symmetric under parity , but we do not make explicit use of this symmetry . in order to find a @xmath123 that satisfies equation ( [ eq : s ] ) , we look for a common solution of the set of equations @xmath218 since @xmath116 is the mask matrix , the linear operator @xmath219 gives the components of the vector along the contaminated galactic region . hence , equation ( [ eq : null ] ) states that the vector @xmath123 should have a vanishing component along such region . in order to find the components of @xmath123 , it is numerically more convenient to work in real space . we fix the values of @xmath220 and @xmath128 and calculate a matrix @xmath81 whose elements are defined by @xmath221 where @xmath192 runs over all pixels in a healpix pixelization of the sphere with @xmath222 , and @xmath223 . then , the set of equations ( [ eq : null ] ) reads @xmath224 , or @xmath225 we find an approximate solution of equation ( [ eq : cv ] ) by singular value decomposition , @xmath226 here , the @xmath227 is a set of @xmath228-dimensional orthonormal vectors , the @xmath229 are the singular values ( arranged in order of decreasing magnitude ) , and the @xmath230 is a set @xmath231-dimensional orthonormal vectors . we choose the vector @xmath123 to be the last right singular vector , that is , @xmath232 , with @xmath233 . we label the corresponding singular eigenvalue @xmath234 . because the vectors @xmath235 and @xmath236 are orthogonal , the singular value @xmath234 is the norm of @xmath237 , @xmath238 in general , this singular value is non - zero , so our solution of equation ( [ eq : cv ] ) is not exact but only approximate . modulo a normalization factor , the value of @xmath234 is then an indicator of the potential degree of contamination , i.e. , the overlap between our vector @xmath123 and the contaminated galactic region . the latter typically increases with increasing @xmath128 , since the number of non - zero elements of @xmath239 freely available to solve equation ( [ eq : cv ] ) decreases with increasing @xmath128 . because the power of the @xmath147 test increases with the number of degrees of freedom , @xmath160 , we choose the maximum possible value of @xmath128 for which @xmath234 , divided by the norm of the mask times the norm of sky encoded in @xmath123 , remains under @xmath240 . indeed , the resulting vector can be represented visually , by defining the sky @xmath241 which captures those regions of the sky that enter our statistic . as an example , the corresponding real space sky for @xmath242 and @xmath243 is shown in figure [ fig : v ] . the same process can be repeated for different choices of @xmath220 and @xmath128 . if the corresponding intervals ( [ eq : interval ] ) do not overlap , we can use the resulting set of vectors @xmath123 to construct a set of mutually independent @xmath147 statistics , as explained in subsection [ sec : extension ] . ) , for @xmath242 and @xmath243 . those regions of the sky with the largest values are those that are more heavily weighted in our statistic , as implied by equation ( [ eq : e and b ] ) . as seen in the figure , the galactic region is basically excluded from our analysis . for other choices of @xmath220 and @xmath128 the structure of the sky is the same , as long as the singular eigenvalue @xmath234 remains sufficiently small.[fig : v],height=302 ] we analyze the seven - year ( version 4 ) , full resolution , foreground - reduced q2 , v2 and w1 differencing assembly cosmic microwave anisotropy maps provided by the wmap mission . since these maps probe different frequencies of the microwave spectrum , a dependence of our results on the particular map would indicate non - thermal foreground contamination . we expect the latter to be smallest for the w and v maps , and largest for the q map . we thus take the w1 assembly to be our fiducial map , and keep the v2 and q2 assemblies just for comparison . the wmap has subtracted the dipole and primordial monopole from the three differencing assemblies , and the latter have been smoothed with a gaussian kernel @xmath71 of @xmath244 . the sky maps are expanded into ( real ) spherical harmonics , and band - limited to a maximum multipole value @xmath245 . the different values of @xmath220 are chosen iteratively to cover the multipole range @xmath246 with non - overlapping intervals . we choose @xmath247 as the absolute maximum for @xmath99 because we expect point sources to significantly contaminate the temperature anisotropies at higher multipoles . note that it is not necessary to mask the sky prior to processing , since the vector @xmath123 has no components along the galactic region by construction . there are two main possible sources of systematic errors in our analysis : galactic contamination due to an insufficiently resolved mask or an imperfect solution of equations ( [ eq : null ] ) , and point source contamination due to unmasked high - latitude point sources . in order to estimate both we basically follow the same approach . let us assume that the values of @xmath220 and @xmath128 have been fixed . to estimate the amount of galactic contamination , we subtract from the cosmic microwave maps the portion of the sky covered by the wmap seven - year temperature analysis mask and run the resulting sky map through the analysis pipeline described above . the change in the @xmath248-value of the @xmath147 statistic is then a measure of galactic contamination . for @xmath242 and @xmath243 for instance , the change in the @xmath147 statistic after subtraction of the galaxy is less @xmath249 for the w1 map . similarly , to estimate the amount of point source contamination we construct a sky map of temperature anisotropies from the point source fluxes listed in the wmap point - source catalog @xcite . we subtract the point source map from the cosmic microwave background and run the resulting sky map through our data analysis pipeline . the change in the corresponding @xmath248-value of the @xmath147 statistic is then a measure of point source contamination . certainly , there are unresolved point sources that the wmap catalog does not contain , but the contribution of these sources is small compared to the contribution of the actually detected sources that we are not able to mask . say , if we choose @xmath242 and @xmath243 , the change in the @xmath147 statistic after point source subtraction is less @xmath250 for the w1 map . point sources do not typically have thermal spectra , so an inspection of our results for different differencing assemblies gives us yet another handle on such contamination . an alternative way to estimate point source contamination involves the ratio @xmath251 where the @xmath252 are the multipoles of the temperature map constructed from detected point sources alone , and the @xmath97 are the spherical harmonic coefficients of the analyzed sky ( say , the @xmath253 map ) . the sums in @xmath254 only run over @xmath114 , because our statistics are sensitive only to a relatively small window of multipoles in @xmath99 space . as shown in figure [ fig : point sources ] , @xmath254 remains below @xmath250 up to @xmath255 , which is in broad agreement with our direct estimate of point source contamination using the @xmath147 statistic . because point source contamination decreases with the frequency of the map @xcite , the w1 differencing assembly is less contaminated than the other two assemblies this is why we take w1 to be our fiducial map . note by the way that none of the procedures described above is an actual attempt to subtract galactic or point source contamination . instead , it is just a way to estimate the contribution of the unmasked foregrounds to the @xmath248-value of our statistic . ) . the point source contribution is subdominant at low multipoles , and reaches @xmath250 at @xmath255.,height=302 ] our results are summarized in table [ tab : results ] and represented graphically in figure [ fig : p values ] . there are two multipole ranges in which the @xmath248-value of our @xmath147 statistic is smaller than @xmath256 . in the first case , for @xmath257 , the @xmath248-value across the three differencing assemblies remains under @xmath258 , so it does not seem that this result is due to residual foregrounds alone . in particular , in this multipole range point source contamination is negligible . in the second case , for @xmath259 , the @xmath248-value of the statistic for the q2 map is normal , so we may tentatively attribute the difference in the values of @xmath147 among maps to foreground contamination . this explanation however is somewhat problematic , because we expect foregrounds to make the actual value of @xmath147 less likely , and because contamination is stronger in the q2 map , whose @xmath147 is normal . can we speak then of statistically significant evidence against the null hypothesis ? to answer this question , we need to realize that we have constructed a set of @xmath260 independent tests of the null hypothesis , one for each multipole range . hence , if we would like the of all individual statistics to be larger than @xmath161 with probability @xmath261 ( under the null hypothesis ) , we should choose the size @xmath161 of each individual test to satisfy @xmath262 for @xmath260 and @xmath263 , this yields @xmath264 . none of the @xmath248-values in table [ tab : results ] is as low . we reach similar conclusions by calculating the value of a statistic often used to combine the results of multiple independent tests of a single hypothesis : stouffer s weighted @xmath265 test @xcite . let @xmath266 be the @xmath248-value of our @xmath147 test in the @xmath192-th multipole range , and let @xmath267 be the @xmath266 point of a standard normal distribution . then , the variate @xmath268 where the @xmath269 are the weights assigned to each test , follows a normal distribution with zero mean and unit variance . we weigh each test by the number of degrees of freedom of the corresponding @xmath147 test , @xmath270 . for the eight @xmath248-values listed in table [ tab : results ] , and the w1 differencing assembly map , the value of stouffer s statistic is @xmath271 clearly under two sigma away from the mean of the standard normal distribution . @xmath272 { r r r@{.}l r@{.}l r@{.}l r@{.}l r@{.}l r@{.}l } \hline \hline \multicolumn{2}{c}{\text{range } } & \multicolumn{4}{c}{q2 } & \multicolumn{4}{c}{v2 } & \multicolumn{4}{c}{w1 } \\ \ell_\text{min } & \ell_\text{max } & \multicolumn{2}{r}{t^2 } & \multicolumn{2}{l}{\text{$p$-value } } & \multicolumn{2}{r}{t^2 } & \multicolumn{2}{l}{\text{$p$-value } } & \multicolumn{2}{r}{t^2 } & \multicolumn{2}{l}{\text{$p$-value } } \\ \hline \hline 1 & 18 & 1&957 & 29&7\% & 2&367 & 26&4\% & 2&457 & 25&8\% \\ 19 & 38 & 0&269 & 60&7\% & 0&208 & 65&1\% & 0&200 & 65&7\% \\ 39 & 60 & 1&963 & 16&5\% & 2&212 & 14&1\% & 2&525 & 11&6\% \\ 61 & 86 & 6&341 & 1&3\% & 6&431 & 1&2\% & 7&397 & 0&7\% \\ 87 & 112 & 0&699 & 40&4\% & 0&275 & 60&1\% & 0&639 & 42&5\% \\ 113 & 142 & 0&473 & 49&2\% & 0&816 & 36&7\% & 0&406 & 52&5\% \\ 143 & 176 & 0&029 & 86&5\% & 0&012 & 91&4\% & 0&025 & 87&5\% \\ 177 & 212 & 2&582 & 10&9\% & 4&615 & 3&2\% & 4&091 & 4&4\% \\ \hline\hline \end{array } $ ] . we plot the @xmath248-values of the @xmath147 statistic under the null - hypothesis for different multipole ranges and different differencing assemblies ( blue for @xmath253 , red for @xmath273 , green for @xmath274 . ) for reference , the horizontal line marks @xmath256 probability . clearly , the value of @xmath147 in the multipole range @xmath275 is anomalously small.,height=302 ] the reader may also wonder how much better the data are fit by a distribution with non - zero mean . in order to find that out , we calculate first an effective @xmath205 by extremizing the likelihood under both the null and the alternative hypothesis , @xmath276 since the variates @xmath203 are normal and independent , the likelihood is simply a product of gaussian density functions . therefore , sample mean sample variance respectively are the maximum - likelihood estimators for the population mean and the population variance . the difference @xmath277 is a measure of how much the fit improves when we relax the assumption of zero mean . because of equation ( [ eq : chisq ] ) , this difference is a monotonic function of the ratio of maximum likelihoods under @xmath37 and @xmath39 , which also happens to be a monotonic function of the @xmath147 statistic ( example 24.1 in ) . for illustration , we list the corresponding values of @xmath278 in table [ tab : ic ] . clearly , since we have an additional parameter to fit the data , we expect a better fit under @xmath39 . to correct for the presence of additional parameters , several model selection measures have been proposed in the literature @xcite . in table [ tab : ic ] we list the difference in the corrected akaike information criterion ( aic@xmath279 ) and the difference in the bayesian information criterion ( bic ) . from a bayesian perspective , the difference in information criteria @xmath66 is a measure of relatively model likelihood @xmath280 this equation allows us then to interpret @xmath281 as a number of standard deviations . again , a distribution with non - zero mean seems to be a better model to describe the data in the multipole range @xmath282 . but as we emphasized above , this is relatively likely to happen if multiple ranges of multipoles are considered . @xmath272 { r r r r@{.}l r@{.}l r@{.}l } \hline \hline \ell_\text{min } & \ell_\text{max } & \text{dof } & \multicolumn{2}{c}{\delta\chi^2 } & \multicolumn{2}{c}{\delta\text{aic}_\text{c } } & \multicolumn{2}{c}{\delta\text{bic } } \\ \hline \hline 1 & 18 & 3 & 2&40 & \multicolumn{2}{c}{- } & 1&31 \\ 19 & 38 & 39 & 0&21 & -2&02 & -3&46 \\ 39 & 60 & 79 & 2&52 & 0&41 & -1&85 \\ 61 & 86 & 123 & 7&24 & 5&17 & 2&43 \\ 87 & 112 & 175 & 0&64 & -1&40 & -4&52 \\ 113 & 142 & 227 & 0&41 & -1&63 & -5&02 \\ 143 & 176 & 287 & 0&02 & -2&00 & -5&63 \\ 177 & 212 & 355 & 4&08 & 2&06 & -1&79 \\ \hline\hline \end{array } $ ] the actual values of the test statistic for our choices of @xmath220 and @xmath128 also allow us to place the first limits on the magnitude of an eventual common mean of the primordial perturbations in the given range of multipoles . these limits are collected in table [ tab : limits ] and graphically represented in figure [ fig : limits ] . at angular scales smaller than about four degrees , the limits are typically one order of magnitude below the standard deviation of the temperature multipoles . in two cases , the confidence interval does not contain zero , which is again an expression of an anomalously high value of the @xmath147 statistic in the corresponding multipole range . but , as before , since these are @xmath283 confidence intervals , the probability that all of them contain the true mean is only @xmath215 . in any case , these limits should not be taken too literally . the assumption of a common mean is somewhat unrealistic , so these intervals should be rather interpreted of an order of magnitude estimate of possible deviations from the zero - mean assumption , even if the means of the anisotropies do not share a common value in the corresponding multipole range . @xmath272 { r r r@{.}l r@{.}l r@{.}l r@{.}l r@{.}l r@{.}l r@{.}l } \hline \hline \multicolumn{2}{c}{\text{range } } & \multicolumn{4}{c}{q2 } & \multicolumn{4}{c}{v2 } & \multicolumn{4}{c}{w1 } & \multicolumn{2}{c } { } \\ \ell_\text{min } & \ell_\text{max } & \multicolumn{2}{r}{\mu_\text{min } } & \multicolumn{2}{l}{\mu_\text{max } } & \multicolumn{2}{r}{\mu_\text{min } } & \multicolumn{2}{l}{\mu_\text{max } } & \multicolumn{2}{r}{\mu_\text{min } } & \multicolumn{2}{l}{\mu_\text{max } } & \multicolumn{2}{r}{\sqrt{c_\ell } } \\ \hline \hline 1 & 18 & -7&061 & 3&596 & -6&989 & 3&308 & -6&962 & 3&244 & 7&20 \\ 19 & 38 & -0&275 & 0&464 & -0&283 & 0&448 & -0&278 & 0&438 & 2&87 \\ 39 & 60 & -0&335 & 0&058 & -0&336 & 0&049 & -0&347 & 0&039 & 1&92 \\ 61 & 86 & -0&278 & -0&033 & -0&278 & -0&034 & -0&286 & -0&045 & 1&46 \\ 87 & 112 & -0&156 & 0&063 & -0&137 & 0&079 & -0&153 & 0&065 & 1&30 \\ 113 & 142 & -0&134 & 0&065 & -0&140 & 0&052 & -0&129 & 0&066 & 1&17 \\ 143 & 176 & -0&112 & 0&094 & -0&106 & 0&095 & -0&108 & 0&092 & 1&09 \\ 177 & 212 & -0&230 & 0&023 & -0&251 & -0&011 & -0&249 & -0&003 & 0&98 \\ \hline\hline \end{array } $ ] confidence limit ( in @xmath284k units ) , as in table [ tab : limits ] . again , blue , red and green label the limits derived from the w1 , v2 and q2 band maps respectively . the extent of the interval on the @xmath99 axis indicates the range in values of @xmath99 for which the limit applies . for comparison we also plot the variance of the multipole components for the best wmap s estimate of the binned power spectrum , @xmath285 . note that the temperature scale is logarithmic , with positive and negative values on either side of the axis . , height=302 ] our results show significant evidence for a non - zero mean of the temperature multipoles in the range @xmath286 to @xmath287 , at the @xmath288 confidence level . taken as a whole however , because this range is just one among eight different multipole bins , the evidence against the zero - mean assumption is statistically insignificant , falling under the @xmath283 confidence level . whatever the case , the limits we have set on the mean of the primordial anisotropies in a set of multipole bins indicate that an eventual non - zero mean has to be about an order of magnitude smaller than the standard deviation of the temperature anisotropies . in that sense , observations constrain the mean to be small . in retrospective , we have therefore partially justified the common assumption of vanishing mean of the cosmological perturbations . this work is supported in part by the nsf grant phy-0855523 . some of the results in this paper have been derived using the healpix @xcite package . we acknowledge the use of the legacy archive for microwave background data analysis ( lambda ) . support for lambda is provided by the nasa office of space science . in this article we expand functions defined on a sphere in _ real _ spherical harmonics @xmath50 . these are related to the conventional complex spherical harmonics @xmath289 by @xmath290 it follows that the real multipole coefficients @xmath97 and their complex counterparts @xmath291 are related to each other by @xmath292 where we have assumed that the function on the sphere being expanded is real . the transformation ( [ eq : sh ] ) is unitary , that is , we can write @xmath293 with @xmath294 a unitary matrix , whose matrix elements are implicitly defined by equation ( [ eq : sh ] ) . because of the unitary transformation , real spherical harmonics are orthonormal , @xmath295 and they also satisfy the addition theorem @xmath296 where @xmath55 is a legendre polynomial . sometimes we need to integrate over the product of three spherical harmonics . we define @xmath297 which clearly is totally symmetric in its three arguments . since the real spherical harmonics are related to the complex spherical harmonics by a unitary transformation , this expression is closely related to the integral of the product of three complex spherical harmonics @xmath298 . the latter can be expressed as a product of clebsch - gordan coefficients ( or wigner symbols ) , so we have @xmath299 with @xmath300 it follows then for instance that @xmath301 . under ( active ) azimuthal rotations by an angle @xmath202 the complex spherical harmonic coefficients transform according to @xmath302 . therefore , it follows from the left equation in ( [ eq : relations ] ) that real spherical harmonic coefficients @xmath97 transform according to @xmath303 n. jarosik _ et al . _ [ wmap collaboration ] , `` first year wilkinson microwave anisotropy probe ( wmap ) observations : on - orbit radiometer characterization , '' astrophys . j. suppl . * 148 * , 29 ( 2003 ) [ arxiv : astro - ph/0302224 ] . e. komatsu _ et al . _ [ wmap collaboration ] , `` five - year wilkinson microwave anisotropy probe ( wmap ) observations : cosmological interpretation , '' astrophys . j. suppl . * 180 * , 330 ( 2009 ) [ arxiv:0803.0547 [ astro - ph ] ] . k. m. gorski , e. hivon , a. j. banday , b. d. wandelt , f. k. hansen , m. reinecke and m. bartelman , `` healpix a framework for high resolution discretization , and fast analysis of data distributed on the sphere , '' astrophys . j. * 622 * , 759 ( 2005 ) [ arxiv : astro - ph/0409513 ] .

a central assumption in our analysis of cosmic structure is that cosmological perturbations have zero ensemble mean . this property is one of the consequences of statistically homogeneity , the invariance of correlation functions under spatial translations . in this article we explore whether cosmological perturbations indeed have zero mean , and thus test one aspect of statistical homogeneity . we carry out a classical test of the zero mean hypothesis against a class of alternatives in which perturbations have non - vanishing means , but homogeneous and isotropic covariances . apart from gaussianity , our test does not make any additional assumptions about the nature of the perturbations and is thus rather generic and model - independent . the test statistic we employ is essentially student s @xmath0 statistic , applied to appropriately masked , foreground - cleaned cosmic microwave background anisotropy maps produced by the wmap mission . we find evidence for a non - zero mean in a particular range of multipoles , but the evidence against the zero mean hypothesis goes away when we correct for multiple testing . we also place constraints on the mean of the temperature multipoles as a function of angular scale . on angular scales smaller than four degrees , a non - zero mean has to be at least an order of magnitude smaller than the standard deviation of the temperature anisotropies .

chain , the weak , @xmath2 and geometric order on @xmath0 , which coincide for @xmath6 ( but not in general ) . ] this paper is about four partial orders on the set @xmath3 of all standard young tableaux of size @xmath1 satisfying : @xmath7 here @xmath8 means that @xmath9 in @xmath10 implies @xmath9 in @xmath11 , in which case we say @xmath11 is _ stronger _ than @xmath10 ( or @xmath10 is _ weaker _ then @xmath11 ) . all four of these orders have appeared in the work of melnikov @xcite , who refers to what we are calling the weak order as the _ induced duflo order_. roughly speaking , 1 . the weak order is induced from the weak bruhat order on the symmetric group @xmath12 via the robinson - schensted insertion map , 2 . the kl order is induced by the kazhdan - lusztig preorder on @xmath12 arising in the theory of kazhdan - lusztig ( right ) cells , 3 . geometric order describes inclusions of certain algebraic varieties indexed by tableaux ( _ orbital varieties _ ) , and 4 . chain order is induced by the dominance order on partitions ; for each interval of values @xmath13 $ ] , one restricts the tableau to these values and compares the insertion shapes in dominance order . all four of these orders on @xmath3 coincide for @xmath14 , and are depicted in figure [ figure1 ] . for @xmath15 , they differ ( see example [ chain - order - shape - agree ] and example [ kl - stronger - example ] ) . after reviewing their definitions in section [ definitions - section ] , we recall some of their known properties in section [ known - properties - section ] . we then prove three main new results . the first result , proven in section [ p - r - theorem - section ] , relates to a hopf algebra defined by poirier and reutenauer @xcite whose basis elements are indexed by standard young tableaux @xmath16 of all sizes . the multiplication in this hopf algebra is somewhat nontrivial to describe , but turns out to be described essentially by any of our four partial orders . [ p - r - theorem ] for any of the four partial order @xmath17 above , one has @xmath18 where @xmath19 and @xmath20 are obtained by sliding @xmath21 over @xmath16 from the left and from the bottom respectively . the second result is about the mbius function and homotopy type of these orders . the weak bruhat order on @xmath12 is well - known to have each interval homotopy equivalent to either a sphere or a point , and hence have mbius function values all in @xmath22 . although it is not true in general for the intervals in the weak , @xmath2 , geometric and chain orders on @xmath3 ( see figure [ mob ] for some examples ) the interval from bottom to top is homotopy equivalent to either a sphere or a point . this result is proven in section [ homotopy - theorem - section ] , by associating descent sets to tableaux and thereby obtaining a poset map to a boolean algebra . ( @xmath23 ) an interval in @xmath24 and ( @xmath25 ) an interval in @xmath26 ordered with @xmath27 , @xmath28 and @xmath29 having mbius function @xmath30 and @xmath31 respectively . ] [ homotopy - theorem ] let @xmath17 be any of the four partial orders . then the map @xmath32}$ ] sending a tableau to its descent set is order - preserving , and induces a homotopy equivalence of the proper parts . in particular , for any such order @xmath33 . the third result , proven in section [ skew - homotopy - theorem - section ] deals with a generalization of the above orders to skew tableaux with fixed inner boundary . the most crucial step in the proof is the application of rambau s suspension lemma @xcite which makes the proof ( compared to the standard methods in topological combinatorics ) much shorter and comprehensible . given a partition @xmath34 , let @xmath35 denote the set of all skew standard tableaux of having @xmath1 cells which are `` skewed by @xmath34 '' , that is , whose shape is @xmath36 for some @xmath37 . it turns out that two of the four orders ( @xmath2 , geometric ) have a property ( the _ inner translation _ property ; see theorem [ vogan - involution ] ) which allows us to generalize them on @xmath35 . each of these skew orders has a top element @xmath38 and a bottom element @xmath39 , so that one can speak of the homotopy of their _ proper parts _ obtained by removing @xmath40 . [ skew - homotopy - theorem ] let @xmath17 be @xmath2 or geometric orders on @xmath3 . then the associated order @xmath17 on @xmath35 has the homotopy type of its proper part equal to that of @xmath41 in particular , for any such order either @xmath42 or @xmath43 , depending on @xmath34 . an illustration of the skew orders on @xmath44 for @xmath45 . when @xmath34 is ( @xmath23 ) rectangular and ( @xmath25 ) nonrectangular , @xmath46 has its proper part homotopy equivalent to a @xmath47-dimensional sphere and a point respectively . ] figure [ skew - diagrams ] provides an illustration for theorem [ skew - homotopy - theorem ] where both posets are considered with @xmath2 or geometric orders . in fact , this theorem follows from a more general statement ( proposition [ skew - mobius - value - theorem ] ) about the homotopy types of certain intervals , which applies to any order between the weak and chain orders ( including the weak order itself ) . .1 in we close this section with some context and motivation for theorems [ p - r - theorem ] and [ homotopy - theorem ] , stemming from two commutative diagrams that appear in the work of loday and ronco @xcite @xmath48 } \\ \end{array } \hskip .4 in \begin{array}{cll } \mathbb{z}{{\mathfrak{s } } } & \longleftarrow & \mathbb{z}y \\ & \nwarrow & \uparrow \\ & & \sigma \\ \end{array}\ ] ] in the left diagram of , @xmath49 denotes the set of planar binary trees with @xmath1 vertices . the horizontal map sends a permutation @xmath50 to a certain tree @xmath51 , and has been considered in many contexts ( see e.g. @xcite , @xcite ) . the southeast map @xmath52}$ ] sends a permutation @xmath50 to its descent set @xmath53 . these maps of sets become order - preserving if one orders @xmath12 by weak order , @xmath49 by the _ tamari order _ ( see @xcite ) , and @xmath4}$ ] by inclusion . indeed , the order preserving maps of the first diagram induce the inclusions of hopf algebras in the second diagram of , in which @xmath54 is the malvenuto - reutenauer algebra , @xmath55 is a subalgebra isomorphic to loday and ronco s free _ dendriform algebra _ on one generator @xcite , and @xmath56 is a subalgebra known as the algebra of _ noncommutative symmetric functions_. in @xcite , loday and ronco proved a description of the product structure for each of these three algebras very much analogous to theorem [ p - r - theorem ] , which should be viewed as the analogue replacing @xmath55 by @xmath57 ; see theorem [ loday - ronco - malvenuto - reutenauer ] below for their description of the product in @xmath54 . the analogy between the standard young tableaux @xmath3 and the planar binary trees @xmath49 is tightened further by recent work of hivert , novelli and thibon @xcite . they show that the planar binary trees @xmath49 can be interpreted as the plactic monoid structure given by a knuth - like relation , similar to the interpretation of the set of standard young tableaux as knuth / plactic classes . we were further motivated in proving theorem [ p - r - theorem ] by the results of aguiar and sottile in @xcite and @xcite where the mbius functions of the weak order on @xmath12 and tamari order on @xmath49 have key roles in understanding the structures of the hopf algebras of permutations and planar binary trees . in ( * ? ? ? * remark 9.12 ) , bjrner and wachs ( essentially ) show that the triangle on the left induces a diagram of homotopy equivalences on the proper parts of the posets involved . theorem [ homotopy - theorem ] gives the analogue of this statement in which one replaces @xmath58 by @xmath59 where @xmath17 is any order between the weak and chain orders . .1 in the first partial order on @xmath3 that will be discussed is the strongest one : _ chain order_. given @xmath60 , we denote by @xmath61 the partition corresponding to the shape of @xmath16 . for @xmath62 , let @xmath63}$ ] be the skew subtableau obtained by restricting @xmath16 to the segment @xmath13 $ ] . let @xmath64})$ ] be the tableau obtained by lowering all entries of @xmath63}$ ] by @xmath65 and sliding it into normal shape by jeu - de - taquin @xcite . the definition of chain order also involves the dominance order . we denote by @xmath66 the set of all partitions of the number @xmath1 ordered by the _ opposite ( or dual ) dominance order _ , that is , @xmath67 if @xmath68 [ chain - order - def ] let @xmath69 and we say @xmath21 is less that @xmath16 in chain order ( @xmath70 ) if for every @xmath62 , @xmath71}))~\leq^{op}_{dom}~{{\mathrm{sh}}}({{\mathrm{std}}}(t_{[i , j]})).\ ] ] .2 in before giving the definition of the weak order it is necessary to recall the robinson - schensted @xmath72 correspondence ; see @xcite for more details and references on @xmath73 . the @xmath73 correspondence is a bijection between @xmath74 and @xmath75 . here @xmath10 and @xmath11 are called the _ insertion _ and _ recording tableau _ respectively . knuth @xcite defined an equivalence relation @xmath76 on @xmath12 with the property that @xmath77 if and only if they have the same insertion tableaux @xmath78 . we will denote the corresponding equivalence classes in @xmath12 by @xmath79 . we now recall the ( _ right _ ) _ weak _ _ bruhat order _ , @xmath27 , on @xmath12 . it is the transitive closure of the relation @xmath80 if @xmath81 for some @xmath82 with @xmath83 , and where @xmath84 is the adjacent transposition @xmath85 . the weak order has an alternative characterization ( * ? ? ? 3.1 ) in terms of _ ( left ) inversion sets _ @xmath86 namely @xmath80 if and only if @xmath87 . .1 in for @xmath62 let @xmath13 $ ] be a segment of the alphabet @xmath88 $ ] and @xmath89}$ ] be the subword of @xmath90 obtained by restricting to the alphabets @xmath13 $ ] and @xmath91})$ ] in @xmath92 be the word obtained from @xmath89}$ ] by subtracting @xmath65 from each letter . in fact @xmath93 gives @xmath94})\subset { { \mathrm{inv}}}_l(w_{[i , j]})$ ] for all @xmath95 and hence @xmath96 } \leq_{weak } w_{[i , j ] } ~\mbox { for all } ~1\leq i < j\leq n.\ ] ] the following basic fact about @xmath73 , knuth equivalence , and jeu - de - taquin are essentially due to knuth and schtzenberger ; see knuth ( * ? ? ? * section 5.1.4 ) for detailed explanations . [ j - d - t - initial - final ] given @xmath97 , let @xmath98 be the insertion tableau of @xmath90 . then for @xmath99 , @xmath100 } ) = p({{\mathrm{std}}}(u_{[i , j]})).\ ] ] .1 in furthermore one can use greene s theorem @xcite for the following fact : @xmath101}))\leq^{op}_{dom}{{\mathrm{sh}}}({{\mathrm{std}}}(p(u)_{[i , j ] } ) ) ~\mbox { for all } ~1\leq i < j\leq n.\ ] ] now and shows that the following order is weaker than chain order on @xmath3 and hence it is well defined . [ weak - order - def ] the _ weak order _ @xmath102 , first introduced by melnikov @xcite under the name _ induced duflo order _ , is the partial order induced by taking transitive closure of the following rule . denoting the knuth class of @xmath16 by @xmath103 , @xmath104 the necessity of taking the transitive closure in the definition of the weak order is illustrated by the following example ( cf . melnikov ( * ? ? ? * example 4.3.1 ) ) . let @xmath105 , .05 in @xmath106 , .05 in @xmath107 with @xmath108 here @xmath109 since @xmath110 , and @xmath111 since @xmath112 . therefore @xmath113 . on the other hand , for every @xmath114 one has @xmath115 , whereas for every @xmath116 one has @xmath117 . this shows that there is no @xmath118 and @xmath116 such that @xmath119 . .2 in it turns out that @xmath73 is closely related to _ kazhdan - lusztig _ preorders on @xmath12 . recall that a _ preorder _ on a set @xmath120 is a binary relation @xmath17 which is reflexive ( @xmath121 ) and transitive ( @xmath122 implies @xmath123 ) . it need not be antisymmetric , that is , the equivalence relation @xmath124 defined by @xmath125 need not have singleton equivalence classes . note that a preorder induces a _ partial order _ on the set @xmath126 of equivalence classes . kazhdan and lusztig @xcite introduced two preorders ( the left and right @xmath2 preorders ) on coxeter groups whose equivalence classes are called the left and right cells repectively . the theory of left ( or right ) cells provides a decomposition of the regular representation of the hecke algebras of coxeter groups ( c.f . * chapter 6 ) ) such that , in case the coxeter group is @xmath12 , each summand is irreducible . in this paper we will denote by @xmath127 the _ opposite _ of the usual @xmath2 right preorder on @xmath12 . for example , with our convention , the identity element @xmath128 and the longest element @xmath129 satisfy @xmath130 . it turns out @xcite ( and explicitly in @xcite ) that the associated equivalence relation for this @xmath2 preorder is the knuth equivalence @xmath76 . hence an equivalence class ( usually called either a _ knuth class _ or _ plactic class _ or a _ kazhdan - lusztig right cell _ in @xmath12 ) corresponds to a tableau @xmath16 in @xmath3 . [ kl - order - def ] _ @xmath2 order _ on @xmath3 is defined by the rule @xmath131 where @xmath132 is the knuth class ( or @xmath2 right cell ) in @xmath12 corresponding to @xmath133 . for later use , we now recall the basic construction of the @xmath2 right preorder on @xmath12 . recall that the _ right descent set _ @xmath134 and the _ left descent set _ @xmath135 of a permutation @xmath97 , are defined by @xmath136 where @xmath137 . in what follows , we will often identify the set @xmath21 of adjacent transpositions with the numbers @xmath138:=\{1,2,\ldots , n-1\}$ ] via the obvious map @xmath139 . .1 in in @xcite , kazhdan and lusztig prove the existence of unique polynomials @xmath140 $ ] indexed by permutations in @xmath12 . denoting by @xmath17 the bruhat order on @xmath12 , @xmath141 the length of the permutation @xmath90 and @xmath142 , these polynomials satisfy : @xmath143 let @xmath144p_{u , w}(q)$ ] denote the coefficient of @xmath145 in @xmath146 and define @xmath147p_{u , sw}(q)&~~ \text { if } l(u , w ) \text { is odd } \\ ~~0&~~ \text { otherwise . } \end{cases}\ ] ] then a recursive formula for these polynomials is given in the following way : for @xmath148 and @xmath149 , @xmath150 where @xmath151 if @xmath152 and @xmath153 otherwise . moreover the dual of right @xmath2 preorder on @xmath12 is given by taking the transitive closure of the following relation : @xmath154 .2 in the final order on @xmath3 to be discussed in this paper relates to the preorder on @xmath12 induced from geometric order on the orbital varieties associated to the lie algebra @xmath155 . the theory of orbital varieties arise from the work of n. spaltenstein @xcite and r. steinberg @xcite on the unipotent variety of a connected complex semi - simple group @xmath156 . they have a key role in the studies of primitive ideals ( i.e. annihilators of irreducible representations ) in the enveloping algebra @xmath157 of lie algebra @xmath158 corresponding to @xmath156 ( c.f . @xcite , @xcite , @xcite ) . they also play an important role in springer s weyl group representations . let @xmath158 be the lie algebra of @xmath156 and @xmath159 be the borel subgroup of @xmath156 given with respect to some triangular decomposition @xmath160 such that @xmath161 is a cartan subalgebra and @xmath162 is the corresponding nilradical . for given @xmath163 , we denote by @xmath164 the nilpotent orbit determined by the adjoint action of @xmath156 on @xmath165 . therefore @xmath166 is an irreducible variety . now an _ orbital variety _ @xmath167 associated to @xmath164 is defined to be an irreducible component of the intersection @xmath168 . given orbital varieties @xmath167 and @xmath169 , the _ geometric order _ is defined by @xmath170 where @xmath171 denotes the zariski closure of @xmath167 inside @xmath162 . the only general description of orbital varieties provided below is due steinberg @xcite . .1 in given a positive root system @xmath172 , recall that @xmath173 where @xmath174 is the root space corresponding to @xmath175 . let @xmath176 be the weyl group of @xmath158 generated by simple roots in @xmath177 , and for @xmath178 let @xmath179 since @xmath159 is an irreducible closed subgroup of @xmath156 , the action of @xmath159 on @xmath180 gives an irreducible locally closed subvariety @xmath181 which , therefore , lies in a unique nilpotent orbit @xmath164 for some @xmath163 and @xmath182 . by the result of steinberg @xmath183 is an orbital variety and the map @xmath184 is a surjection . moreover geometric order induces a preorder on @xmath176 such that , for @xmath185 @xmath186 .1 in according to steinberg @xcite , the fibers of the map @xmath187 for @xmath188 are the knuth classes of @xmath12 and therefore each orbital variety @xmath167 in @xmath155 can be identified with some @xmath60 i.e. , @xmath189 . this leads to the following definition . the _ geometric order _ on @xmath3 , @xmath190 , is given by the following rule : @xmath191 when @xmath188 , an explicit description of orbital varieties can be given in the following way . let @xmath159 to be the borel subgroup of invertible upper triangular @xmath192 matrices given by the cartan decomposition of @xmath158 with cartan subalgebra @xmath161 of trace @xmath47 diagonal matrices and nilradicals @xmath162 and @xmath193 , whose elements are strictly upper and strictly lower triangular matrices respectively . then the set of matrices @xmath194 ( and @xmath195 ) , where @xmath196 has @xmath128 on the position @xmath197 and @xmath47 elsewhere , provides a basis for @xmath162 ( respectively @xmath198 ) . the action of the weyl group @xmath12 on @xmath196 can be described by @xmath199 where @xmath200 is the permutation matrix of @xmath201 and this leads to the following characterization @xmath202 on the other hand the adjoint action of @xmath159 on @xmath196 sweeps the corner at @xmath197 to the northeast direction . in other words @xmath203 consists of all matrices of rank @xmath128 , having a nonzero entry at @xmath197 and all other nonzero entries are located at some positions to the northeast of @xmath197 . therefore all matrices in @xmath204 have their nonzero entries in some boundary provided by @xmath205 , and @xmath206 consists of all those matrices in @xmath204 whose jordan form is the same as that of @xmath165 . recall that @xmath165 is uniquely determined by the condition @xmath182 . actually one can show that the partition determined by the jordan form of @xmath165 and the partition obtained from @xmath50 through the @xmath73 correspondence are the same . .1 in there is also a bijection , revealed by steinberg @xcite , between the orbital varieties determined by @xmath165 and springer fiber @xmath207 of the complete flag variety @xmath208 . moreover geometric order results in an ordering between the irreducible components of @xmath207 . we next discuss this connection . let @xmath209 be the jordan form of @xmath165 , @xmath210 be the @xmath211-orbit of @xmath165 and @xmath212 here @xmath211 acts on @xmath213 and @xmath208 by conjugation and left translation respectively ; therefore it acts on @xmath214 , and the projections onto @xmath213 and @xmath208 are equivariant maps . we have the following diagram : @xmath215 in this diagram , the fiber of any @xmath216 is equal to @xmath217 . since @xmath211 is irreducible and its action on @xmath218 is transitive , the irreducible components of this _ springer fiber _ @xmath207 are in bijection with the irreducible components of @xmath214 . on the other hand for any @xmath219 , let @xmath159 be the borel subgroup of @xmath211 which fixes @xmath220 and let @xmath162 be nilradical of the corresponding borel algebra @xmath221 . then the fiber of @xmath220 is equal to @xmath222 and again the transitivity of the action of @xmath211 on @xmath208 implies that the irreducible components of @xmath223 are in bijection with the irreducible components of @xmath214 . these two bijections determine the correspondence between the orbital varieties and the irreducible components of springer fibers in the flag variety . the geometric order describes the inclusions among ( the closures of ) these components as one varies @xmath37 , in either context . .1 in in this section we recall some of the main properties of these four orders which we need later in proving our main results . these properties also can be found in or deduced from the works of melnikov @xcite and barbash and vogan @xcite . in order to make these posets more understandable we provide the proofs of those which are combinatorially approachable , while for those which need theoretical approaches the reader is directed to the references . .1 in for @xmath97 and @xmath60 recall the definitions of @xmath91})$ ] and @xmath64})$ ] from section [ weak - order - subsection ] and section [ chain - order - subsection ] respectively . .1 in say that a family of preorders @xmath17 on @xmath12 _ restricts to segments _ if @xmath224 } ) \leq { { \mathrm{std}}}(w_{[i , j]})~ \text { for all } ~1\leq i < j \leq n.\ ] ] melnikov shows in ( * ? ? ? * page 45 ) the preorder @xmath29 on the weyl group @xmath176 of any reductive lie algebra restricts to @xmath225 , where @xmath226 is any subset of simple roots generating @xmath225 . therefore geometric order on @xmath12 restricts to segments . the same fact about @xmath2 preorder was first shown by barbash and vogan @xcite for arbitrary finite weyl groups ( see also work by lusztig @xcite ) whereas the generalization to coxeter groups is due to geck ( * ? ? ? * corollary 3.4 ) . on the other hand this result for the weak order on @xmath12 follows from an easy analysis on the ( left ) inversion sets . .1 in we say the order @xmath17 on @xmath3 _ restricts to segments _ if @xmath227 } ) \leq { { \mathrm{std}}}(t_{[i , j]})~~ \text { for all } ~~1\leq i < j \leq n.\ ] ] the following result for the weak , @xmath2 and geometric order on @xmath3 is an easy consequence of the above discussion together with lemma [ j - d - t - initial - final ] , whereas for chain order it follows directly from its definition . [ segment - restriction - lemma ] on @xmath3 all of the four orders restrict to segments of standard young tableaux . .1 in in fact any order @xmath17 on @xmath3 which is stronger than the weak order and which restricts to segments shares a crucial property that we describe now . recall that _ ( left ) descent set _ of a permutation @xmath228 is defined by @xmath229 as a consequence of a well - known properties of @xmath73 , the left descent set @xmath230 is constant on knuth classes @xmath103 ; the _ descent set _ of the standard young tableau @xmath16 is described intrinsically by @xmath231 we let @xmath232 } , \subseteq)$ ] be the boolean algebra of all subsets of @xmath138 $ ] ordered by inclusion . .2 in [ tableau - descent - lemma ] let @xmath17 be any order on @xmath3 which is stronger than the weak order and restricts to segments . then the map @xmath233 } , \subseteq)\ ] ] sending any tableau @xmath16 to its descent set @xmath234 is order preserving . for @xmath45 , such an order is isomorphic to weak order on @xmath235 and the statement follows directly by examination of figure [ figure1 ] . for @xmath236 , one can use the fact that @xmath237})\cup{{\mathrm{des}}}_l(t_{[2,n]})\ ] ] to get the desired result by induction . .2 in for the record , we note here some symmetries and order - preserving maps of @xmath238 , @xmath27 , @xmath127 and @xmath29 on @xmath3 , to other posets . [ order - preserving - maps ] let @xmath17 represent to any of the orders @xmath27 , @xmath127 @xmath29 or @xmath238 on @xmath3 . then the following maps are order preserving : 1 . the map @xmath239 } , \subseteq)\ ] ] sending a tableau @xmath16 to its descent set @xmath234 . 2 . the map @xmath240 sending @xmath16 to its shape @xmath241 . on the other hand for @xmath17 equal to any of the orders @xmath27 , @xmath127 , @xmath29 or @xmath238 1 . the schtzenberger s evacuation map @xmath242 sending @xmath16 to its evacuation tableau @xmath243 is an poset automorphism , whereas for @xmath17 equal to @xmath27 , @xmath127 or @xmath238 1 . the map @xmath242 sending @xmath16 to its transpose @xmath244 is a poset anti - automorphism . the first assertion follows from lemma [ tableau - descent - lemma ] , since all of the four orders are stronger than the weak order and restrict to segments . second assertion for @xmath238 follows from its definition . for @xmath27 , as it mentioned earlier , one can apply greene s theorem @xcite . if @xmath245 then there are orbital varieties given by @xmath246 and @xmath247 such that @xmath248 . now the nilpotent orbits that these orbital varieties belong to can be characterized by the partition given by @xmath249 and @xmath61 . moreover we have @xmath250 . by the result of gerstenhaber , see ( * ? ? ? * chapter 6 ) for example , last inclusions implies @xmath251 , proving the statement for geometric order . for @xmath2 order the proof based on the theory that relates the kazhdan - lusztig cells to the primitive ideals : let @xmath158 be a semisimple algebra with universal enveloping lie algebra @xmath252 and weyl group @xmath176 . as it is shown in @xcite and @xcite , for any primitive ideal @xmath226 of @xmath252 , the set of the form @xmath253 can be characterized by a kazhdan - lusztig left cell . moreover @xmath254 ( right dual @xmath2 order ) if and only if @xmath255 , whence the associated variety of the primitive ideal @xmath256 is contained in that of @xmath257 . on the other hand by the result of borho and brylinski @xcite and joseph @xcite associated variety of a primitive ideal is the closure of a nilpotent orbit in @xmath258 . in our case @xmath188 , @xmath259 and the nilpotent orbits are characterized by partitions of @xmath1 , therefore the result of gerstenhaber reveals the desired property on the shapes of the corresponding tableaux of @xmath260 and @xmath50 . the assertions about transposition and evacuation for @xmath127 and @xmath27 , follow from the fact that the involutive maps @xmath261 are antiautomorphisms of both @xmath262 @xcite and @xmath263 . hence @xmath264 is an automorphism of both . on the other hand @xmath265 is just the transpose tableau of @xmath266 @xcite and @xmath267 is nothing but the evacuation of @xmath266 @xcite . indeed @xmath268 and @xmath269 correspond reversing the value and the order of numbers in @xmath50 respectively . therefore by greene s theorem they reverse the dominance order on the @xmath73 insertion shapes which then gives the desired property for @xmath270 . the assertion that schtzenberger s evacuation map gives a poset automorphism of @xmath190 follows from melnikov s work ( * ? ? ? * page 1718 ) . .1 in ( see discussion by van leeuwen @xcite ) . is the map which sends a tableau to its transpose an anti - automorphism of the geometric order ? .1 in by part ( ii ) of the proposition [ order - preserving - maps ] , if @xmath271 under the weak , @xmath2 , geometric or chain orders then @xmath272 . actually we have a stronger condition for the first three orders which is given in proposition [ change - of - shapes ] below . on the other hand example [ chain - order - shape - agree ] shows that this property is not satisfied by chain order . [ change - of - shapes ] let @xmath17 be any of @xmath27 , @xmath127 or @xmath29 on @xmath3 . then @xmath273 @xmath274 , under these orders the shape of the tableaux is not fixed . for @xmath127 , this property can be induced from the work of lusztig @xcite which result in the conclusion that , for @xmath12 right cells given by the tableaux of the same shape form an antichain in the @xmath2 order . for @xmath29 , gerstenhaber s result mentioned in the proof of proposition [ order - preserving - maps](ii ) gives the required property ; if @xmath275 , the orbital varieties @xmath247 and @xmath246 lie in the same nilpotent orbit @xmath276 . as being the irreducible components of @xmath277 they satisfy neither @xmath278 nor @xmath279 . therefore @xmath16 and @xmath21 are not comparable under @xmath29 and this proves the hypothesis . now @xmath27 satisfy the hypothesis since it is weaker then @xmath2 and geometric orders . [ chain - order - shape - agree ] the following tableaux have @xmath280 although they have the same shape . @xmath281 .15 in it is known that the ( right ) weak order on @xmath12 is weaker than the ( right ) @xmath2 preorder on @xmath12 @xcite . as it is described , for instance in @xcite , the weak order is also weaker than geometric order on @xmath12 . therefore by the virtue of its definition @xmath102 embeds in @xmath282 and @xmath190 . on the other hand by corollary [ segment - restriction - lemma ] and by proposition [ order - preserving - maps]@xmath283 the weak , @xmath2 and geometric orders on @xmath3 are weaker then chain order . the following important result , which reveals that @xmath2 order embeds in geometric order on @xmath3 , can be deduced from the work of melnikov ( * ? ? ? * corollary 1.2 ) , borho and brylinski @xcite and vogan @xcite . [ kl - weaker - than - geometric ] on @xmath12 , @xmath2 order is weaker than geometric order . therefore for all @xmath69 , @xmath284 it happens that all these four orders coincide for @xmath14 , but they start to differ for @xmath285 . proposition [ change - of - shapes ] and the example [ chain - order - shape - agree ] provided above show that @xmath286 differs from all the other orders for @xmath285 . the following examples reveals the same fact for @xmath287 . melnikov ( * ? ? ? * example 4.1.6 ) ) . [ kl - stronger - example ] let @xmath288 computer calculations show that @xmath289 , but @xmath290 . by using the anti - automorphism of @xmath127 and @xmath291 that transposes a standard young tableau ( see proposition [ order - preserving - maps ] ) one obtains two more examples of pairs of tableaux which are comparable in @xmath127 , but not in @xmath27 . these are the _ only _ such examples in @xmath292 . to summarize we have the following diagram : @xmath293 .2 in do @xmath294 and @xmath295 coincide ? .3 in in this section we discuss two order preserving maps which embed @xmath3 into @xmath296 under any of the four orders . denoted by @xmath297 and @xmath298 , these maps are given by the following rule : for each @xmath299 , @xmath300 concatenates @xmath301 to the first row of @xmath16 from the right whereas @xmath302 concatenates @xmath301 to the first column of @xmath16 from the bottom i.e. , @xmath303 any partial order @xmath17 on @xmath3 is said to have the property of _ extension from segments _ if the maps @xmath304 are order preserving . in what follows we will prove that all of the four orders have the extension from segments property . [ embedding - from - initial - segments - theorem]the maps @xmath297 and @xmath298 are order preserving under the weak , @xmath2 , the geometric and chain orders . for any @xmath60 and @xmath305 , let @xmath306 and @xmath307 be the words obtained by concatenating @xmath301 to @xmath228 from the right and respectively from the left . the @xmath73 insertion algorithm yields that @xmath308 conventionally , we use the following notation : @xmath309 .1 in _ chain order _ : let @xmath310 in @xmath3 , i.e. , for any @xmath311 one has @xmath71 } ) ) \leq_{dom}^{op } { { \mathrm{sh}}}({{\mathrm{std}}}(t_{[i , j]})).\ ] ] now concatenating @xmath301 to the first row of @xmath21 and @xmath16 from the right ( after applying jeu de taquin slides ) obviously does not affect @xmath312}))$ ] and @xmath313}))$ ] if @xmath314 , and both have @xmath301 added to first row if @xmath315 . therefore @xmath316 on the other hand by proposition [ order - preserving - maps](@xmath317 ) one has : @xmath318 and since @xmath319 for any tableau @xmath21 , now @xmath298 is also order preserving . .1 in _ weak order _ : for this it is enough to consider the covering relations of @xmath320 . if @xmath21 is covered by @xmath16 then there exist two permutations @xmath321 and @xmath305 such that @xmath322 . equivalently @xmath323 . on the other hand the last assertion implies @xmath324 therefore in either case the weak order relation is preserved and we have @xmath325 .1 in _ @xmath2 order _ : this fact for @xmath2 order can be deduced easily by considering @xmath12 as a parabolic subgroup of @xmath326 : any two permutations @xmath327 satisfying @xmath328 in the parabolic subgroup @xmath12 still have the same relation in @xmath326 . if @xmath329 then there exist @xmath321 and @xmath330 satisfying @xmath331 in @xmath12 . then concatenating @xmath301 to the right side of both words still yields @xmath332 in @xmath326 . hence @xmath333 and by proposition [ order - preserving - maps](@xmath317 ) @xmath334 . .1 in _ geometric order _ : this fact follows from the result of melnikov ( * ? ? ? * proposition 6.6 ) . .2 in in @xcite , melnikov indicates another extension property of the weak and geometric order on @xmath3 which also generalize the property of extension from segments . .1 in let @xmath17 be any order on @xmath3 , @xmath335 and @xmath336 and @xmath337 are some tableaux on @xmath88-\{i\}$ ] . suppose @xmath21 and @xmath16 are the tableaux in @xmath338 obtained by standardizing @xmath336 and @xmath337 respectively . define an order on @xmath336 and @xmath337 in the following way @xmath339 then @xmath17 is said to have the property of _ extension by @xmath73 insertions _ if the @xmath73 insertion of the element @xmath82 into both tableaux @xmath336 and @xmath337 from above ( or from the left ) still preserves the order , in other words , denoting the resulting tableaux by @xmath340 and @xmath341 , if one has @xmath342 the property of extension by @xmath73 insertions for the weak order and geometric order was first proven by melnikov in @xcite and @xcite respectively . the same fact for @xmath2 order can be deduced from the work of barbash and vogan @xcite by using the theory that relates kazhdan - lusztig ( left ) cells to primitive ideals . below , independently from this theory , we provide a proof that shows @xmath2 order has the property of extension by @xmath73 insertions . on the other hand the following example shows that chain order does not have this property . @xmath343 [ kl - rsk - insertions - property lemma ] @xmath2 order on @xmath3 has the extension by @xmath73 insertions property . let @xmath336 and @xmath337 be two tableaux on @xmath88-\{i\}$ ] such that @xmath344 . in other words for @xmath21 and @xmath16 which are obtained by standardizing @xmath336 and @xmath337 respectively , we have @xmath345 . we may assume that @xmath21 is covered by @xmath16 . then there exist @xmath346 and @xmath228 in the knuth classes of @xmath21 and @xmath16 respectively such that @xmath347 . since @xmath348 is a parabolic subgroup of @xmath12 , as lemma [ embedding - from - initial - segments - theorem ] for the @xmath2 order shows , concatenating @xmath1 to the right side of both permutations yields @xmath349 in @xmath12 . therefore we have @xmath350 where @xmath17 denotes bruhat order . without lost of generality we assume @xmath351 and @xmath352 . .1 in consider the permutations @xmath353 which are obtained by multiplying @xmath354 and @xmath355 from the left by the transpositions @xmath356 in this order . it is easy to check that the @xmath73 insertion tableaux of @xmath357 and @xmath358 are nothing but @xmath340 and respectively @xmath341 . then @xmath359 follows , once it is shown that @xmath360 let @xmath361 and for each @xmath362 such that @xmath363 , let @xmath364 obviously for each @xmath365 , analysis on the ( left ) inversion sets yields @xmath366 and one can check that @xmath367 by using a basic characterization of bruhat order . that is : @xmath368 in @xmath12 if and only if for each @xmath369 , the sets of the form @xmath370 can be compared in the manner that after ordering their elements from the smallest to the biggest , the @xmath82-th element of the first set is always smaller than or equal to the @xmath82-th element of the second set for each @xmath371 . on the other hand multiplying @xmath354 and @xmath355 by @xmath372 from the left does not change the right descents of these permutations on the first @xmath373 positions . in other words , when restricted to the first @xmath373 positions @xmath354 and @xmath374 ( similarly @xmath355 and @xmath375 ) share the same right descents . therefore @xmath376 now we will show that @xmath377 obviously @xmath378 and therefore it is enough to prove that @xmath379 , since then the required equality follows by induction . .05 in observe that @xmath380 , @xmath381 i.e. , both of them are permutations in @xmath12 ending with the number @xmath362 . so @xmath382 lies both in @xmath383 and @xmath384 and by @xmath385 since @xmath374 ends with @xmath362 and @xmath386 ends with @xmath387 , from the characterization of the bruhat order it follows that @xmath388 and furthermore there exist no permutation @xmath260 satisfying @xmath389 . then by , all the summation terms on the right hand side , except @xmath390 , are equal to @xmath47 . henceforth @xmath391 and @xmath392 follows by induction . this result together with and imply that @xmath393 therefore by , , and we have @xmath394 for eack @xmath395 and so is true . hence @xmath396 .3 in .1 in malvenuto and reutenauer , in @xcite construct two graded hopf algebra structures on the @xmath397 module of all permutations @xmath398 which are dual to each other , and shown to be free as associative algebras by poirier and reutenauer in @xcite . the product structure of the one that concerns us here is given by @xmath399 where @xmath400 is obtained by increasing the indices of @xmath50 by the length of @xmath90 and @xmath401 denotes the shuffle product . .1 in poirier and reutenauer also show that @xmath397 module of all plactic classes @xmath402 , where @xmath403 becomes a hopf subalgebra of permutations whose product ( also shown in @xcite and @xcite ) is given by the formula @xmath404 then the bijection sending each plactic class to its defining tableau gives us a hopf algebra structure on the @xmath397 module of all standard young tableaux , @xmath405 . .1 in for example , @xmath406 another approach to calculate the product of two tableaux is given in @xcite where poirier and reutenauer explain this product using jeu de taquin slides . our goal is to show that it can also be described by a formula using partial orders , analogous to a result of loday and ronco ( * ? ? ? to state their result , given @xmath407 and @xmath408 , with @xmath409 , let @xmath410 be obtained from @xmath228 by adding @xmath362 to each letter . then let @xmath411 and @xmath412 denote the concatenations of @xmath413 and of @xmath414 , respectively . [ loday - ronco - malvenuto - reutenauer ] for @xmath415 and @xmath416 , with @xmath409 , one has in the malvenuto - reutenauer hopf algebra @xmath417 equivalently , the shuffles @xmath418 are the interval @xmath419_{\leq_{weak}}.$ ] .1 in the following facts are crucial for transporting the loday and ronco result to @xmath3 . let @xmath420 . when @xmath421 and @xmath422 , let @xmath423 denote the result of adding @xmath362 to every entry of @xmath16 . it is easily seen that @xmath424 where @xmath425 ( respectively , @xmath426 ) is the tableaux whose columns ( resp . rows ) are obtained by concatenating the columns ( resp . rows ) of @xmath21 and @xmath423 . note also that lemma [ j - d - t - initial - final ] shows for @xmath427 $ ] @xmath428 .2 in the following theorem is a consequence of lemma [ j - d - t - initial - final ] , corollary [ segment - restriction - lemma ] and theorem [ loday - ronco - malvenuto - reutenauer ] . [ p - r - theorem-0 ] let @xmath17 be a partial order on @xmath3 , for all @xmath429 , that 1 . is stronger than @xmath27 and 2 . restricts to segments . then in the poirier - reutenauer hopf algebra , @xmath430 let @xmath17 be a partial order on @xmath431 satisfying hypothesis . from ( [ p - r - multiplication ] ) , ( [ p - r - sileds ] ) and theorem [ loday - ronco - malvenuto - reutenauer ] it follows that any tableau @xmath432 appearing in the product @xmath433 satisfies : @xmath434 . therefore we have @xmath435 and this proves one direction . let @xmath432 be any tableau such that @xmath436 . also let @xmath427 $ ] where @xmath362 is the size of the tableau @xmath21 . by hypothesis @xmath437 @xmath438 i.e. , @xmath439 and @xmath440 and this shows that @xmath432 can be found by shuffling @xmath21 and @xmath16 in a certain way . therefore @xmath432 lies in the product @xmath433 . [ proof of theorem [ p - r - theorem ] ] all four orders @xmath238 , @xmath27 , @xmath28 and @xmath29 on @xmath3 satisfy the hypotheses of theorem [ p - r - theorem-0 ] by corollary [ segment - restriction - lemma ] . therefore the result follows . let @xmath441 and @xmath442 . then @xmath443 , @xmath444 and gives @xmath445 on the other hand , when considered with any of the four orders , the hasse diagram of @xmath446 in figure [ figure1 ] shows that the product above is equal to the sum of all tableaux in the interval @xmath447 $ ] . .1 in in this section , we prove theorem [ homotopy - theorem ] . we will view the commutative diagram @xmath448 } \\ \end{array}\ ] ] as an instance of the following set - up , involving closure relations , equivalence relations , order - preserving maps , and the topology of posets . for background on poset topology , see @xcite . let @xmath10 be a partially ordered set @xmath449 and @xmath450 a closure relation on @xmath10 , that is , @xmath451 it is well - known ( * ? ? ? * corollary 10.12 ) that in this instance , the order - preserving closure map @xmath452 has the property that its associated simplicial map of order complexes @xmath453 is a strong deformation retraction . now assume @xmath454 is an equivalence relation on @xmath10 such that , as maps of sets , the closure map @xmath455 factors through the quotient map @xmath456 . equivalently , the vertical map below is well - defined , and makes the diagram commute : @xmath457 [ closure - factorization ] in the above situation , partially order @xmath458 by the restriction of @xmath459 , and assume that @xmath460 has been given a partial order @xmath17 in such a way that the horizontal and vertical maps in the are also order - preserving . then the commutative diagram of associated simplicial maps of order complexes are all homotopy equivalences . obviously one can define a closure relation on @xmath460 such that @xmath461 , and the result follows . [ descent - fibers ] given any subset @xmath462 $ ] , there exists a maximum element @xmath463 in @xmath464 for the descent class @xmath465 consequently , the map @xmath466 defined by @xmath467 is a closure relation which also restricts to the proper parts and its image is isomorphic to @xmath232},\subseteq)$ ] . it is known that @xcite @xmath468 is actually an interval of the weak bruhat order on @xmath469 . therefore the map @xmath470 is a closure relation and since @xmath471 and @xmath472)$ ] consist of respectively @xmath39 and @xmath38 in ( @xmath473 ) , it restricts to the proper parts . now it is easy to see that its image is isomorphic to @xmath232},\subseteq)$ ] . [ syt - homotopy - corollary ] order @xmath12 by @xmath27 and @xmath4}$ ] by @xmath474 . let @xmath17 be any order on @xmath3 such that the commuting diagram has all the maps order - preserving . then these restrict to a commuting diagram of order - preserving maps on the proper parts , each of which induces a homotopy equivalence of order complexes . consequently , @xmath475 . the fact that the maps restrict to the proper parts follows because we know the maps explicitly as maps of sets , and the images of @xmath476 in @xmath464 must be exactly the @xmath476 in @xmath477 ( namely the single - row and single - column tableaux ) because the horizontal map is order - preserving . the fact that they induce homotopy equivalences follows from proposition [ closure - factorization ] applied to the three proper parts , using the closure relation in lemma [ descent - fibers ] and letting @xmath454 be knuth equivalence @xmath76 . one must observe that @xmath478 depends only on the knuth class of @xmath346 . the fact that @xmath475 for the boolean algebra @xmath232 } , \subseteq)$ ] is well - known ( * ? ? ? * prop.3.8.4 ) . .1 in by proposition [ order - preserving - maps ] all four orders on @xmath3 satisfy the hypotheses of corollary [ syt - homotopy - corollary ] . .1 in in figure [ mob ] , the first interval in @xmath479 has mbius function value @xmath30 , whereas mbius function value of the second interval which is found in @xmath480 , @xmath481 and @xmath482 is @xmath31 . therefore the mbius function values of the proper intervals in @xmath238 , @xmath27 , @xmath28 and @xmath29 on @xmath3 need not all lie in @xmath483 as they do in @xmath263 . .1 in in this section we describe the _ inner translation property _ of @xmath2 and geometric order on @xmath3 which enable us to generalize these orders to the skew standard young tableaux @xmath484 of size @xmath1 with some fixed inner boundary @xmath34 . to do this first we need to recall the notion of _ dual knuth _ relations on @xmath12 : permutations @xmath485 are said to be _ differ by a single dual knuth relation _ if for some @xmath486 $ ] , @xmath487 and @xmath488 whereas @xmath489 and @xmath490 . in this case @xmath491 we say @xmath492 are _ knuth equivalent _ written as @xmath493 , if @xmath228 can be generated from @xmath346 by a sequence of single dual knuth relations . observe that @xmath493 if and only if @xmath494 . since left descent sets are all equal for the permutations in a knuth class @xmath103 , @xmath60 , a single dual knuth relation gives the following action on tableaux : let @xmath495 be the row number of @xmath82 in @xmath16 from the top . _ case 1_. if @xmath496 and @xmath497 then @xmath498 the resulting tableau is found by interchanging @xmath499 and @xmath500 in the first case and interchanging @xmath82 and @xmath500 in the second case . _ if @xmath501 and @xmath502 then @xmath503 this time interchanging @xmath499 and @xmath500 in the first case and interchanging @xmath82 and @xmath500 in the second case gives us the resulting tableau under the action of the single dual knuth relation given with the triple @xmath504 . we say @xmath505 if @xmath506 can be obtained from @xmath16 by applying a sequence of single dual knuth relations as described above . the following theorem , see ( * proposition 3.8.1 ) for example , is a nice characterization of this relation . [ dual - knuth ] let @xmath69 . then @xmath507 .1 in let @xmath508 and @xmath509}$ ] be a subset of @xmath3 given by @xmath510}:=\ { t \in syt_n \mid \alpha \in { { \mathrm{des}}}(t ) , \beta \not \in { { \mathrm{des}}}(t)\}.\ ] ] as we described above we can apply a single dual knuth relation determined with the triple @xmath511 on each @xmath512}$ ] and this gives us the following _ inner translation map _ : @xmath513 } : syt_n^{[\alpha,\beta]}\mapsto syt_n^{[\beta,\alpha]},\ ] ] where @xmath514}\circ \mathcal{v}_{[\alpha,\beta]}$ ] and @xmath515}\circ \mathcal{v}_{[\beta,\alpha]}$ ] are just identity maps on their domains . .1 in any order @xmath17 on @xmath3 is said to have the _ inner translation property _ if the inner translation map @xmath513 } : ( syt_n^{[\alpha,\beta]},\leq)\mapsto ( syt_n^{[\beta,\alpha]},\leq)\ ] ] is order preserving . .1 in now we give the following corollary which is crucial in the sense that it provides the sufficient tool for generalizing any partial order on standard young tableaux to the skew standard tableaux . .1 in for @xmath516 , let @xmath432 be a tableau in @xmath517 and @xmath518}=r\}.\ ] ] .1 in [ equal - inner - shapes ] suppose @xmath69 and @xmath519 satisfy @xmath520}=t_{[1,k]}=r\ ] ] @xmath521 moreover suppose @xmath522 and @xmath506 are the tableaux in @xmath3 obtained by replacing @xmath432 by @xmath523 in @xmath21 and @xmath16 respectively . then for @xmath17 having the inner translation property on @xmath3 , one has @xmath524 in particular @xmath525 and @xmath526 are isomorphic subposets of @xmath59 . as a consequence of theorem [ dual - knuth ] , by applying to @xmath21 and @xmath16 the same sequence of dual knuth relations on their subtableau @xmath432 , one can generate @xmath522 and @xmath506 respectively . on the other hand , since @xmath17 has inner translation property at each step the order is preserved . [ vogan - involution ] @xmath2 and geometric order on @xmath3 have the inner translation property . therefore for any @xmath527 such that @xmath528 and @xmath529 , @xmath530 are isomorphic subposet of @xmath3 in @xmath2 and geometric orders . this map is first introduced by vogan in @xcite for @xmath2 order , where he also shows the desired property . for geometric order this result is due to melnikov ( * ? ? ? * proposition 6.6 ) . the example given below shows that chain and the weak order do not satisfy the inner translation property . ( see also remark [ inner - translation - remark ] ) . @xmath531 where the latter pair is obtained from the former by applying a single dual knuth relation on the triple @xmath532 . @xmath533 where the latter pair is obtained from the former by applying a single dual knuth relation on the triple @xmath534 . let @xmath535 , @xmath536 and @xmath537 such that @xmath538 . for @xmath539 of shape @xmath37 , define @xmath540 to be the skew standard tableau on @xmath88 $ ] of shape @xmath541 obtained by standardizing the skew segment of @xmath16 having shape @xmath541 . [ skew - orders - definition ] let @xmath17 be partial order on @xmath3 having inner translation property . for @xmath542 and @xmath543 be two skew standard tableaux in @xmath44 , we set @xmath544 if there exist two tableaux @xmath21 and @xmath16 in @xmath545 of shape @xmath37 and @xmath546 respectively which satisfy : @xmath547 [ kl - geom - skew - poset - remark ] as a consequence of theorem [ vogan - involution ] , the skew orders , @xmath28 and @xmath29 on @xmath44 becomes well defined . .1 in in what follows we first prove a result , namely proposition [ skew - mobius - value - theorem ] below , which is about the mbius function of the subposet @xmath548 of @xmath3 ordered by any order that is stronger than @xmath27 , restricts to segments and has the property of extension from the segments . consequently theorem [ skew - homotopy - theorem ] follows from this results together with theorem [ embedding - from - initial - segments - theorem ] , theorem [ vogan - involution ] and definition [ skew - orders - definition ] . .1 in let @xmath432 be a tableau in @xmath517 and @xmath535 . recall that @xmath549}=r\}.\ ] ] since the weak order restricts to segment , it can be induced on @xmath548 . moreover the analysis made by comparing the left inversion sets yields that any tableau @xmath550 , under the weak order , lies between two tableaux @xmath551 and @xmath552 given below . @xmath553 .1 in therefore @xmath554_{\leq_{weak}}$ ] and for any order @xmath17 which is stronger than the weak order and which restricts to segments we have @xmath555_{\leq}\ ] ] .1 in [ skew - mobius - value - theorem ] let @xmath17 be any order on @xmath545 with the following properties 1 . @xmath17 is stronger than @xmath27 2 . @xmath17 restricts to segments 3 . @xmath17 extends from segments . then for @xmath551 and @xmath552 as above one has @xmath556,\ ] ] and the proper part of @xmath548 is homotopy equivalent to @xmath557 .1 in below we recall rambau s suspension lemma about bounded posets @xcite , which will be used to prove proposition [ skew - mobius - value - theorem ] . [ suspension - lemma ] let @xmath558 and @xmath559 be two bounded posets such that @xmath560 . assume @xmath558 is the disjoint union of its two subsets @xmath561 and @xmath562 where @xmath561 forms an order ideal and @xmath562 forms an order filter of @xmath558 . assume further that there are order preserving maps @xmath563 satisfying the following properties : 1 . the image of @xmath82 lies in @xmath561 and the image of @xmath564 lies in @xmath562 2 . the maps @xmath565 and @xmath566 are identity on @xmath559 3 . for every @xmath567 , @xmath568 4 . the fiber @xmath569 lies in @xmath562 except for @xmath570 and the fiber @xmath571 lies in @xmath561 except for @xmath572 . then the proper part @xmath573 of @xmath558 is homotopy equivalent to the suspension of the proper part of @xmath559 . .1 in for @xmath574 , let @xmath575~ \mbox { and } ~\mathcal{q}=[\hat{0}_{r , n-1},\hat{1}_{r , n-1}]\ ] ] together with the subposets of @xmath558 given as @xmath576 moreover let @xmath577 where the map @xmath220 restricts any @xmath578 to its initial segment @xmath579}$ ] and the map @xmath82 concatenates @xmath580 to the first row of any @xmath581 from right whereas @xmath564 concatenates @xmath580 to the first column of @xmath21 from the bottom . first we will show that @xmath561 is an order ideal of @xmath558 . let @xmath582 and @xmath583 . then by lemma [ tableau - descent - lemma ] , @xmath584 and therefore @xmath585 does not belong to @xmath586 . this shows that @xmath587 and @xmath561 is an order ideal . a similar argument also shows that @xmath562 is an order filter of @xmath558 . on the other hand it can be easily seen that @xmath558 is the disjoint union of @xmath588 and @xmath562 . since the tableau @xmath432 is common for both @xmath558 and @xmath559 and @xmath17 restricts to the initial segments , the map @xmath589 is well defined and order preserving . by virtue of their definitions the maps @xmath590 are also well defined . on the other hand since @xmath17 has the property of extension from segments therefore they both are order preserving . now part @xmath591 follows from the fact that the map @xmath82 concatenates @xmath580 to the right of the first row of @xmath592 , which provides no possibility that @xmath580 appears below @xmath585 in @xmath593 . therefore @xmath594 and @xmath595 . on the other hand in @xmath596 , @xmath580 always appears below @xmath585 and this shows that @xmath597 . for part @xmath598 , let @xmath599 be the row word of @xmath578 . the analysis on the ( left ) inversion sets gives : @xmath600 and by @xmath73 correspondence @xmath601 and hence @xmath602 . one can check the hypotheses @xmath603 and @xmath604 easily . therefore by lemma [ suspension - lemma ] , the proper part of @xmath558 is homotopy equivalent to the suspension of the proper part of @xmath559 . .1 in in the rest we proceed by induction : let @xmath605 . then all tableaux in the poset @xmath606 $ ] are obtained by placing @xmath580 in some outer corner of @xmath432 , i.e , in an empty cell along the boundary of @xmath432 whose addition to @xmath432 still gives a young tableau shape . moreover it can be easily checked , for example by comparing the left inversion sets of their row words , that these tableaux form a saturated chain in @xmath607 . on the other hand since @xmath17 is stronger then the @xmath27 and restricts to segments this chain remains saturated in @xmath608 . the following diagram illustrates the case when @xmath432 has three outer corners . .1 in @xmath609 now if @xmath432 has rectangular shape then it has two outer corners and the poset @xmath606 $ ] consists of two tableaux . it has the mbius function from the bottom to the top elements to be @xmath610 and moreover the proper part of @xmath558 is homotopy equivalent to the empty set i.e , @xmath611-dimensional sphere . if @xmath432 is non rectangular then as in the above diagram @xmath558 is a saturated chain having more than two elements . hence its mbius function is @xmath47 from the bottom to the top elements and it is homotopic to a point . now assume that for @xmath612 the poset @xmath613 $ ] satisfies the hypothesis i.e. , the proper part of @xmath559 is homotopic to a @xmath614-sphere in case @xmath432 is rectangular and it is homotopic to a point otherwise . on the other hand we already see that the proper part of @xmath615 $ ] is homotopy equivalent to the suspension of the proper part of @xmath559 , so that the former becomes homotopy equivalent to a ( @xmath616)-sphere if @xmath432 is rectangular and to a point otherwise . therefore the assertion of proposition [ skew - mobius - value - theorem ] follows . .1 in by remark [ kl - geom - skew - poset - remark ] , @xmath2 and geometric orders are well defined on @xmath44 . on the other hand they restrict segments and have the property of embedding from initial segments by lemma [ embedding - from - initial - segments - theorem ] . so the required statement follows from proposition [ skew - mobius - value - theorem ] . .1 in by observing figure [ figure1 ] , one can see that the posets of @xmath3 with all these orders are not lattices and not ranked . on the other hand we can still say something about the size of their shortest or longest chains , where by convention @xmath617 has size @xmath82 . 1 . 2 . the size of a shortest saturated chain in @xmath287 is @xmath1 . the size of a longest chain in @xmath320 , @xmath618 and @xmath619 is equal to the size of the longest chain in @xmath620 , which is asymptotically @xmath621 . observe that if @xmath346 is covered by @xmath228 in @xmath622 then the size of the ( left ) descent set @xmath623 of @xmath228 is at most one bigger than the size of @xmath478 . this fact is also true for @xmath320 : if @xmath21 is covered by @xmath16 in @xmath320 then @xmath624 this shows that the size of a shortest saturated chain must be at least @xmath1 . on the other hand it can be seen by an easy induction that there exist a saturated chain in @xmath320 of size @xmath1 with the following form : @xmath625 therefore the statement about shortest chains in @xmath287 follows . for longest chains the proof is based on two facts : a result of greene and kleitman which calculates the size of longest chain in the lattice of integer partitions ordered by the dominance order and the result of melnikov ( * ? ? ? * proposition 4.1.8 ) which shows that for any tableau @xmath21 of shape @xmath34 in @xmath0 and for any partition @xmath626 such that @xmath627 , there is a tableau @xmath60 such that @xmath628 and @xmath629 . these two facts enable us to calculate the longest chain in @xmath0 ordered by the weak order . since @xmath630 and @xmath29 also change the shapes of the tableaux , the longest chain of @xmath287 still remains saturated in @xmath618 and @xmath295 . by an easy induction one can see that chain in ( [ a - shortest - chain ] ) still remains saturated in @xmath3 for @xmath2 , geometric and chain orders . therefore if it were known ( [ covering - descent - relation2 ] ) is satisfied by these three orders , we could deduce the same conclusion about their shortest chains . .1 in theorem [ homotopy - theorem ] also follows from proposition [ skew - mobius - value - theorem ] by taking @xmath631 . the original proof is kept here for indicating different approaches to the subject . the order complex of the proper part of @xmath59 under any of the four orders is not homeomorphic to a sphere . one can observe @xmath632 in figure [ figure1 ] to see the smallest example . moreover since these posets are not ranked for @xmath633 , the order complex of their proper parts are not pseudomanifolds . [ inner - translation - remark ] although the weak order on @xmath3 does not have the inner translation property , it might still satisfy corollary [ equal - inner - shapes ] without this property , which would then make it possible to define weak order on skew standard tableaux . for chain order , two pairs of tableaux given below where the inner tableau @xmath634 common to the first pair is replaced by @xmath635 in the second pair , show that corollary [ equal - inner - shapes ] is not satisfied by chain order : @xmath636 one might ask to what extent the definitions and results in this paper apply to other lexicographic coxeter systems @xmath637 . the weak order on @xmath176 is well - defined , as are @xmath2 and the geometric order , where the former still remains weaker than the latter ( @xcite ; see ( * ? ? ? * fact 7 ) ) . definition [ weak - order - def ] makes sense and remains valid , and so does proposition [ order - preserving - maps]@xmath638 for @xmath2 order ( @xcite ; see ( * ? ? ? * fact 7 ) ) . for geometric order the same property follows from ( * ? ? ? * theorem 6.11 ) or ( * ? ? ? * theorem 9.9 ) . for the analysis of mbius function and homotopy types , the crucial lemma [ descent - fibers ] was proven by bjorner and wachs ( * ? ? ? * theorem 6.1 ) for all _ finite _ coxeter groups @xmath176 . hence corollary [ syt - homotopy - corollary ] and theorem [ homotopy - theorem ] are valid also in this generality , with the same proof . the author is grateful to v. reiner for his helpful questions and comments throughout this work . the author also thanks a. melnikov for allowing access to her unpublished preprints , w. mcgovern and m. geck for helpful comments .

let @xmath0 be the set of all standard young tableaux with @xmath1 cells . after recalling the definitions of four partial orders , the weak , @xmath2 , geometric and chain orders on @xmath3 and some of their crucial properties , we prove three main results : 1 . intervals in any of these four orders essentially describe the product in a hopf algebra of tableaux defined by poirier and reutenauer . 2 . the map sending a tableau to its descent set induces a homotopy equivalence of the proper parts of all of these orders on tableaux with that of the boolean algebra @xmath4}$ ] . in particular , the mbius function of these orders on tableaux is @xmath5 . 3 . for two of the four orders , one can define a more general order on skew tableaux having fixed inner boundary , and similarly analyze their homotopy type and mbius function .

recently , cleo collaboration have improved their previous measurements of @xmath0 @xcite : @xmath4 these branching ratios are much larger than the estimations under the standard theorectical frame which is based on the effective hamiltonian and general factorization approximation . now it is commonly believed that these large branching ratios are due to the special properties of @xmath5 , and several new mechanisms have been proposed to enhance the decay rates of @xmath6 and @xmath7 . in the following we only discuss the standard model(sm ) mechanisms because we think that the contribution of the sm should be carefully examined first . halperin and zhitnisky @xcite proposed an interesting mechanism : @xmath5 can be directly produced through @xmath8 if @xmath9 . but if this mechanism dominates , one easily find that @xmath10 which is in contradiction with the stringent upper limit of @xmath11 given by cleo @xcite : @xmath12 so this mechanism is unlikely to be dominant . because the effective vertex of @xmath13 in sm is very small , @xmath14 @xcite is also impossible to account for the large @xmath6 branching ratios . the authors of ref @xcite proposed that @xmath15 and @xmath16 via qcd anomaly can account for the large semi - inclusive branching ratios of @xmath7 . cleo @xcite have measured the @xmath5 momentum spectrum in @xmath7 and this measurement favors the mechanism of ref@xcite . but because it has an extra gluon in the final state , unless the gluon is soft and absorbed into the @xmath5 wave functions @xcite , it seems difficult to realize its contribution to two - body exclusive decay @xmath6 . motivated by the idea of ref @xcite , the authors of ref @xcite proposed di - gluon fusion mechanism , which is depicted in fig.1 . because both of the gluons are hard , it seems reasonable to give a perturbative qcd estimation . but because of our ignorance about the form factor of @xmath1 vertex , there are large uncertainties in calculations . so in the following we would reanalyze di - gluon fusion mechanism in detail using several kinds of @xmath17 form factors and compare our numerical results with experimental data . we first give a brief introduction on how to estimate contributions of di - gluon fusion mechanism in perturbative qcd . @xmath18 vertex can be parameterized as @xcite : @xmath19 where @xmath20 is a constant and can be extracted from @xmath21 or @xmath22 , @xmath23 is qcd coupling constant , @xmath24 is @xmath17 form factor and @xmath25 . neglecting the transverse momentum of quarks and taking the wave functions for @xmath26 and @xmath27 meson as @xmath28 where @xmath29 is an identity in color space , and the integration of the distribution amplitude is related to the meson decay constant , @xmath30 in this paper , we take the distribution amplitudes as@xcite : @xmath31 then we can write the amplitude of fig.1 as @xmath32 4 \epsilon^{\mu \nu \alpha \beta } k_{1\alpha } k_{2\beta } a_{\eta^{\prime } } f(k_1 ^ 2,k_2 ^ 2 ) c_f g_s^3 } { \sqrt{2 } \sqrt{2 } k_1 ^ 2 k_2 ^ 2 } \phi_k(y),\ ] ] where @xmath33 is the effective @xmath34 vertex@xcite @xmath35 and the color factor @xmath36\delta^{ab}=\frac{4}{3}$ ] . finally we obtain @xmath37 + f_2^i m_b m_b[(q\cdot k_2 ) k_1 ^ 2-(q\cdot k_1)(k_1\cdot k_2 ) ] } { k_1 ^ 2 k_2 ^ 2 } .\end{aligned}\ ] ] in the above integral , there are two singularities from the gluon propagators and sometimes there would be another singularity from the @xmath38 form factor ( for example , @xmath39 ) . we add a small pure imaginary number @xmath40 ( @xmath41 ) to the denominator to get a convergent integral . to evaluate the numerical result of eq.(10 ) , we take the form factors @xmath42 and @xmath43 according to ref@xcite as @xmath44 where @xmath45 are wilson coefficients at the nll level.(because qcd corrections maybe large , we do not take @xcite @xmath46 and @xmath47 which are derived from the sm without qcd corrections . ) it is always subtle to choose the scale of @xmath48 , because the average momenta squared of the gluons are @xmath49 we prefer to choose @xmath50 though we also give the branching ratios when @xmath51 or @xmath52 in tab.1 and tab.2 . as to the @xmath17 form factor @xmath24 , because of our complete ignorance there are large uncertainties in the numerical evaluation of eq.(10 ) . in the following we try several kinds of form factors . in ref@xcite , the authors have assumed that @xmath53-dependence of the form factor is weak and as an approximation they take form factor as a constant to explain large semi - inclusive decay @xmath54 . the difference between ref@xcite and ref@xcite is that the running of @xmath55 with the scale is considered in @xcite but not in @xcite . we also use their ansatz to estimate exclusive decays @xmath0 and @xmath2 . using eq.(10 ) , we give numerical results in tab.1 . from the table it seems that the constant form factor can account for the large branching ratios of @xmath0 , but unfortunately we would get much larger branching ratios of @xmath2 , which is strongly in contradiction with the cleo s measurements@xcite ( see , for instance , the case of @xmath56 in tab.1 ) . from eq.(10 ) , we notice that the amplitude must be integrated over a wide range of @xmath57 and @xmath58 , therefore the effects of @xmath57 , @xmath58 dependence of @xmath24 must be taken into account . the authors of ref@xcite choose the form factor as @xmath39 because such a form factor works well in @xmath59 @xcite . we have followed their calculations and it seems that their numerical results are overestimated than ours(in tab.2 ) by a factor about three . from tab.2 , we can see that our estimations of @xmath60 are about @xmath3 , but when considering that the standard theoretical frame ( based on effective hamiltonian and general factorization approximation ) can give@xcite @xmath61 , it is still possible to account for the experimental data only if the contributions from di - gluon fusion mechanism and the standard theoretical frame constructively interfere . if this is true , then because the contributions to @xmath62 from the standard theoretical frame are negligible , the di - gluon fusion mechanism would be dominant in @xmath63 , _ i.e. _ , @xmath64 which can be tested by future measurements of cleo or b factories . kagan and petrov @xcite proposed a model of the @xmath18 vertex in which a pseudoscalar current is coupled to two gluons through quark loops . their perturbative calculations yield a form factor : @xmath65 where we take @xmath66 , @xmath67 and normalize @xmath24 with the normal condition @xmath25 . we use this form factor in eq.(10 ) with @xmath56 and get @xmath68 which is too small to account for the experiments . in ref @xcite , the authors calculate the transition form factor in @xmath69 coupling to @xmath70 in the frame of a perturbative qcd based on the modified factorization formula . they find that numerically their results are extremely similar to that obtained by applying the interpolation procedure in the manner of brodsky and lepage in the case of two off - shell photons : @xmath71 with @xmath72 and @xmath73 . we assume that the structure of the transition of @xmath74 is similar to that of @xmath75 : @xmath76 where we approximate @xmath77 . by using this form factor , we calculate the branching ratios of @xmath0 which are listed in tab.3 . the results are about fifty times smaller than the experimental data . in experimental fit , pole approximation is often used to fit the momentum square dependence of form factor @xcite . as a try , we also assume @xmath78 for our calculations , we list the results in tab.4 . and we can see that this kind of form factor would make the digluon fusion mechanism completely negligible in @xmath6 . to examine the validity of perturbative qcd in the above caculations , or in other words , whether the amplitude calculated by perturbative qcd is dominated by hard gluon contributions , we set cut on @xmath58 and compare the results with those without cut . taking @xmath79 , we list the numerical results in tab.v . we can see that hard gluon contributions are really dominant in the case of constant form factor . but if we take form factor as @xmath80 , hard gluon contributions are small , this does not mean that non - perturbative contributions are dominant in this case . this is due to the fact that the singularity in the form factor is accidently close to the other singularities in @xmath57 and @xmath58 and then enhance the contributions of soft gluon . the authors of ref @xcite proposed digluon fusion mechanism to explain the large branching ratios of @xmath6 , but because of our ignorance about effective @xmath1 vertex , there are large uncertainties in perturbative qcd estimations . we try several kinds of @xmath38 form factors , and through our calculations , we find that constant form factor is consistent with the data of @xmath6 , but inconsistent with the data of @xmath81 in some cases of different @xmath48 or @xmath82 . if we take @xmath80 as the authors of ref @xcite have done , di - gluon fusion mechanism is important in @xmath6 but not dominant . as a consequence , we could anticipate that @xmath83 would be about @xmath3 which can be tested by future experiments . we also try the other three kinds of form factors and they all give very small contributions to the branching ratios . we conclude that , though we know little about @xmath17 form factor , if di - gluon fusion mechanism is important in @xmath6 , the branching ratios of @xmath81 would be definitely around @xmath3 . this work is supported in part by the national natural science foundation of china and the grant of china state commission of science and technology . a. szczepaniak , e. henley and s. j. brodsky , phy.lett.b * 152 * , 380(1990 ) ; + c. e. carlson and j. milana , phys.rev.d * 49 * , 5908(1994 ) ; phys.lett.b * 301 * , 237(1993 ) ; + h. simma and d. wyler , phys.lett.b * 272 * , 395(1991 ) ; + c. d. l and d. x. zhang , phys.lett.b * 400 * , 188(1997 ) .

di - gluon fusion mechanism might account for the large branching ratios of @xmath0 . but because we know little about the effective @xmath1 vertex , there are large uncertainties in perturbative qcd estimations . in this paper , we try several kinds of @xmath1 form factors and compare the numerical results with the experiment . we find that , though we know little about @xmath1 form factor , if di - gluon fusion mechanism is important in @xmath0 , the branching ratios of the decays @xmath2 would be around @xmath3 which can be tested by future experiments . * pacs numbers 13.25.hw 13.20.he *

in ads - cft black brane is a thermal state of the boundary conformal field theory living on the minkowski space - time . this is not a relevant deformation of the cft hamiltonian and there is no renormalization group flow in the ordinary sense . therefore the question of the existence of a @xmath1-function , in the sense of zamolodchikov @xcite , does not naturally arise in this situation . moreover zamolodchikov @xmath1-function is constant at a fixed point and independent of the state of the cft . the purpose of this note is to point out that ads - cft duality and the thermodynamic nature of classical gravity allows us to introduce a generalized notion of @xmath1-function , at least for large-@xmath4 theories with classical gravity dual . this generalized @xmath1-function can not be interpreted as an off - shell central charge . rather it can be interpreted as a measure of quantum entanglement that exists at different energy scales in the given state . we will construct this @xmath1-function holographically when the cft is in thermal state and the gravity dual is an empty black brane geometry . we focus on four dimensional field theories only . our choice of the thermal state is motivated by the fact that the gravity dual has a curvature singularity and the lorentz invariance is broken everywhere except near the uv boundary of ads . so it can teach us some lessons about rg - flow interpretation of more general geometries . * throughout this paper we will assume that the bulk theory is einstein gravity coupled minimally to a set of matter fields . * the holographic picture is based on the fact that the gravity dual of @xmath1-theorem is the second law of causal horizon thermodynamics in asymptotically ads spaces @xcite . in a nutshell , second law for causal horizons say that if we consider the future bulk light - cone of a boundary point then the expansion of the null geodesic generators of the light - cone is negative @xcite . now one can assign bekenstein - hawking entropy to the causal horizon . the fact that the expansion is negative then implies that as we move away from the boundary the entropy density decreases monotonically . this is essentially holographic @xmath1-theorem @xcite if we specialize to a domain - wall geometry . the bulk future light - cone interpolates between the uv - ads and the ir - ads and the monotonically decreasing bekenstein - hawking entropy density gives the holographic @xmath1-function @xcite . if we focus on domain - wall geometry then the second law has the interpretation of holographic @xmath1-theorem . but what about other asymptotically ads ( aads ) geometries ? second law of causal horizon thermodynamics holds in any aads geometry and in fact holographic rg @xcite applies to any such setup . it has been argued that the holographic rg in the bulk is dual to the wilsonian rg in the boundary @xcite . so is it possible to associate a notion of irreversibility to any classical aads geometry ? it seems that the existence of second law for classical gravity allows us to do precisely this thing . in the field theory side its interpretation will require us to generalize the concept of zamolodchikov - type @xmath1-function . we will have almost nothing to say on it in this paper . to gain some experience with such generalized @xmath1-functions we will work out a reasonably simple but interesting example of empty black brane geometry in ads@xmath5.-function by viewing the black brane background as rg flow , was also considered in @xcite . @xmath1-function for attractor flows were considered in @xcite . ] it has curvature singularity hidden behind the black brane horizon and the geometry is not lorentz invariant except near the ads@xmath5 boundary . the @xmath1-function we construct is just the bekenstein - hawking entropy density of a causal horizon in the black brane geometry @xcite . the causal horizon originates at some point of the ads boundary and terminates at the curvature singularity . nothing depends on the choice of the boundary point where the causal horizon originates because of space - time translation invariance . second law guarantees that our function monotonically decreases as we move away from the boundary along the null geodesic generators of the causal horizon . we will see that the @xmath1-function monotonically decreases from @xmath2 to zero at the curvature singularity . the causal horizon is just the future bulk light - cone of a boundary point . we take the boundary point , @xmath6 , to have coordinates , @xmath7 . the metric of the five - dimenional black brane is , @xmath8\ ] ] our job is to construct the ingoing null geodesics in this geometry which originate from the boundary point @xmath6 . black brane @xcite . we have shown only the radial null geodesic coming out from the boundary point @xmath6 . other non - radial null geodesics from @xmath6 are not shown here . in this paper we do not need the setup of the two sided black brane . we have drawn it for the sake of completeness.,width=566 ] let us define the ingoing eddington - finkelstein coordinate as , @xmath9 where @xmath10 , so that the metric takes the form @xmath11\ ] ] there is no singularity at the horizon , @xmath12 , and so we can follow the null geodesics all the way to the curvature singularity at , @xmath13 . since we want to find out the null geodesics we can as well work with the conformally transformed metric given by , @xmath14 let @xmath15 denote the affine parameter along a null geodesic in the conformally transformed metric @xmath16 . by using the affine parameter corresponding to the original geometry . but this does not affect the physics . this is just a change in scheme . it will of course be better to solve this in terms of the original affine parameter ] so we have , @xmath17 we also have four conserved charges corresponding to the translations in @xmath18 and the @xmath19 s . @xmath20 where @xmath21 . @xmath22 and @xmath23 are the conserved charges along a null geodesic . here we are assuming that the affine parameter @xmath15 increases as we move away from the boundary at @xmath24 . we will be working with the future bulk light - cone of the boundary point @xmath6 and so with our convention for the affine parameter , @xmath25 . so @xmath26 . now using ( @xmath27 ) and ( @xmath28 ) we get , @xmath29 so we can see that the null geodesics which can reach the boundary point must satisfy the constraint , @xmath30 . this constraint together with the constraint @xmath31 , allow us to parametrize the conserved charges as , @xmath32 where @xmath33 , @xmath34 and @xmath35 is a unit vector in @xmath36 . now it is easy to see that @xmath37 is redundant because it can be absorbed by an affine reparametrization , @xmath38 . therefore we will set @xmath39 . so the equation for @xmath40 simplifies to , @xmath41 we have chosen the positive root because our convention is @xmath42 . so we can write , @xmath43 where the boundary condition , @xmath44 has been imposed . the solution of this equation is , in the argument of @xmath45 should be replaced by @xmath46 . ] @xmath47 where @xmath45 is one of the jacobian elliptic functions . its properties are well studied although a closed form expression in terms of elementary functions does not exist . given the solution for @xmath48 we can in principle determine @xmath49 from ( [ cc ] ) , but we were unable to do so in any convenient way . in any case the complete set of solutions can be written as , @xmath50 where we have defined , @xmath51 and @xmath52 . we have imposed boundary conditions such that , @xmath53 for all values of @xmath54 and @xmath55 . this corresponds to the fact that the null geodesics are all coming out of the point @xmath6 with coordinates @xmath56 . note that @xmath57 at the boundary @xmath24 . for any fixed values of @xmath54 and @xmath58 , the above equation ( @xmath59 ) reduces to the equation of the null geodesic parametrised by the affine parameter @xmath15 and coming out of the fixed boundary point @xmath60 . as we vary @xmath54 and @xmath58 , we scan over all the geodesics coming out of the point @xmath6 . all these null geodesics form a null hyper surface whose parametric equation is given by ( @xmath59 ) . the intrinsic coordinates on the null hyper surface are ( @xmath61 ) . ( @xmath62 ) are comoving coordinates along a null geodesic parametrised by @xmath15 . this null hyper surface is the sought for bulk future light - cone or the past causal horizon of the point @xmath6 . our next job is to find out the induced metric on the null - hypersurface ( @xmath59 ) . to find out the induced metric we have to use the original black brane metric ( @xmath63 ) . using this we get , [ i m ] ds_ind^2 = + d^2 + ^2 ^2d_2 ^ 2 where @xmath64 is the metric of a unit two - sphere parametrised by @xmath55 . the induced metric is degenerate as it should be because ( @xmath59 ) is a null - hypersurface . ( @xmath65 ) is the metric on a @xmath66 space - like slice of the causal horizon ( @xmath59 ) , parametrised by the coordinates ( @xmath62 ) . the volume form can be written as , dv_ind = c ( , ) dv_h^3 where we have defined , c(,)= @xmath67 is the volume form on a unit three dimensional hyperbolic space given by , @xmath68 where we have parametrised @xmath55 as ( @xmath69 ) . the fact that @xmath1 is a function only of @xmath15 and @xmath54 is a consequence of the rotational symmetry of the metric . in the more standard domain - wall geometry @xmath1 is function only of @xmath15 because of the lorentz invariance of the metric . in the black brane geometry lorentz invariance is broken down to the spatial rotation group and so the @xmath54 dependence is non - trivial . now second law for causal horizons is the statement that , [ sl ] here we have used the fact that @xmath67 is a comoving volume element and @xmath54 is a comoving coordinate i.e , @xmath54 is constant along a null geodesic generator of the causal horizon . the bekenstein - hawking entropy density associated to the volume element @xmath70 is , ds_bh = = dv_h^3 we can put in the ads radius @xmath71 by replacing @xmath72 . this gives , ds_bh = = c ( , ) dv_h^3 so our @xmath1-function is , we get a family of @xmath1-functions parametrised by @xmath54 ( fig-2 ) . we check in the appendix using perturbation theory for small @xmath15 that @xmath73 as @xmath74 for * all * values of @xmath54 i.e , @xmath75 . it will be true for any aads geometry , not just the black brane . note that @xmath76 is the ads boundary and @xmath15 increases as we move away from the boundary along the null geodesics . so for any fixed value of @xmath54 the @xmath1-function @xmath77 starts at the uv value @xmath2 and decreases monotonically as a result of the second law ( @xmath78 ) . it turns out that in the case of the black brane the @xmath1-function becomes zero at the curvature singularity for * all * values of @xmath54 . so for black brane in five dimensions , the @xmath1-function monotonically decreases from the uv central charge to zero at the curvature singularity . it does not show any characteristic behavior while crossing the black brane horizon . -function for three different values of @xmath54 . all of them start at the uv value @xmath79 and monotonically decreases to zero at the curvature singularity . the values of @xmath15 at the singularity for different values of @xmath54 can be obtained from ( @xmath80 ) by setting @xmath13 . of course from a physical point of view going to the singularity with gr is meaningless . but if we forget about any stringy physics for the time being , then as a classical theory gr holds everywhere except at the singularity . , width=377 ] we would like to emphasise that the fact that we have obtained a family of @xmath1-functions parametrized by @xmath54 , instead of just one , is no cause for concern . @xmath1-function is not unique . for example in two dimensions one can construct the standard zamolodchikov @xmath1-function @xcite and also the entanglement entropy @xmath1-function due to casini and huerta @xcite . it is know that they are not the same , but they both monotonically interpolate between the uv and the ir central charges . in fact if we can construct one @xmath1-function then we can construct an infinite family all of which contain the same physical information @xcite . the plot of the @xmath1-function in fig-2 shows that it is not stationary at the singularity . this is not a problem because strictly speaking the function is not analytic there . we do not know how to extend the function beyond the singularity . but the fact that it is zero at the singularity shows that the flow comes to an end at the singularity . the @xmath1-function is an element of area of the causal horizon and so it is positive semidefinite by construction . so the flow saturates the lower bound at the singularity . this is similar to what sometimes happens in case of the @xmath1-function constructed out of entanglement entropy . for example in three dimensions , the entropic @xmath1-function for a massive scalar is not stationary at the uv fixed point @xcite . this is attributed to the fact that a scalar field with negative mass squared is pathological and the entropic @xmath1-function knows about that @xcite . also another is that in our case the geometry is not lorentz - invariant anywhere except near the boundary and so our standard intuition about @xmath1-function may need some modification . before we conclude we would like to mention an important point . in einstein gravity one can not really distinguish between the @xmath81 and @xmath1 central charges . in order to do that one has to include higher - derivative terms in the bulk gravity action . in the presence of higher derivative terms instead of bekenstein - hawking entropy we have to use the entropy expression which satisfies the second law in the bulk and reduces to the wald entropy when evaluated on a killing horizon @xcite . if we do this we will recover the @xmath81-charge at the asymptotic uv boundary as was shown in @xcite . that means the @xmath81-function will start decreasing from @xmath2 . the important point is the fate of this @xmath81-charge in the deep ir i.e , when the causal horizon reaches the singularity . we expect it to go to zero because the thermal state has a finite correlation length even in the presence of the higher - derivative terms , but proving this in general seems to be a complicated thing . empty black brane in ads is dual to a thermal state of the boundary conformal field theory ( cft ) @xcite . this is not a relevant deformation of the cft hamiltonian and there is no renormalization group ( rg ) flow in the ordinary sense . so it is unlikely that the holographic @xmath1-function is an off - shell central charge . to make further progress , it will be useful to take note of the fact that a thermal state is effectively massive with a gap set by the temperature . there is a finite correlation length of the order of inverse temperature . the ir behavior of the holographic @xmath1-function that we have constructed shows the presence of this effective mass gap . it is monotonically decreasing from the central charge of the uv - cft , @xmath2 , to zero at the curvature singularity which is in the deep ir and space - time ends there . _ therefore the causal - horizon @xmath1-function faithfully quantifies the amount of pure quantum correlation or the effective number of `` quantum degrees of freedom '' that exists at different scales in the thermal state . _ can this be related to renormalized entanglement entropy in the boundary theory ? first of all space - like slices of the causal horizon are not in general extremal surfaces in the bulk @xcite . in the field theory side , suppose we consider a ball in @xmath36 of radius @xmath82 . this is our subsystem for which we want to compute the renormalized entanglement entropy @xcite when the field theory is in the thermal state . since the the theory is scale invariant the renormalized entanglement entropy will have the functional form @xmath83 , where @xmath84 is the temperature . it is known that as @xmath85 , @xmath86 @xcite . this matches with the behavior of our @xmath1-function in the same limit . in the opposite limit of @xmath87 on the other hand the renormalized entanglement entropy @xmath88 is nonzero and dominated by thermal entropy of the system @xcite . this does not match with the behavior of the @xmath1-function . this is not surprising because entanglement entropy is not an entanglement measure in a mixed state . in the high temperature limit it is contaminated by classical correlations and fails to capture the quantum part , which should go to zero . on the contrary the behavior of the causal horizon @xmath1-function shows that it is sensitive only to quantum correlations . is there a candidate for such a quantity in the field theory ? as we have discussed entanglement entropy at finite temperature is not a candidate for this generalized @xmath1-function because it is not an entanglement measure in a mixed state . one such measure which can be calculated in field theory is entanglement negativity @xcite . entanglement negativity was studied from a holographic point of view in @xcite , but to the best of our knowledge a geometric prescription of computing this in gravity does not exist so far . entanglement negativity at finite temperature in a two dimensional cft was computed in @xcite . they calculated this for a single interval of length @xmath71 when the total system lives on an infinite line and the temperature is @xmath89 . in this case the answer is given by , [ n ] e = - + f(e^- ) + 2c _ where @xmath81 is the short distance cutoff , @xmath1 is the central charge of the cft and @xmath90 is a constant . @xmath91 is a universal scaling function which depends on the full operator content of the cft such that @xmath92 and @xmath93constant . given this we can calculate its value in the uv and the ir . uv is the region where @xmath94 and we get , e_uv = + 2c _ which is the correct zero temperature result . similarly in the ir , @xmath95 and we get , e_ir = + f(0 ) + 2c _ so in the ir this becomes a non - universal constant independent of the length @xmath71 of the subsystem @xcite . the second term in ( @xmath96 ) is very important in the high temperature limit because it cancels the contribution to the negativity which is extensive in @xmath71 . this is the principal difference from entanglement entropy which is useful for us . now if we define a renormalized negativity , @xmath97 , just like renormalized entanglement entropy @xcite , as , e_r = l | _ e then we get , @xmath98 @xmath97 is a uv - finite quantity . therefore we can see that the renormalized entanglement negativity at least satisfies the asymptotic conditions , i.e , in the uv it is given by the central charge of the theory and in the ir this is zero . the reason that it is going to zero in the ir or in the high temperature limit is that it is an entanglement measure and at very high temperature quantum entanglement goes to zero because the system should crossover to a classical one @xcite . this is a non - trivial constraint . anything that is sensitive to classical correlations may fail to satisfy the ir - condition . therefore the question is does it satisfy the monotonicity condition , i.e , t|_l e_r 0 ? if this condition is satisfied then it is a generalized @xmath1-function . in four dimensions we expect the same thing to happen in the uv . we have to compute the logarithmic negativity for a ball of radius @xmath82 when the field theory is in a thermal state with temperature @xmath84 . the structure of the uv divergences of the negativity is the same as that of entanglement entropy in the same dimension @xcite . so if we apply the liu - mezei operator then we will get a uv finite quantity . the main question is what happens in the ir . does the renormalized negativity go to zero ? this will be the case if negativity becomes independent of the size of the ball in the high temperature limit . this is a reasonable thing to expect given that there is a finite correlation length of order @xmath99 . so we expect the same thing to happen but we can not prove this right now . it will be fascinating to prove the monotonicity of the negativity at least in two dimensions . in the large @xmath1 limit we expect some simplifications @xcite . in fact negativity in the large @xmath1 limit was considered in @xcite . their calculation was for the vacuum sector of the cft . it will be fascinating to extend the calculation to the thermal state using technology of @xcite . before we end this section we would like to emphasize that we are not saying that the causal horizon entropy density is computing some entanglement measure in a thermal state . that may turn out to be the case but our calculation does not show that . what we can infer from this is the existence of such a monotonic function in field theory which is most likely an entanglement measure . in two dimensional cft we have shown a potential candidate for this . causal horizon entropy density represents that quantity in the bulk but perhaps in a different choice of scheme . so numerically they can be different but they will have the same physical content just like in more conventional @xmath1-theorem . there is a different aspect to this problem . our results can be thought of as a realization of the paradigm that space - time is built out of entanglement @xcite , but in a different setting . in the ir there is no quantum correlation or entanglement because of the effective mass gap in the thermal state . in the bulk our holographic @xmath1-function is monotonically decreasing and nonzero everywhere except at the curvature singularity . the curvature singularity is the end of space - time and represents the extreme ir of the dual field theory . therefore the behavior of our @xmath1-function correlates the two facts : loss of quantum correlation / entanglement in the ir field theory and the end of geometry which in this case is the formation of curvature singularity behind the horizon . in fact this is one of our main motivations for interpreting the @xmath1-function as an effective bulk measure of quantum correlation or quantum entanglement between the field theory degrees of freedom at different scales . there is another thing which we would like to point out is that since the causal horizon goes behind the black brane horizon and reaches the singularity , the holographic @xmath1-function is affected by things behind the horizon . therefore the corresponding boundary @xmath1-function knows something about physics behind the horizon . if it turns out that the entanglement negativity indeed satisfies the monotonicity condition then this function will have some information about the interior . at infinite temperature when the negativity is zero we are on the singularity because there is no quantum entanglement . as we lower the temperature we are moving away from the singularity but space - time is still very curved because there is only a very small amount of entanglement . so high temperature expansion is an expansion around the singularity . this is a difficult expansion because negativity depends on the full operator content of the theory , bur this may be a virtue of the function for many purposes . in @xcite behind the horizon physics was explored using the analytically continued correlation functions in the cft . the entanglement negativity ( or any candidate thermal @xmath1-function ) does not seem to have any simple expression in terms of thermal correlators . it is a highly non - local object . it will be interesting to see if there are more fine - grained characterisations of rg - flow which can tell us about the physics behind the horizon . let us now go back to the issue of irreversibility associated to a particular geometry . in a black hole geometry there is a natural notion of irreversibility , which is crossing the horizon or falling into the the black hole . anything that goes into the black hole does not come out . nothing comes out of the black hole singularity . how is that irreversibility encoded in the field theory this is a very difficult question and so we will only try to make a guess . first of all , our @xmath1-function does not show any particular sharp feature which can be used to predict the existence of horizon . so a natural guess will be that this is a quantity which is associated with an infalling observer . in gr an infalling observer does not see anything special happening while crossing the horizon . so let us make the assumption that the rg - flow or coarse - graining of the thermal state of the cft describes an infalling observer . we can not make this statement more precise right now . this assumption together with the fact that this coarse - graining is an irreversible process due to the existence of the @xmath1-theorem seem to imply that that the observer can never come out of the black hole . the coarse graining starts in the uv when the observer is near the ads boundary . as we lower the energy scale the observer moves deeper into the bulk . in the extreme ir when the @xmath1-function hits zero the observer hits the singularity . this is consistent with the fact that our holographic @xmath1-function reaches zero at the curvature singularity . things can not come out of the black hole singularity because in the field theory there is no unitary rg - flow which starts at @xmath100 and go to @xmath101 . this is forbidden by @xmath1-theorem .- theorem to the thermal state . we have proved such a theorem only in the bulk . ] no unitary rg - flow can start at @xmath100 because along the rg - flow @xmath1 has to decrease . so in rg - time there is an ordering in which the @xmath100 theory always lives in the future . this is also the ordering of time for the infalling observer for whom the black hole singularity is always in the future . this is not quantitative and many things need to be checked before one can say anything conclusive , but at least it is clear that the existence of the @xmath1-theorem imposes an ordering among different scales in the field theory which , it looks like , can be translated to the bulk under certain assumption and does not immediately produce a contradiction . there is another reason to suspect that this may be a correct interpretation . this is related to the tensor network representation of the thermofield double of a scale invariant theory after time evolution . this representation was proposed by hartman and maldacena @xcite . in this picture the tensor network has a scale - invariant uv region and a gapped ir region . the gapped region arises due to the effective mass gap of the thermal state and this represents the interior of the black brane . this resonates well with the behavior of our holographic @xmath1-function because it shows the extreme thinning of the `` effective number of degrees of freedom '' near the curvature singularity . a better understanding of this will probably require a more covariant formulation of tensor network ideas . overall , it seems that mera @xcite might be a proper framework to think about such generalized holographic @xmath1-functions . the function we have constructed measures the quantum correlation that exists at different scales in the thermal density matrix . mera does a coarse - 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in this paper we revisit the question that in what sense empty @xmath0 black brane geometry can be thought of as rg - flow . we do this by first constructing a holographic @xmath1-function using causal horizon in the black brane geometry . the uv value of the @xmath1-function is @xmath2 and then it decreases monotonically to zero at the curvature singularity . intuitively , the behavior of the @xmath1-function can be understood if we recognize that the dual cft is in a thermal state and thermal states are effectively massive with a gap set by the temperature . in field theory , logarithmic entanglement negativity is an entanglement measure for mixed states . for example , in two dimensional cfts at finite temperature the renormalized entanglement negativity of an interval has uv ( low- t ) value @xmath3 and ir ( high - t ) value zero . so this is a potential candidate for our @xmath1-function . in four dimensions we expect the same thing to hold on physical grounds . now since the causal horizon goes behind the black brane horizon the holographic @xmath1-function is sensitive to the physics of the interior . correspondingly the field theory @xmath1-function should also contain information about the interior . so our results suggest that high temperature ( ir ) expansion of the negativity ( or any candidate @xmath1-function ) may be a way to probe part of the physics near the singularity . negativity at finite temperature depends on the full operator content of the theory and so perhaps this can be be done in specific cases only . the existence of this @xmath1-function in the bulk is an extreme example of the paradigm that space - time is built out of entanglement . in particular the fact that the @xmath1-function reaches zero at the curvature singularity correlates the two facts : loss of quantum entanglement in the ir field theory and the end of geometry in the bulk which in this case is the formation of curvature singularity .

makespan minimization is a fundamental and extensively studied problem in scheduling theory . consider a sequence of jobs @xmath0 that has to be scheduled on @xmath1 identical parallel machines . each job @xmath11 is specified by a processing time @xmath12 , @xmath13 . preemption of jobs is not allowed . the goal is to minimize the makespan , i.e. the maximum completion time of any job in the constructed schedule . we focus on the online version of the problem where the jobs of @xmath2 arrive one by one . each incoming job @xmath11 has to be assigned immediately to one of the machines without knowledge of any future jobs @xmath14 , @xmath15 . online algorithms for makespan minimization have been studied since the 1960s . in an early paper graham @xcite showed that the famous _ list _ scheduling algorithm is @xmath16-competitive . the best online strategy currently known achieves a competitiveness of about 1.92 . makespan minimization has also been studied with various types of _ resource augmentation _ , giving an online algorithm additional information or power while processing @xmath2 . the following scenarios were considered . ( 1 ) an online algorithm knows the optimum makespan or the sum of the processing times of @xmath2 . ( 2 ) an online strategy has a buffer that can be used to reorder @xmath2 . whenever a job arrives , it is inserted into the buffer ; then one job of the buffer is removed and placed in the current schedule . ( 3 ) an online algorithm may migrate a certain number or volume of jobs . in this paper we investigate makespan minimization assuming that an online algorithm is allowed to build several schedules in parallel while processing a job sequence @xmath2 . each incoming job is sequenced in each of the schedules . at the end of the scheduling process the best schedule is selected . we believe that this is a natural form of resource augmentation : in classical online makespan minimization , studied in the literature so far , an algorithm constructs a schedule while jobs arrive one by one . once all jobs have arrived , the schedule may be executed . hence in this standard framework there is a priori no reason why an algorithm should not be able to construct several solutions , the best of which is finally chosen . our new proposed setting can be viewed as providing an online algorithm with extra space , which is used to maintain several solutions . very little is known about the value of extra space in the design of online algorithms . makespan minimization with parallel schedules is of particular interest in parallel processing environments where each processor can take care of a single or a small set of schedules . we develop algorithms that require hardly any coordination or communication among the schedules . last not least the proposed setting is interesting w.r.t . to the foundations of scheduling theory , giving insight into the value of multiple candidate solutions . makespan minimization with parallel schedules was also addressed by kellerer et al . @xcite . however , the paper focused on the restricted setting with @xmath17 machines . in this paper we explore the problem for a general number @xmath1 of machines . as a main result we show that a constant number of schedules suffices to achieve a significantly improved competitiveness , compared to the standard setting without resource augmentation . the competitive ratios obtained are at least as good and in most cases better than those attained in the other models of resource augmentation mentioned above . the approach to grant an online algorithm extra space , invested to maintain multiple solutions , could be interesting in other problems as well . the approach is viable in applications where an online algorithm constructs a solution that is used when the entire input has arrived . this is the case , for instance , in basic online graph coloring and matching problems @xcite . the approach is also promising in problems that can be solved by a set of independent agents , each of which constructs a separate solution . good examples are online navigation and exploration problems in robotics @xcite . some results are known for graph search and exploration , see e.g. @xcite , but the approach has not been studied for geometric environments . * problem definition : * we investigate the problem _ makespan minimization with parallel schedules ( mps)_. as always , the jobs of a sequence @xmath0 arrive one by one and must be scheduled non - preemptively on @xmath1 identical parallel machines . each job @xmath11 has a processing time @xmath18 . in mps , an online algorithm @xmath19 may maintain a set @xmath20 of schedules during the scheduling process while jobs of @xmath2 arrive . each job @xmath11 is sequenced in each schedule @xmath21 , @xmath22 . at the end of @xmath2 , algorithm @xmath23 selects a schedule @xmath24 having the smallest makespan and outputs this solution . the other schedules of @xmath25 are deleted . as we shall show mps can be reduced to the problem variant where the optimum makespan of the job sequence to the processed is known in advance . hence let mps@xmath26 denote the variant of mps where , prior to the arrival of the first job , an algorithm @xmath23 is given the value of the optimum makespan @xmath27 for the incoming job sequence @xmath2 . an algorithm @xmath19 for mps or mps@xmath26 is @xmath28-competitive if , for every job sequence @xmath2 , it outputs a schedule whose makespan is at most @xmath28 times @xmath27 . we present a comprehensive study of mps . we develop a @xmath3-competitive algorithm , for any @xmath4 , using a constant number of @xmath5 schedules . furthermore , we give a @xmath6-competitive algorithm , for any @xmath4 , that uses a polynomial number of schedules . the number is @xmath7 , which depends on @xmath1 but is independent of the job sequence @xmath2 . these performance guarantees are nearly best possible . the algorithms are obtained via some intermediate results that may be of independent interest . first , in section [ sec : redu ] we show that the original problem mps can be reduced to the variant mps@xmath26 in which the optimum makespan is known . more specifically , given any @xmath28-competitive algorithm @xmath19 for mps@xmath26 we construct a @xmath29-competitive algorithm @xmath30 , for any @xmath31 . if @xmath19 uses @xmath32 schedules , then @xmath30 uses @xmath33 schedules . the construction works for any algorithm @xmath19 for mps@xmath26 . in particular we could use a 1.6-competitive algorithm by chen et al . @xcite that assumes that the optimum makespan is known and builds a single schedule . we would obtain a @xmath34-competitive algorithm that builds at most @xmath35 schedules . we proceed and develop algorithms for mps@xmath26 . in section [ sec : ptas ] we give a @xmath6-competitive algorithm , for any @xmath4 , that uses @xmath36 schedules . in section [ sec:4/3 ] we devise a @xmath3-competitive algorithm , for any @xmath4 , that uses @xmath5 schedules . combining these algorithms with @xmath30 , we derive the two algorithms for mps mentioned in the above paragraph ; see also section [ sec : mps ] . the number of schedules used by our strategies depends on @xmath37 and exponentially on @xmath38 or @xmath37 . such a dependence seems inherent if we wish to explore the full power of parallel schedules . the trade - offs resemble those exhibited by ptases in offline approximation . recall that the ptas by hochbaum and shmoys @xcite for makespan minimization achieves a @xmath6-approximation with a running time of @xmath39 . in section [ sec : lb ] we present lower bounds . we show that any online algorithm for mps that achieves a competitive ratio smaller than 4/3 must construct more than @xmath40 schedules . hence the competitive ratio of 4/3 is best possible using a constant number of schedules . we show a second lower bound that implies that the number of schedules of our @xmath6-competitive algorithm is nearly optimal , up to a polynomial factor . our algorithms make use of novel guessing schemes . @xmath30 works with guesses on the optimum makespan . guessing and _ doubling _ the value of the optimal solution is a technique that has been applied in other load balancing problems , see e.g. @xcite . however here we design a refined scheme that carefully sets and readjusts guesses so that the resulting competitive ratio increases by a factor of @xmath10 only , for any @xmath41 . moreover , the readjustment and job assignment rules have to ensure that scheduling errors , made when guesses were to small , are not critical . our @xmath3-competitive algorithm works with guesses on the job processing times and their frequencies in @xmath2 . in order to achieve a constant number of schedules , we have to sparsify the set of all possible guesses . as far as we know such an approach has not been used in the literature before . all our algorithms have the property that the parallel schedules are constructed basically independently . the algorithms for mps@xmath26 require no coordination at all among the schedules . in @xmath30 a schedule only has to report when it fails , i.e. when a guess on the optimum makespan is too small . the competitive ratios achieved with parallel schedules are considerably smaller than the best ratios of about 1.92 known for the scenario without resource augmentation . our ratio of @xmath3 , for small @xmath42 , is lower than the competitiveness of about 1.46 obtained in the settings where a reordering buffer of size @xmath43 is available or @xmath43 jobs may be reassigned . skutella et al . @xcite gave an online algorithm that is @xmath6-competitive if , before the assignment of any job @xmath11 , jobs of processing volume @xmath44 may be migrated . hence the total amount of extra resources used while scheduling @xmath2 depends on the input sequence . * related work : * makespan minimization with parallel schedules was first studied by kellerer et al . they assume that @xmath17 machines are available and two schedules may be constructed . they show that in this case the optimal competitive ratio is 4/3 . we summarize results known for online makespan minimization without resource augmentation . as mentioned before , _ list _ is @xmath16-competitive . deterministic online algorithms with a smaller competitive ratio were presented in @xcite . the best algorithm currently known is 1.9201-competitive @xcite . lower bounds on the performance of deterministic strategies were given in @xcite . the best bound currently known is 1.88 , see @xcite . no randomized online algorithm whose competitive ratio is provably below the deterministic lower bound is currently known for general @xmath1 . we next review the results for the various models of resource augmentation . articles @xcite study makespan minimization assuming that an online algorithm knows the optimum makespan or the sum of the processing times of @xmath2 . chen et al . @xcite developed a 1.6-competitive algorithm . azar and regev @xcite showed that no online algorithm can attain a competitive ratio smaller than 4/3 . the setting in which an online algorithm is given a reordering buffer was explored in @xcite . englert et al . @xcite presented an algorithm that , using a buffer of size @xmath43 , achieves a competitive ratio of @xmath45 , where @xmath46 is the lambert @xmath47 function . no algorithm using a buffer of size @xmath48 can beat this ratio . makespan minimization with job migration was addressed in @xcite . an algorithm that achieves again a competitiveness of @xmath45 and uses @xmath43 job reassignments was devised in @xcite . no algorithm using @xmath48 reassignments can obtain a smaller competitiveness . sanders et al . @xcite study a scenario in which before the assignment of each job @xmath11 , jobs up to a total processing volume of @xmath49 may be migrated , for some constant @xmath50 . for @xmath51 , they present a 1.5-competitive algorithm . they also show a @xmath6-competitive algorithm , for any @xmath41 , where @xmath52 . as for memory in online algorithms , sleator and tarjan @xcite studied the paging problem assuming that an online algorithm has a larger fast memory than an offline strategy . raghavan and snir @xcite traded memory for randomness in online caching . * notation : * throughout this paper it will be convenient to associate schedules with algorithms , i.e. a schedule @xmath21 is maintained by an algorithm @xmath53 that specifies how to assign jobs to machines in @xmath21 . thus an algorithm @xmath19 for mps or mps@xmath26 can be viewed as a family @xmath54 of algorithms that maintain the various schedules . we will write @xmath55 . if @xmath19 is an algorithm for mps@xmath26 , then the value @xmath27 is of course given to all algorithms of @xmath54 . furthermore , the _ load _ of a machine always denotes the sum of the processing times of the jobs already assigned to that machine . in this section we will show that any @xmath28-competitive algorithm @xmath19 for mps@xmath26 can be used to construct a @xmath29-competitive algorithm @xmath56 for mps , for any @xmath4 . the main idea is to repeatedly execute @xmath19 for a set of guesses on the optimum makespan . the initial guesses are small and are increased whenever a guess turns out to be smaller than @xmath27 . the increments are done in small steps so that , among the final guesses , there exists one that is upper bounded by approximately @xmath57 . in the analysis of this scheme we will have to bound machine loads caused by scheduling `` errors '' made when guesses were too small . unfortunately the execution of @xmath19 , given a guess @xmath58 , can lead to undefined algorithmic behavior . as we shall show , guesses @xmath59 are not critical . however , guesses @xmath60 have to be handled carefully . so let @xmath55 be a @xmath28-competitive algorithm for mps@xmath26 that , given guess @xmath61 , is executed on a job sequence @xmath2 . upon the arrival of a job @xmath11 , an algorithm @xmath62 may _ fail _ because the scheduling rules of @xmath53 do not specify a machine where to place @xmath11 in the current schedule @xmath21 . we define two further conditions when an algorithm @xmath53 fails . the first one identifies situations where a makespan of @xmath63 is not preserved and hence @xmath28-competitiveness may not be guaranteed . more precisely , @xmath53 would assign @xmath11 to a machine @xmath64 such that @xmath65 , where @xmath66 denotes @xmath64 s machine load before the assignment . the second condition identifies situations where @xmath61 is not consistent with lower bounds on the optimum makespan , i.e. @xmath61 is smaller than the average machine load or the processing time of @xmath11 . formally , an algorithm @xmath53 _ fails _ if a job @xmath11 , @xmath13 , has to be scheduled and one of the following conditions holds . a. @xmath53 does not specify a machine where to place @xmath11 in the current schedule @xmath21 . b. there holds @xmath67 , for the machine @xmath64 to which @xmath53 would assign @xmath11 in @xmath21 . c. there holds @xmath68 or @xmath69 . we first show that guesses @xmath70 are not problematic . if a @xmath28-competitive algorithm @xmath55 for mps@xmath26 is given a guess @xmath59 , then there exists an algorithm @xmath62 that does not fail during the processing of @xmath2 and generates a schedule whose makespan is at most @xmath63 . this is shown by the next lemma . [ lem : guess1 ] let @xmath55 be a @xmath28-competitive algorithm for mps@xmath26 that , given guess @xmath61 , is executed on a job sequence @xmath2 with @xmath59 . then there exists an algorithm @xmath62 that does not fail during the processing of @xmath2 and generates a schedule whose makespan is at most @xmath63 . let @xmath71 be an optimal schedule for the job sequence @xmath0 . moreover , let @xmath66 denote the load of machine @xmath64 in @xmath71 , @xmath72 . for any @xmath73 with @xmath74 , define a job @xmath75 of processing time @xmath76 . let @xmath77 be the job sequence consisting of @xmath2 followed by the new jobs @xmath75 . these up to @xmath1 jobs may be appended to @xmath2 in any order . obviously @xmath78 . hence when @xmath23 using guess @xmath61 is executed on @xmath77 , there must exist an algorithm @xmath79 that generates a schedule with a makespan of at most @xmath63 . since @xmath2 is a prefix of @xmath77 , this algorithm @xmath80 does not fail and generates a schedule with a makespan of at most @xmath63 , when @xmath19 given guess @xmath61 is executed on @xmath2 . * algorithm for mps : * we describe our algorithm @xmath81 for mps , where @xmath4 and @xmath82 may be chosen arbitrarily . the construction takes as input any algorithm @xmath55 for mps@xmath26 . for a proper choice of @xmath83 , @xmath81 will be @xmath29-competitive , provided that @xmath19 is @xmath28-competitive . at any time @xmath81 works with @xmath83 guesses @xmath84 on the optimum makespan for the incoming job sequence @xmath2 . these guesses may be adjusted during the processing of @xmath2 ; the update procedure will be described in detail below . for each guess @xmath85 , @xmath86 , @xmath81 executes @xmath19 . hence @xmath81 maintains a total of @xmath87 schedules , which can be partitioned into subsets @xmath88 . subset @xmath89 contains those schedules generated by @xmath19 using @xmath85 , @xmath86 . let @xmath90 denote the schedule generated by @xmath53 using @xmath85 . a job sequence @xmath2 is processed as follows . initially , upon the arrival of the first job @xmath91 , the guesses are initialized as @xmath92 and @xmath93 , for @xmath94 . each job @xmath11 , @xmath13 , is handled in the following way . of course each such job is sequenced in every schedule @xmath95 , @xmath86 and @xmath96 . algorithm @xmath81 checks if @xmath53 using @xmath85 fails when having to sequence @xmath11 in @xmath95 . we remark that this check can be performed easily by just verifying if one of the conditions ( i iii ) holds . if @xmath53 using @xmath85 does not fail and has not failed since the last adjustment of @xmath85 , then in @xmath95 job @xmath11 is assigned to the machine specified by @xmath53 using @xmath85 . the initialization of a guess is also regarded as an adjustment . if @xmath53 using @xmath85 does fail , then @xmath11 and all future jobs are always assigned to a least loaded machine in @xmath95 until @xmath85 is adjusted the next time . suppose that after the sequencing of @xmath11 all algorithms of @xmath55 using a particular guess @xmath85 have failed since the last adjustment of this guess . let @xmath97 be the largest index @xmath98 with this property . then the guesses @xmath99 are adjusted . set @xmath100 and @xmath93 , for @xmath101 . for any readjusted guess @xmath85 , @xmath102 , algorithm @xmath19 using @xmath85 ignores all jobs @xmath14 with @xmath103 when processing future jobs of @xmath2 . specifically , when making scheduling decisions and determining machine loads , algorithm @xmath53 using @xmath85 ignores all job @xmath14 with @xmath103 in its schedule @xmath95 . these jobs are also ignored when @xmath81 checks if @xmath53 using guess @xmath85 fails on the arrival of a job . furthermore , after the assignment of @xmath11 , machines in @xmath95 machines are renumbered so that @xmath11 is located on a machine it would occupy if it were the first job of an input sequence . when guesses have been adjusted , they are renumbered , together with the corresponding schedule sets @xmath89 , such that again @xmath84 . hence at any time @xmath104 and @xmath105 , for @xmath94 . we also observe that whenever a guess is adjusted , its value increases by a factor of at least @xmath106 . a summary of @xmath81 is given in figure [ fig:1 ] . we obtain the following theorem . [ th : guess1 ] let @xmath107 be a @xmath28-competitive algorithm for mps@xmath26 . then for any @xmath4 and @xmath108 , algorithm @xmath109 for mps is @xmath29-competitive and uses @xmath110 schedules . for the analysis of @xmath81 we need the following lemma . [ lem : guess2 ] after @xmath81 has processed a job sequence @xmath2 , there holds @xmath111 . at any time @xmath81 maintains @xmath83 guesses . we can view these guesses as being stored in @xmath83 variables . a variable is updated whenever its current guess is increased . hence during the processing of @xmath2 a variable may take any position in the sorted sequence of guesses . we analyze the steps in which @xmath81 adjusts guesses . we first show that when @xmath81 adjusts a guess @xmath61 , then @xmath60 . so suppose that after the arrival of a job @xmath11 , @xmath81 adjust guesses @xmath112 , where @xmath97 is the largest index @xmath98 such that all algorithms @xmath54 using @xmath85 have failed . we prove @xmath113 , which implies the desired statement because guesses are numbered in order of increasing value . let @xmath114 , with @xmath115 , be the most recent time when the variable storing @xmath116 was updated last . if the variable has never been updated since its initialization , then let @xmath117 . all the algorithms @xmath54 using @xmath116 ignore the jobs having arrived before @xmath118 when making scheduling decisions for @xmath119 . let @xmath120 . there holds , @xmath121 . if @xmath122 held true , then by lemma [ lem : guess1 ] there would be an algorithm @xmath123 that , using guess @xmath116 , does not fail when handling @xmath124 . this contradicts the fact that at time @xmath125 all algorithms @xmath54 using @xmath116 fail or have failed since the arrival of @xmath118 . let @xmath126 denote the value of the smallest guess when @xmath81 has finished processing @xmath2 . we distinguish two cases depending on whether or not the variable storing @xmath126 has ever been updated since its initialization . if the variable has never been updated , then @xmath127 , for some @xmath128 . if @xmath129 , there is nothing to show because @xmath130 . if @xmath131 , then the initial guess of value @xmath132 must have been adjusted this implies , as shown above , @xmath133 and the lemma follows because @xmath134 . in the remainder of the proof we assume that the variable @xmath135 storing @xmath126 has been updated . consider the last update of @xmath135 before the end of @xmath2 and suppose that it took place on the arrival of job @xmath118 . first assume that @xmath135 stores the smallest guess , among the @xmath83 guesses , before the update . then @xmath136 , where @xmath137 is the largest guess before the update . if @xmath137 is also adjusted on the arrival of @xmath118 , then we are done because , as shown above , @xmath138 and thus @xmath139 . if @xmath137 is not adjusted on the arrival of @xmath118 , then @xmath126 is the smallest guess greater than @xmath137 after the update . by the end of @xmath2 guess @xmath137 must be adjusted since otherwise @xmath126 can not become the smallest guess . again @xmath138 and we are done . finally assume that before the update @xmath135 does not store the smallest guess . let @xmath140 be the variable that stores the largest guess smaller than that in @xmath135 . after the update there holds @xmath141 , where @xmath61 is the guess stored in @xmath140 after the update . until the end of @xmath2 , @xmath61 must be adjusted again since otherwise @xmath126 can not become the smallest guess . again @xmath142 and hence @xmath143 . throughout the proof let @xmath108 and @xmath109 . consider an arbitrary job sequence and let @xmath144 be the smallest of the @xmath83 guesses maintained by @xmath56 at the end of @xmath2 . let @xmath145 be the set of schedules associated with @xmath144 , i.e. @xmath145 was generated by @xmath107 using a series of guesses ending with @xmath144 . let @xmath146 , with @xmath147 , be this series and @xmath135 be the variable that stored these guesses . here @xmath148 is one of the initial guesses and @xmath149 . a first observation is that at the end of @xmath2 there exists an algorithm @xmath123 that using @xmath144 has not failed . this holds true if @xmath135 was set to @xmath150 upon the arrival of a job @xmath11 with @xmath151 because the failure of all algorithms @xmath54 using @xmath144 would have caused an adjustment of @xmath144 . this also holds true if @xmath135 was set to @xmath144 upon the arrival of @xmath152 because in this case none of the algorithms @xmath54 using @xmath144 has failed at the end of @xmath2 . so let @xmath123 be an algorithm that using @xmath144 has not failed and let @xmath153 be the associated schedule . we prove that the load of every machine in @xmath153 is upper bounded by @xmath154 . this establishes the theorem . let @xmath155 . if the variable @xmath135 was updated during the processing of @xmath2 , then let @xmath156 be these points in time , i.e. the arrival of @xmath157 caused an update of @xmath135 and the variable was set to @xmath158 , @xmath159 . for any machine @xmath64 , @xmath160 , in @xmath153 let @xmath66 denote its final load at the end of @xmath2 . moreover , let @xmath161 denote its load due to jobs @xmath11 with @xmath162 , for @xmath163 . obviously @xmath164 we first show that @xmath165 . immediately after @xmath166 has been scheduled @xmath64 s load consisting of jobs @xmath14 with @xmath167 is at most @xmath168 . since @xmath135 was set to @xmath169 on the arrival of @xmath166 , the guess adjustment rule ensures @xmath170 . until the end of @xmath2 algorithm @xmath80 using @xmath144 does not fail and hence condition ( ii ) specifying the failure of algorithms implies that the assignment of each further job does not create a machine load greater than @xmath171 in @xmath153 . we next show @xmath172 , for each @xmath173 . the latter difference is the load on machine @xmath64 caused by jobs of the subsequence @xmath174 . hence it suffices to show that after the assignment of any @xmath11 , with @xmath175 , @xmath64 s load due to jobs @xmath14 , with @xmath176 , is at most @xmath177 . after the assignment of @xmath157 @xmath64 s respective load @xmath161 is at most @xmath178 and this value is upper bounded by @xmath158 as ensured by the guess adjustment rule . at times @xmath179 , while @xmath80 using @xmath158 has not failed , @xmath64 s load due to jobs @xmath14 with @xmath176 does not exceed @xmath180 as ensured by condition ( ii ) specifying the failure of algorithms . finally consider a time @xmath125 , @xmath181 , at which @xmath80 fails or has failed . the incoming job @xmath11 is assigned to a least loaded machine . hence if @xmath11 is placed on @xmath64 , then the resulting machine load due to jobs @xmath14 with @xmath176 is upper bounded by @xmath182 . observe that after the arrival of @xmath11 there exists an algorithm @xmath62 that using @xmath158 has not yet failed , since otherwise @xmath158 would be adjusted before time @xmath183 . condition ( iii ) defining the failure of algorithms ensures that @xmath184 and @xmath185 . we obtain that @xmath64 s machine load is at most @xmath186 . we conclude that ( [ eq : b1 ] ) is upper bounded by @xmath187 by lemma [ lem : guess2 ] , @xmath188 . at the end of the description of @xmath81 we observed that whenever a guess is adjusted it increases by a factor of at least @xmath106 . hence @xmath189 . it follows that @xmath190 , for every @xmath191 . hence ( [ eq : b2 ] ) is upper bounded by @xmath192 here ( [ eq : xb2 ] ) uses the fact that @xmath193 and , as mentioned above , is a consequence of lemma [ lem : guess2 ] . line ( [ eq : b3 ] ) follows from the geometric series and , finally , ( [ eq : b4 ] ) is by the choice of @xmath83 and the assumption @xmath31 . we present an algorithm @xmath195 for mps@xmath26 that attains a competitive ratio of @xmath10 , for any @xmath41 . the number of parallel schedules will be @xmath36 . the algorithms will yield a @xmath6-competitive strategy for @xmath196 and , furthermore , will be useful in the next section where we develop a @xmath3-competitive algorithm for mps@xmath26 . there @xmath195 will be used as subroutine for a small , constant number of @xmath1 . * description of @xmath195 : * let @xmath41 be arbitrary . recall that in mps@xmath26 the optimum makespan @xmath27 for the incoming job sequence @xmath2 is initially known . assume without loss of generality that @xmath197 . then all job processing times are in @xmath198 $ ] . set @xmath199 . first we partition the range of possible job processing times into intervals @xmath200 such , within each interval @xmath201 with @xmath202 , the values differ by a factor of at most @xmath203 . such a partitioning is standard and has been used e.g. in the ptas for offline makespan minimization @xcite . let @xmath204 . set @xmath205 $ ] and @xmath206 $ ] , for @xmath207 . obviously @xmath208 $ ] and @xmath198 \subseteq ( 0 , ( 1+{\varepsilon}')^l{\varepsilon}']$ ] . a job is _ small _ if its processing time is at most @xmath209 and hence contained in @xmath210 ; otherwise the job is _ large_. each job sequence @xmath2 with @xmath211 contains at most @xmath212 large jobs . for each possible distribution of large jobs over the processing time intervals @xmath213 , algorithm @xmath195 prepares one algorithm / schedule . let @xmath214 . there holds @xmath215 . let @xmath216 . for any vector @xmath217 , algorithm @xmath218 works as follows . it assumes that the incoming job sequence @xmath2 contains exactly @xmath219 jobs with a processing time in @xmath201 , for @xmath207 . moreover , it pessimistically assumes that each processing time in @xmath201 takes the largest possible value @xmath220 . hence , initially @xmath218 computes an optimal schedule @xmath221 for a job sequence consisting of @xmath219 jobs with a processing time of @xmath220 , for @xmath207 . small jobs are ignored . since running time is not an issue in the design of online algorithms , such a schedule @xmath221 can be computed exactly . alternatively , an @xmath222-approximation to the optimal schedule can be computed using the ptas by hochbaum and shmoys @xcite . let @xmath223 denote the number of jobs with a processing time of @xmath224 assigned to machine @xmath64 in @xmath221 , where @xmath225 and @xmath160 . moreover , let @xmath226 be the load on machine @xmath64 in @xmath221 , @xmath72 . when processing the actual job sequence @xmath2 and constructing a real schedule @xmath227 , @xmath218 uses @xmath221 as a guideline to make scheduling decisions . at any time during the scheduling process , let @xmath228 be the number of jobs with a processing time in @xmath201 that have already been assigned to machine @xmath64 in @xmath227 , where again @xmath225 and @xmath160 . each incoming job @xmath11 , @xmath229 , is handled as follows . if @xmath11 is large , then let @xmath201 with @xmath225 be the interval such that @xmath230 . algorithm @xmath218 checks if there is a machine @xmath64 such that @xmath231 , i.e. there is a machine that can still accept a job with a processing time in @xmath201 as suggested by the optimal schedule @xmath221 . if such a machine @xmath64 exists , then @xmath11 is assigned to it ; otherwise @xmath11 is scheduled on an arbitrary machine . if @xmath11 is small , then @xmath11 is assigned to a machine @xmath64 with the smallest current value @xmath232 . here @xmath233 denotes the current load on machine @xmath64 caused by small jobs in @xmath227 . a summary of @xmath195 is given in figure [ fig:2 ] . subsequently we show theorem [ th : guess2 ] . [ th : guess2 ] for any @xmath234 , @xmath195 is @xmath6-competitive and uses at most @xmath36 schedules . the bound on the number of schedules simply follows from the fact that @xmath195 maintains @xmath235 schedules where @xmath236 and @xmath204 . let @xmath2 be an arbitrary job sequence and let @xmath219 be the number of jobs with a processing time in @xmath201 , for @xmath207 . since any @xmath219 is upper bounded by @xmath237 , the resulting vector @xmath238 is in @xmath239 . for this vector @xmath240 , consider the associated algorithm @xmath218 . we prove that when @xmath218 has finished processing @xmath2 , the resulting schedule @xmath227 has a makespan of at most @xmath241 . recall again that we assume without loss of generality that @xmath197 . we analyze the steps in which @xmath218 assigns jobs @xmath11 , @xmath229 , to machines in @xmath227 . if @xmath11 is large with @xmath230 , @xmath242 , then there must exist a machine @xmath64 in the current schedule @xmath227 such that @xmath243 . algorithm @xmath218 will assign @xmath11 to such a machine . hence after the processing of @xmath2 , for any @xmath64 in @xmath227 , the total load caused by large jobs is upper bounded by @xmath244 . we next argue that this value is at most @xmath245 . consider an optimal schedule @xmath71 for @xmath2 . modify this schedule by ( a ) deleting all small jobs and ( b ) rounding each job processing time in @xmath201 to @xmath220 , for @xmath246 . the resulting schedule schedule @xmath247 has a makespan of at most @xmath245 . furthermore @xmath247 is a schedule for an input sequence consisting of @xmath219 jobs of processing time @xmath220 . since @xmath248 is an optimal schedule for this input , each machine load @xmath244 is upper bounded by @xmath245 . we finally show that when @xmath218 has to sequence a small job @xmath11 , then there is a machine @xmath64 such that @xmath232 is upper bounded by @xmath245 . this implies that the assignment of @xmath11 causes a machine load of at most @xmath249 in the final schedule @xmath227 . so suppose that upon the arrival of a small job @xmath11 there holds @xmath250 for all machines @xmath64 , @xmath160 . recall that @xmath233 is the load on machine @xmath64 caused by small jobs in the current schedule @xmath227 . note that @xmath251 is the total processing time of large jobs in @xmath2 if processing times in @xmath201 are rounded up to @xmath220 , for @xmath207 . hence @xmath252 is a lower bound on the total processing time of large jobs in @xmath2 . it follows that the total processing time of all jobs in @xmath2 is at least @xmath253 . the assumption that @xmath250 holds for all machines @xmath64 implies that the total processing time of jobs in @xmath2 is at least @xmath254 , which contradicts the fact that @xmath27 is the optimum makespan . we develop an algorithm @xmath256 for mps@xmath26 that is @xmath3-competitive , for any @xmath4 , if the number @xmath1 of machines is not too small . we then combine @xmath256 with @xmath195 , presented in the last section , and derive a strategy @xmath257 that is @xmath3-competitive , for arbitrary @xmath1 . the number of required schedules is @xmath5 , which is a constant independent of @xmath258 and @xmath1 . we firstly present a description of the algorithm ; the corresponding analysis is given thereafter . before describing @xmath256 in detail , we explain the main ideas of the algorithm . one concept is identical to that used by @xmath195 : partition the range of possible job processing times into intervals or _ job classes _ and consider distributions of jobs over these classes . however , in order to achieve a constant number of schedules we have to refine this scheme and incorporate new ideas . first , the job classes have to be chosen properly so as to allow a compact packing of jobs on the machines . an important , new aspect in the construction of @xmath256 is that we will not consider the entire set @xmath239 of tuples specifying how large jobs of an input sequence @xmath2 are distributed over the job classes . instead we will define a suitable sparsification @xmath259 of @xmath239 . each @xmath260 represents an estimate or guess on the number of large jobs arising in @xmath2 . more specifically , if @xmath261 , then it is assumed that @xmath2 contains at least @xmath219 jobs with a processing time of job class @xmath98 . obviously , the job sequence @xmath2 may contain more large jobs , the exact number of which is unknown . furthermore , it is unknown which portion of the total processing time of @xmath2 will arrive as small jobs . in order to cope with these uncertainties @xmath256 has to construct robust schedules . to this end the number of machines is partitioned into two sets @xmath262 and @xmath263 . for the machines of @xmath262 , the algorithm initially determines a good assignment or _ configuration _ assuming that @xmath219 jobs of job class @xmath98 will arrive . the machines of @xmath263 are reserve machines and will be assigned additional large jobs as they arise in @xmath2 . small jobs will always be placed on machines in @xmath262 . the initial configuration determined for these machines has the property that , no matter how many small jobs arrive , a machine load never exceeds @xmath264 times the optimum makespan . we proceed to describe @xmath256 in detail . let @xmath4 . moreover , set @xmath265 . again we assume without loss of generality that , for an incoming job sequence , there holds @xmath197 . hence the processing time of any job is upper bounded by 1 . * job classes : * a job @xmath11 , @xmath13 , is _ small _ if @xmath266 ; otherwise @xmath11 is _ large_. we divide the range of possible job processing times into job classes . let @xmath267 $ ] be the interval containing the processing times of small jobs . let @xmath268 and @xmath269 , where the logarithm is taken to base 2 . for @xmath207 , let @xmath270 it is easy to verify that @xmath271 and @xmath272 , for @xmath207 . furthermore @xmath273 and @xmath274 . for @xmath207 define @xmath275 $ ] . there holds @xmath276 $ ] . moreover , for @xmath277 , let @xmath278 $ ] . intuitively , @xmath279 contains the processing times that are twice as large as those in @xmath201 , @xmath280 . there holds @xmath281 $ ] . hence @xmath282 $ ] . in the following @xmath201 represents _ job class _ @xmath98 , for @xmath283 . we say that @xmath11 is a _ class-@xmath98 job _ if @xmath230 , where @xmath284 . * definition of target configurations : * as mentioned above , for any incoming job sequence @xmath2 , @xmath256 works with estimates on the number of class-@xmath98 jobs arising in @xmath2 , @xmath284 . for each estimate , the algorithm initially determines a virtual schedule or _ target configuration _ on a subset of the machines , assuming that the estimated set of large jobs will indeed arrive . hence we partition the @xmath1 machines into two sets @xmath262 and @xmath263 . let @xmath285 . moreover , let @xmath286 and @xmath287 . set @xmath262 contains the machines for which a target configuration will be computed ; @xmath263 contains the reserve machines . the proportion of @xmath288 to @xmath289 is roughly @xmath290 . a target configuration has the important property that any machine @xmath291 contains large jobs of only one job class @xmath98 , @xmath284 . therefore , a target configuration is properly defined by a vector @xmath292 . if @xmath293 , then @xmath64 does not contain any large jobs in the target configuration , @xmath294 . if @xmath295 , where @xmath296 , then @xmath64 contains class-@xmath98 jobs , @xmath294 . the vector @xmath297 implicitly also specifies how many large jobs reside on a machine . if @xmath295 with @xmath225 , then @xmath64 contains two class-@xmath98 jobs . note that , for general @xmath298 , a third job can not be placed on the machine without exceeding a load bound of @xmath264 . if @xmath295 with @xmath299 , then @xmath64 contains one class-@xmath98 job . again , the assignment of a second job is not feasible in general . given a configuration @xmath297 , @xmath64 is referred to as a _ class-@xmath98 machine _ if @xmath295 , where @xmath300 and @xmath284 . with the above interpretation of target configurations , each vector @xmath301 encodes inputs containing @xmath302 class-@xmath98 jobs , for @xmath246 , as well as @xmath303 class-@xmath98 jobs , for @xmath304 . hence , for an incoming job sequence , instead of considering estimates on the number of class-@xmath98 jobs , for any @xmath284 , we can equivalently consider target configurations . unfortunately , it will not be possible to work with all target configurations @xmath305 since the resulting number of schedules to be constructed would be @xmath306 . therefore , we will work with a suitable sparsification of the set of all configurations . * sparsification of the set of target configurations : * let @xmath307 and @xmath308 . we will show that @xmath309 if @xmath1 is not too small ( see lemma [ lem : kappa ] ) . this property in turn will ensure that any job sequence @xmath2 can be mapped to a @xmath310 . for any vector @xmath311 , we define a target configuration @xmath312 that contains @xmath313 class-@xmath98 machines , for @xmath283 , provided that @xmath314 does not exceed @xmath315 . more specifically , for any @xmath311 , let @xmath316 and @xmath317 , be the partial sums of the first @xmath98 entries of @xmath318 , multiplied by @xmath319 , for @xmath283 . let @xmath320 . first construct a vector @xmath321 of length @xmath322 that contains exactly @xmath313 class-@xmath98 machines . that is , for @xmath283 , let @xmath323 for @xmath324 . we now truncate or extend @xmath325 to obtain a vector of length @xmath315 . if @xmath326 , then @xmath312 is the vector consisting of the first @xmath315 entries of @xmath325 . if @xmath327 , then @xmath328 , i.e.the last @xmath329 entries are set to 0 . let @xmath330 be the set of all target configurations constructed from vectors @xmath310 . * the algorithm family : * let @xmath331 . for any @xmath332 , algorithm @xmath333 works as follows . initially , prior to the arrival of any job of @xmath2 , @xmath333 determines the target configuration specified by @xmath334 and uses this virtual schedule for the machines of @xmath262 to make scheduling decisions . consider a machine @xmath291 and suppose @xmath335 , i.e. @xmath64 is a class-@xmath98 machine for some @xmath202 . let @xmath336 and @xmath337 be the targeted minimal and maximal loads caused by large jobs on @xmath64 , according to the target configuration . more precisely , if @xmath298 , then @xmath338 and @xmath339 . recall that in a target configuration a class-@xmath98 machine contains two class-@xmath98 jobs if @xmath225 . if @xmath340 and hence @xmath341 for some @xmath342 , then @xmath343 and @xmath344 . if @xmath345 is a machine with @xmath346 , then @xmath347 . while the job sequence @xmath2 is processed , a machine @xmath291 may or may not be _ again assume that @xmath64 is a class-@xmath98 machine with @xmath202 . if @xmath348 , then at any time during the scheduling process @xmath64 is admissible if it has received less than two class-@xmath98 jobs so far . analogously , if @xmath349 , then @xmath64 is admissible if it has received no class-@xmath98 job so far . finally , at any time during the scheduling process , let @xmath66 be the current load of machine @xmath64 and let @xmath233 be the load due to small jobs , @xmath160 . algorithm @xmath333 schedules each incoming job @xmath11 , @xmath13 , in the following way . first assume that @xmath11 is a large job and , in particular , a class-@xmath98 job , @xmath284 . the algorithm checks if there is a class-@xmath98 machine in @xmath262 that is admissible . if so , @xmath11 is assigned to such a machine . if there is no admissible class-@xmath98 machine available , then @xmath11 is placed on a machine in @xmath263 . there jobs are scheduled according to the _ best - fit _ policy . more specifically , @xmath333 checks if there exists a machine @xmath350 such that @xmath351 . if this is the case , then @xmath11 is assigned to such a machine with the largest current load @xmath66 . if no such machine exists , @xmath11 is assigned to an arbitrary machine in @xmath263 . next assume that @xmath11 is small . the job is a assigned to a machine in @xmath262 , where preference is given to machines that have already received small jobs . algorithm @xmath333 checks if there is an @xmath291 with @xmath352 such that @xmath353 . if this is the case , then @xmath11 is assigned to any such machine . otherwise @xmath333 considers the machines of @xmath262 which have not yet received any small jobs . if there exists an @xmath291 with @xmath354 such that @xmath355 , then among these machines @xmath11 is assigned to one having the smallest targeted load @xmath336 . if again no such machine exists , @xmath11 is assigned to an arbitrary machine in @xmath262 . a summary of @xmath256 , which focuses on the job assignment rules , is given in figure [ fig:3 ] . we obtain the following result . [ th : alg2 ] @xmath256 is @xmath3-competitive , for any @xmath31 and @xmath356 . the algorithm uses @xmath5 schedules . @xmath256 is @xmath3-competitive if , for the chosen @xmath42 , the number of machines is at least @xmath357 . if the number of machines is smaller , we can simply apply algorithm @xmath195 with an accuracy of @xmath358 . let @xmath257 be the following combined algorithm . if for the chosen @xmath42 , @xmath359 , execute @xmath360 . otherwise execute @xmath256 . [ cor : a3 ] @xmath257 is @xmath3-competitive , for any @xmath31 , and uses @xmath5 schedules . if @xmath360 is executed for a machine number @xmath359 , then by theorem [ th : guess2 ] the number of schedules is @xmath361 , which is @xmath362 . in the remainder of this section we prove theorem [ th : alg2 ] . the stated number of schedules follows from the fact that @xmath256 consists of @xmath363 algorithms . recall that @xmath307 and @xmath364 . hence @xmath365 and @xmath366 , which gives that @xmath367 is @xmath5 . hence it suffices to show that , for any job sequence @xmath2 , @xmath256 generates a schedule whose makespan is at most @xmath368 , which we will do in the remainder of this section . more specifically we will prove that , for any @xmath2 , there exists a target configuration @xmath332 that accurately models the large jobs arising in @xmath2 . we will refer to such a vector as a valid target configuration . then we will show that the corresponding algorithm @xmath333 builds a schedule with a makespan of at most @xmath368 . we introduce some notation . consider any job sequence @xmath2 . for any @xmath98 , @xmath284 , let @xmath369 be the number of class-@xmath98 jobs arising in @xmath2 , i.e. @xmath369 is the number of jobs @xmath11 with @xmath230 . furthermore , for any target configuration @xmath370 and any @xmath98 with @xmath284 , let @xmath371 be the number of class-@xmath98 machines in @xmath297 , i.e. @xmath372 . let @xmath373 be the total number of class-@xmath98 machines with @xmath348 . similarly , @xmath374 is the total number of class-@xmath98 machines with @xmath349 . given @xmath2 , vector @xmath332 will be a valid target configuration if , for any @xmath283 , @xmath2 contains as many class-@xmath98 jobs as specified in @xmath297 and , moreover , if all the additional large jobs can be feasibly scheduled on the @xmath375 reserve machines . recall that in a configuration @xmath297 , any class-@xmath98 machine with @xmath225 is supposed to contain two class-@xmath98 jobs . formally , @xmath332 is a _ valid target configuration _ if the following three conditions hold . a. for @xmath246 , there holds @xmath376 . b. for @xmath304 , there holds @xmath377 . c. @xmath378 conditions ( i ) and ( ii ) represent the constraint that @xmath2 contains as many class-@xmath98 jobs as specified in @xmath297 , @xmath284 . condition ( iii ) models the requirement that extra large jobs can be feasibly packed on the reserve machines . here @xmath379 is the extra number of class-@xmath98 jobs with @xmath348 in @xmath2 . any two of these can be packed on one machine since the processing time of any of these jobs is upper bounded by @xmath380 . hence two jobs incur a machine load of at most @xmath381 . analogously , @xmath382 is the extra number of class-@xmath98 jobs with @xmath349 , which can not be combined together because their processing times are greater than @xmath383 . in order to prove that , for any @xmath2 , there exists a valid target configuration we need two lemmas . [ lem : jobs ] for any @xmath2 , there holds @xmath384 . consider any optimal schedule @xmath385 for @xmath2 and recall that we assume without loss of generality that @xmath197 . in @xmath385 any machine containing a class-@xmath98 job with @xmath349 can not contain an additional large job : the class-@xmath98 job causes a load greater than @xmath386 and any additional large job , having a processing time greater than @xmath387 , would generate a total load greater than 1 . furthermore , any machine containing a class-@xmath98 job with @xmath298 can contain at most one additional job of the job classes @xmath388 because two further jobs would generate a total load greater than @xmath389 . [ lem : kappa ] for any @xmath390 , there holds @xmath391 if @xmath392 . there holds @xmath393 where the last line follows because of @xmath356 and @xmath394 , for any @xmath395 . the next lemma establishes the existence of valid target configurations . [ lem : config ] for any @xmath2 , there exists a valid target configuration @xmath332 if @xmath356 . in this proof let @xmath396 . given @xmath2 , we first construct a vector @xmath310 . lemma [ lem : jobs ] implies that for any job class @xmath98 , @xmath225 , there holds @xmath397 . for any job class @xmath98 , @xmath299 , there holds @xmath398 . by lemma [ lem : kappa ] , @xmath399 , which is equivalent to @xmath400 . for any @xmath98 with @xmath225 , set @xmath401 . for any @xmath98 with @xmath299 , set @xmath402 . then @xmath403 , for @xmath404 , and the resulting vector @xmath405 is element of @xmath406 . we next show that the vector @xmath312 constructed by @xmath256 is a valid target configuration . when @xmath256 constructs @xmath312 , it first builds a vector @xmath407 of length @xmath408 containing exactly @xmath409 entries with @xmath323 , for @xmath404 . if @xmath326 , then @xmath312 contains the first @xmath315 entries of @xmath325 . if @xmath410 , then @xmath312 is obtained from @xmath325 by adding @xmath329 entries of value 0 . in either case @xmath312 contains at most @xmath409 entries of values @xmath98 , for @xmath404 . hence for the target configuration @xmath312 , there holds @xmath411 , for @xmath404 , where @xmath371 is again the total number of class-@xmath98 machines in @xmath312 . if @xmath412 , then @xmath413 , which is equivalent to @xmath376 . similarly , if @xmath349 , then @xmath414 . therefore , conditions ( i ) and ( ii ) defining valid target configurations are satisfied and we are left to verify condition ( iii ) . first assume @xmath326 . in this case the vector @xmath312 contains no entries of value 0 and hence @xmath415 . recall that @xmath373 is the total number of class-@xmath98 machines with @xmath298 specified in @xmath312 . similarly , @xmath374 is the total number of class-@xmath98 machines with @xmath340 . by lemma [ lem : jobs ] , @xmath384 . subtracting the equation @xmath416 , we obtain @xmath417 there holds @xmath418 because @xmath419 is an integer . hence condition ( iii ) defining valid target configurations is satisfied . it remains to study the case @xmath420 . for any @xmath98 with @xmath340 , there holds @xmath421 and hence @xmath422 , which is equivalent to @xmath423 . hence @xmath424 the sum @xmath425 is the total number of entries @xmath426 with @xmath427 in @xmath325 . since @xmath428 , none of these entries is deleted when @xmath312 is derived from @xmath325 . hence @xmath429 is the total number of class-@xmath98 machines with @xmath340 specified in @xmath312 . we conclude @xmath430 for any @xmath98 with @xmath298 , there holds @xmath431 and hence @xmath432 . this implies @xmath433 . since @xmath434 is an integer we obtain @xmath435 . thus @xmath436 again @xmath437 because @xmath325 contains exactly @xmath438 entries @xmath426 with @xmath439 and all of these entries are contained in @xmath312 representing class-@xmath98 machines for @xmath298 . inequalities ( [ eq : n1 ] ) and ( [ eq : n2 ] ) together with the identity @xmath396 imply @xmath440 since again @xmath441 , condition ( iii ) defining valid target configurations holds . we next analyze the scheduling steps of @xmath256 . [ lem : sched1 ] let @xmath333 be any algorithm of @xmath256 processing a job sequence @xmath2 . at any time there exists at most one machine @xmath291 with @xmath352 and @xmath442 in the schedule maintained by @xmath333 . consider any point in time while @xmath333 sequences @xmath2 . suppose that there exists a machine @xmath291 with @xmath352 and @xmath442 . we show that if a small job @xmath11 arrives and @xmath333 assigns it to a machine @xmath443 with @xmath444 , then @xmath445 so that no new machine with the property specified in the lemma is generated . a first observation is that @xmath64 is not a class-@xmath32 machine because in this case @xmath336 would be @xmath446 . also , if @xmath447 is a class-@xmath32 machine , there is nothing to show because , again , in this case @xmath448 . so assume that @xmath333 assigns @xmath11 to a machine @xmath443 , which is not a class-@xmath32 machine , and @xmath444 prior to the assignment . we first show that @xmath449 . consider the scheduling step in which @xmath333 assigned the first small job @xmath14 to @xmath64 . since @xmath64 is not a class-@xmath32 machine @xmath450 , for some @xmath451 and the assignment of @xmath14 to @xmath64 led to a load of at most @xmath452 . since @xmath447 is not a class-@xmath32 machine either , @xmath14 could have also been assigned to @xmath447 incurring a resulting load of at most @xmath453 on this machine . note that when an algorithm @xmath333 can not assign a small job to a machine @xmath291 with @xmath352 and instead has to resort to machines @xmath454 with @xmath455 , it chooses a machine having the smallest @xmath456 value . we conclude @xmath457 . next consider the assignment of @xmath11 . algorithm @xmath333 would prefer to place @xmath11 on @xmath64 as it already contains small jobs . since this is impossible , there holds @xmath458 and thus @xmath459 . since by assumption @xmath460 it follows @xmath461 . suppose that @xmath450 , for some @xmath451 . then @xmath338 . since @xmath449 we obtain @xmath462 as desired . the following lemmas focus on algorithms @xmath333 such that @xmath297 is a valid target configuration for @xmath2 . [ lem : sched2 ] let @xmath2 be any job sequence and @xmath333 be an algorithm such that @xmath297 is a valid target configuration for @xmath2 . let @xmath356 . consider any point in time during the scheduling process . if the schedule of @xmath333 contains at most one machine @xmath291 with @xmath463 , then no further small job can arrive . since @xmath297 is a valid target configuration for @xmath2 , the job sequence contains as many class-@xmath98 jobs , for any @xmath298 , as indicated by @xmath297 . hence the total processing time of large jobs in @xmath2 is lower bounded by @xmath464 . hence the total processing time of jobs in @xmath2 is at least @xmath465 , where the machine loads due to small jobs may be considered at an arbitrary point in time . hence if there exists a time such that @xmath466 for at most one @xmath345 , we obtain @xmath467 the last inequality holds because @xmath468 , for any @xmath469 . hence no further small job can arrive . [ lem : sched3 ] let @xmath2 be any job sequence and @xmath333 be an algorithm such that @xmath297 is a valid target configuration for @xmath2 . let @xmath356 . then in the final schedule constructed by @xmath333 , each machine in @xmath262 has a load of at most @xmath470 . we consider the scheduling steps in which @xmath333 assigns a job @xmath11 to a machine in @xmath262 . first suppose that @xmath11 is large . let @xmath11 be a class-@xmath98 job , where @xmath284 . if @xmath11 is assigned to an @xmath345 , then @xmath64 must be an admissible class-@xmath98 machine , i.e. prior to the assignment of @xmath11 it contains fewer class-@xmath98 jobs as specified by the target configuration . this implies that for any machine @xmath345 , its load due to large jobs is always at most @xmath337 . the latter value is upper bounded by @xmath471 . hence , in order to establish the lemma it suffices to show that whenever a small job is assigned to a machine @xmath345 , the resulting load @xmath472 on @xmath64 is at most @xmath264 . suppose on the contrary that a small job @xmath11 arrives and @xmath333 schedules it on a machine in @xmath262 such that the resulting load is greater than @xmath264 . algorithm @xmath333 first tries to place @xmath11 on a machine @xmath345 with @xmath352 , which has already received small jobs . by lemma [ lem : sched1 ] , among these machines there exists at most one having the property that @xmath442 . since an assignment to those machines is impossible without exceeding a load of @xmath264 , @xmath333 tries to place @xmath11 on a machine @xmath291 with @xmath473 . since this is also impossible without exceeding a load of @xmath264 , any @xmath291 with @xmath474 must be a class-@xmath32 machine . this holds true because for any class-@xmath98 machine with @xmath475 , there holds @xmath476 and an assignment of a small job would result in a total load of at most @xmath477 . observe that any class-@xmath32 machine has a targeted minimal load of @xmath478 . we conclude that immediately before the assignment of @xmath11 the schedule of @xmath333 contains at most one machine @xmath345 with @xmath442 . lemma [ lem : sched2 ] implies that the incoming job @xmath11 can not be small , and we obtain a contradiction . [ lem : sched4 ] let @xmath2 be any job sequence and @xmath333 be an algorithm such that @xmath297 is a valid target configuration for @xmath2 . then in the final schedule constructed by @xmath333 , each machine in @xmath263 has a load of at most @xmath470 . algorithm @xmath333 assigns only large jobs to machines in @xmath263 . a first observation is that whenever there exists an @xmath479 that contains only one class-@xmath98 job with @xmath298 but no further jobs , then an incoming class-@xmath480 job with @xmath481 will not be assigned to an empty machine . this holds true because the two jobs can be combined , which results in a total load of at most @xmath482 . the observation implies that at any time while @xmath333 processes @xmath2 , the number of machines of @xmath263 containing at least one job is upper bounded by @xmath483 . here @xmath484 denotes the total number of class-@xmath98 jobs with @xmath298 that have been assigned to machines of @xmath263 so far . analogously , @xmath485 is the total number of class-@xmath98 jobs with @xmath340 currently residing on machines in @xmath263 . since @xmath297 is a valid target configuration for @xmath2 conditions ( i ) and ( ii ) defining those configurations imply @xmath486 and @xmath487 . moreover , since @xmath333 assigns large jobs preferably to machines in @xmath262 , there holds @xmath488 and @xmath489 . by condition ( iii ) defining valid target configurations , @xmath490 . hence , while @xmath491 there holds @xmath492 and thus exists an empty machine @xmath263 to which an incoming class-@xmath98 jobs with @xmath340 can be assigned . similarly , while @xmath493 , there must exist an empty machine or a machine containing only one class-@xmath480 job with @xmath481 to which in incoming class-@xmath98 job with @xmath298 can be assigned . in either case , the assignment generates a load of at most @xmath264 on the selected machine . theorem [ th : alg2 ] now follows from lemmas [ lem : config ] , [ lem : sched3 ] and [ lem : sched4 ] . we derive our algorithms for mps . the strategies are obtained by simply combining @xmath56 , presented in section [ sec : redu ] , with @xmath195 and @xmath257 . in order to achieve a precision of @xmath42 in the competitive ratio , the strategies are combined with a precision of @xmath494 in its parameters . for any @xmath31 , let @xmath495 be the algorithm obtained by executing @xmath496 in @xmath497 . for any @xmath31 , let @xmath498 be the algorithm obtained by executing @xmath499 in @xmath497 . [ cor:2 ] @xmath495 is a @xmath3-competitive algorithm for mps and uses no more than @xmath5 schedules , for any @xmath4 . theorem [ th : guess1 ] and corollary [ cor : a3 ] imply that @xmath495 is @xmath3-competitive , for any @xmath4 , and that the total number of schedules is the product of @xmath5 and @xmath500 , where @xmath501 . by the taylor series for @xmath502 , @xmath503 , we obtain @xmath504 , for any @xmath505 . hence the second term of the product is @xmath362 . [ cor:3 ] @xmath498 is a @xmath6-competitive algorithm for mps and uses no more than @xmath7 schedules , for any @xmath4 . by theorems [ th : guess1 ] and [ th : guess2 ] algorithm @xmath498 is @xmath6-competitive , for any @xmath4 . the total number of schedules is the product of @xmath506 and @xmath500 , where @xmath507 . again , by the taylor series , @xmath504 , for any @xmath505 . hence both terms of the product are upper bounded by @xmath7 . we develop lower bounds that apply to both mps and mps@xmath26 . let @xmath19 be any deterministic online algorithm for mps or mps@xmath26 that maintains at most @xmath509 schedules . we show that @xmath19 s competitive ratio is at least @xmath8 . to this end we construct an adversarial job sequence @xmath2 such that each schedule maintained by @xmath19 has a makespan of at least @xmath510 . the job sequence @xmath2 is composed of two subsequences @xmath511 and @xmath512 , i.e.@xmath513 . subsequence @xmath511 consists of @xmath1 jobs of processing time @xmath514 each . subsequence @xmath512 will consist of jobs having a processing time of either 2/3 or 1 . the exact number of these jobs depends on the schedules constructed by @xmath19 and will be determined later . consider the schedules that @xmath19 may have built after all jobs of @xmath511 have been assigned . each such schedule contains @xmath1 jobs of processing time 1/3 . for the moment we concentrate on schedules in which each machine contains either zero , one or three jobs , i.e. there exists no machine containing two or more than three jobs . each such schedule @xmath515 can be represented by a pair @xmath516 , where @xmath517 denotes the number of machines containing exactly one job and @xmath518 is the number of machines containing three jobs . here @xmath517 and @xmath518 are non - negative integers such that @xmath519 . let @xmath520 be the set of all these pairs . set @xmath521 has @xmath508 elements because @xmath518 can take any value between 0 and @xmath509 and @xmath522 . let @xmath515 be an arbitrary schedule containing @xmath1 jobs of processing time 1/3 and @xmath523 . we say that @xmath515 is an _ @xmath516-schedule _ if the number of machines containing exactly one job equals @xmath517 and the number of machines containing exactly three jobs equals @xmath518 . let @xmath524 be the set of schedules constructed by @xmath19 when the entire subsequence @xmath511 has arrived . by assumption @xmath19 maintains at most @xmath509 schedules , i.e. @xmath525 . hence there must exist a pair @xmath526 such that no schedule of @xmath524 is an @xmath527-schedule . on the other hand , let @xmath528 be an @xmath527-schedule . in @xmath528 we number the machines in order of non - decreasing load such that @xmath529 . schedule @xmath528 contains @xmath530 machines with a load smaller than 1 and , in particular , @xmath531 empty machines . now the subsequence @xmath512 consists of @xmath532 jobs , where the @xmath73-th job has a processing time of @xmath533 , for @xmath534 . hence @xmath512 contains @xmath531 jobs of processing time 1 followed by @xmath535 jobs of processing time @xmath536 . obviously , the makespan of an optimal schedule for @xmath2 is 1 : the jobs of @xmath511 are sequenced so that an @xmath527-schedule is obtained . again , after @xmath511 has arrived , the machines are numbered in order of non - decreasing load . while @xmath512 arrives , the @xmath73-th job is assigned to machine @xmath64 , having a load of @xmath244 , for @xmath534 . in the remainder of this proof we consider any schedule @xmath537 and show that after @xmath512 has been sequenced , the resulting makespan is at least 4/3 . this establishes the theorem . so let @xmath538 be any schedule and recall that @xmath515 contains @xmath1 jobs of processing time 1/3 each . if in @xmath515 there exists a machine that contains at least four of these jobs , then the makespan is already 4/3 and there is nothing to show . therefore , we restrict ourselves to the case that every machine in @xmath515 contains at most three jobs . again we number the machines in @xmath515 in order of non - decreasing load so that @xmath539 . consider the @xmath527-schedule @xmath528 in which the machines loads satisfy @xmath529 . there must exist a machine @xmath540 , @xmath541 , such that @xmath542 : for , if @xmath543 held for all @xmath544 , then @xmath545 for all @xmath544 because @xmath515 and @xmath385 both contain jobs with a total processing time of @xmath546 . thus @xmath515 would be an @xmath527-schedule and we obtain a contradiction . the last @xmath547 machines in @xmath385 have a load of 1 . it follows that @xmath548 because otherwise @xmath540 in @xmath515 contained at least four jobs . the property @xmath542 implies @xmath549 because @xmath515 and @xmath385 only contain jobs of processing time @xmath514 . we finally show that sequencing of @xmath512 leads to a makespan of at least @xmath8 in @xmath515 . if @xmath19 assigns two jobs of @xmath512 to the same machine , then the resulting machine load is at least 4/3 because each job of @xmath512 has a processing time of at least @xmath536 . so assume that @xmath19 assigns the jobs of @xmath512 to different machines . the first @xmath550 jobs of @xmath512 each have a processing time of at least @xmath551 because the jobs arrive in order of non - increasing processing times . in @xmath515 there exist at most @xmath552 machines having a load strictly smaller than @xmath553 . hence , after the first @xmath550 jobs have been scheduled in @xmath515 , there exists a machine having a load of at least @xmath554 . this concludes the proof . the next theorem gives a lower bound on the number of schedules required by a @xmath6-competitive algorithm , where @xmath555 . it implies that , for any fixed @xmath42 , the number asymptotically depends on @xmath556 , as @xmath1 increases . for instance , any algorithm with a competitive ratio smaller than @xmath557 requires @xmath558 schedules . any algorithm with a competitive ratio smaller than @xmath559 needs @xmath560 schedules . [ th : lb2 ] let @xmath19 be a deterministic online algorithm for mps or mps@xmath26 . if @xmath19 attains a competitive ratio smaller than @xmath10 , where @xmath561 , then it must maintain at least @xmath562 schedules , where @xmath563 and @xmath564 . the binomial coefficient increases as @xmath42 decreases and is at least @xmath565 . we extend the proof of theorem [ th : lb1 ] . let @xmath561 . furthermore , let @xmath566 and @xmath83 be defined as in the theorem . there holds @xmath567 . let @xmath568 and note that @xmath569 . we will define a set @xmath570 whose cardinality is at least @xmath562 , and show that if @xmath19 maintains less than @xmath571 schedules , then its competitive ratio is at least @xmath203 . we specify a job sequence @xmath2 and first assume that @xmath1 is even . later we will describe how to adapt @xmath2 if @xmath1 is odd . again @xmath2 is composed of two partial sequences @xmath511 and @xmath512 so that @xmath572 . subsequence @xmath511 consists of @xmath573 jobs of processing time @xmath209 each . subsequence @xmath512 depends on the schedules constructed by @xmath19 and will be specified below . consider the possible schedules after @xmath511 has been sequenced on the @xmath1 machines . we restrict ourselves to schedules having the following property : each machine has a load of exactly 1 or a load that is at most @xmath574 . observe that each machine of load 1 contains @xmath575 jobs . each machine of load at most @xmath574 contains up to @xmath576 jobs because @xmath577 . therefore any schedule with the stated property can be described by a vector @xmath578 , where @xmath579 is the number of machines having a load of 1 and @xmath371 is the number of machines containing exactly @xmath98 jobs , for @xmath580 . the vector @xmath581 satisfies @xmath582 and @xmath583 . the last equation specifies the constraint that the schedule contains @xmath573 jobs . let @xmath570 be the set of all these vectors , i.e.@xmath584 we remark that each @xmath585 uniquely identifies one schedule with our desired property . let @xmath515 be any schedule containing exactly @xmath573 jobs of processing time @xmath209 and @xmath586 . we say that @xmath515 is an _ @xmath581-schedule _ if in @xmath515 there exist @xmath579 machines of load 1 and @xmath371 machines containing exactly @xmath98 jobs , for @xmath580 . now suppose that @xmath19 maintains less than @xmath571 schedules . let @xmath524 be the set of schedules constructed by @xmath19 after all jobs of @xmath511 have arrived . since @xmath587 there must exist an @xmath588 such that no schedule of @xmath524 is an @xmath589-schedule . let @xmath385 be an @xmath589-schedule in which machines are numbered in order of non - decreasing load such that @xmath529 . subsequence @xmath512 consists of @xmath590 jobs , where job @xmath73 has a processing time of @xmath533 , for @xmath591 . hence @xmath512 consists of @xmath592 jobs of processing time @xmath593 , for @xmath580 . these jobs arrive in order of non - increasing processing time . each job has a processing time of at least @xmath594 because @xmath595 . the makespan of an optimal schedule for @xmath2 is 1 . the jobs of @xmath511 are sequenced so that an @xmath589-schedule is obtained . machines are again numbered in order of non - decreasing load . then , while the jobs of @xmath512 arrive , the @xmath73-th job of the subsequence is assigned to machine @xmath64 in @xmath385 , @xmath596 . we next show that after @xmath19 has sequenced @xmath512 , each of its schedules has a makepan of at least @xmath203 . so consider any @xmath537 and , as always , number the machines in order of non - decreasing load such that @xmath597 . if in @xmath515 there exists a machine that has a load of at least @xmath203 and hence contains at least @xmath598 jobs , then there is nothing to show . so assume that each machine in @xmath515 contains at most @xmath575 jobs and thus has a load of at most 1 . we study the assignment of the jobs of @xmath512 to @xmath515 . if @xmath19 places two jobs of @xmath512 on the same machine , then we are done because each job has a processing time of at least @xmath594 . therefore we focus on the case that @xmath19 assigns the jobs of @xmath512 to different machines . schedules @xmath515 and @xmath385 both contain jobs of total processing time @xmath599 . since @xmath515 is not an @xmath589-schedule there must exist a @xmath550 , @xmath600 , such that @xmath542 and hence @xmath601 . each machine in @xmath515 has a load of at most 1 while the last @xmath590 machines in @xmath385 have a load of exactly 1 . this implies @xmath602 . the first @xmath550 jobs of @xmath512 each have a processing time of at least @xmath551 . however , there exist at most @xmath552 machines in @xmath515 having a load strictly smaller than @xmath603 . hence after @xmath19 has sequenced the first @xmath550 jobs of @xmath512 there must exist a machine in @xmath515 with a load of at least @xmath604 . so far we have assumed that @xmath1 is even . if @xmath1 is odd , we can easily modify @xmath2 . the first job of @xmath2 is a job of processing time 1 . then @xmath511 and @xmath512 follow . these subsequences are defined as above , where @xmath1 is replaced by the even number @xmath605 . in this case @xmath606 the analysis presented above carries over because the first job of @xmath2 , having a processing time of 1 , must be scheduled on a separate machine and can not be combined with any job of @xmath511 or @xmath512 if a competitive ratio smaller than @xmath203 is to be attained . we next lower bound the cardinality of @xmath570 . again we first focus on the case that @xmath1 is even . in the definition of @xmath570 the critical constraint is @xmath583 , which implies that not every vector of @xmath607 represents a schedule that can be built of @xmath573 jobs . in particular , the vector @xmath608 of length @xmath609 would require @xmath610 jobs . therefore , we introduce a set @xmath611 and show @xmath612 . set @xmath611 contains vectors of length @xmath609 in which the first @xmath613 entries as well as the last one are equal to 0 . the other entries sum to at most @xmath614 , i.e.@xmath615 we show that each @xmath616 can be mapped to a @xmath585 . the mapping has the property that any two different vectors of @xmath611 are mapped to different vectors of @xmath570 . this implies @xmath612 . consider any @xmath617 . let @xmath578 be defined as follows . for @xmath618 , let @xmath619 . for @xmath620 , let @xmath621 . finally , let @xmath622 . note that @xmath623 . we next show that @xmath585 . there holds @xmath624 . furthermore , @xmath625 it follows , as desired , @xmath585 . note that the last @xmath83 entries of @xmath581 are identical to the last @xmath83 entries of @xmath626 . hence no two vectors of @xmath611 that differ in at least one entry are mapped to the same vector of @xmath570 . hence @xmath612 . if the number @xmath1 of machines is odd , then in the definition of @xmath611 the entries of a vector sum to at most @xmath627 . the rest of the construction and analysis is the same . thus , for a general number @xmath1 of machines @xmath628 this set contains exactly @xmath562 elements , where again @xmath629 . in the remainder of this proof we lower bound this binomial coefficient . e. angelelli , m.g . speranza and z. tuza . new bounds and algorithms for on - line scheduling : two identical processors , known sum and upper bound on the tasks . _ discrete mathematics & theoretical computer science _ , 8:116 , 2006 .

online makespan minimization is a classical problem in which a sequence of jobs @xmath0 has to be scheduled on @xmath1 identical parallel machines so as to minimize the maximum completion time of any job . in this paper we investigate the problem with an essentially new model of resource augmentation . more specifically , an online algorithm is allowed to build several schedules in parallel while processing @xmath2 . at the end of the scheduling process the best schedule is selected . this model can be viewed as providing an online algorithm with extra space , which is invested to maintain multiple solutions . the setting is of particular interest in parallel processing environments where each processor can maintain a single or a small set of solutions . as a main result we develop a @xmath3-competitive algorithm , for any @xmath4 , that uses a constant number of schedules . the constant is @xmath5 . we also give a @xmath6-competitive algorithm , for any @xmath4 , that builds a polynomial number of @xmath7 schedules . this value depends on @xmath1 but is independent of the input @xmath2 . the performance guarantees are nearly best possible . we show that any algorithm that achieves a competitiveness smaller than @xmath8 must construct @xmath9 schedules . our algorithms make use of novel guessing schemes that ( 1 ) predict the optimum makespan of a job sequence @xmath2 to within a factor of @xmath10 and ( 2 ) guess the job processing times and their frequencies in @xmath2 . in ( 2 ) we have to sparsify the universe of all guesses so as to reduce the number of schedules to a constant . the competitive ratios achieved using parallel schedules are considerably smaller than those in the standard problem without resource augmentation . furthermore they are at least as good and in most cases better than the ratios obtained with other means of resource augmentation for makespan minimization .

the lowest order chiral lagrangian supplemented by the chiral symmetry breaking terms @xmath4 of o(@xmath5 ) and @xmath6 of o(@xmath7 ) ( which model the mass - dependence of the chiral slope for the meson decay constants ) is given by@xcite : . @xmath8 where @xmath9 we use explicit formulas for the pseudoscalar masses , and pseudoscalar and axial vector decay constants , consistent through order @xmath10 ( for details see @xcite ) . the coefficients @xmath11 follow the notation of gasser and leutwyler @xcite . the expression for the pseudoscalar mass squared up to first order in the hairpin mass , @xmath12 and @xmath13 is : @xmath14 + \frac{1}{f^2}8 l_5 \chi_{ij } \delta \tilde{j}_{ij } \ } \end{aligned}\ ] ] where @xmath15 is a slope parameter , @xmath16 , @xmath17 and @xmath18 . here @xmath19 and @xmath20 . the loop integrals @xmath21 and @xmath22 are defined in @xcite . for the pseudoscalar decay constants : @xmath23 \nonumber \\ & & ~-~\frac{4}{f^2}l_{5}\delta \chi_{ij } [ ( \frac{\tilde{i}_{ij}}{2 } + i_{ij } ) - ( \tilde{j}_{ii}+\tilde{j}_{jj})]\ } \end{aligned}\ ] ] while for the axial vector decay constants : @xmath24 = 0.45 = 0.45 = 0.45 = 0.45 = 0.45 = 0.45 we study 350 configurations on a @xmath25 lattice with @xmath26 . clover fermions ( @xmath27 ) and the mqa procedure@xcite was used . we consider six @xmath28 values ( @xmath29 ) with @xmath30 . the fits are shown in figures [ fig : diag]-[fig : ratio ] . in the quenched theory the disconnected part of the eta prime correlator ( hairpin term ) has a double pole @xmath31 whose coefficient determines @xmath32 . the correlator for @xmath33 and a double pole fit(@xmath34 and @xmath35 ) is shown in figure [ fig : etap ] . the @xmath2 parameter is given by : @xmath36 = 0.42 as the quenched chiral limit is approached the isovector scalar correlator shows negative norm behaviour . the correlators for the lightest and heaviest @xmath28 values are shown in figure [ fig : scalar ] . properly accounting for the effects of the hairpin - pion bubble@xcite allows a good fit of the isovector scalar correlator for all quark masses . one output is the ( @xmath37 ) mass . using the mqa technique , meson properties ( masses and decay constants ) can be extracted with sufficient accuracy to allow a fit of higher order chiral parameters , @xmath4 and @xmath6 . the physical results for @xmath38 ( @xmath39 gev @xmath40 ) are compared with our @xmath0 ( @xmath41 gev @xmath42 ) results@xcite . the physical values of @xmath32 ( in gev ) extracted from the hairpin analysis ( at @xmath38 ) are @xmath43 , @xmath44 , @xmath45 , @xmath46 , @xmath47 and @xmath48 for @xmath49 and @xmath50 gev respectively . extrapolating to @xmath51 we obtain @xmath52 . the corresponding value for @xmath0 is @xmath53 . extracting @xmath37 and @xmath54 masses ( at @xmath55 ) from scalar and axial vector propagators gives @xmath56 and @xmath57 gev and @xmath58 and @xmath59 gev for @xmath38 and @xmath60 respectively . the value for @xmath61 gev suggests that the observed @xmath62 resonance is a @xmath63 `` molecule '' and not an ordinary @xmath3 meson . 16 w. bardeen , a. duncan , e. eichten , and h. thacker , phys . d62 ( 2000 ) 114505 . j. gasser and h. leutwyler , nucl . b250 ( 1985 ) 465 . w. bardeen , a. duncan , e. eichten , and h. thacker , phys . d57 ( 1998 ) 1633 ; phys . d59 ( 1999 ) 014507 . w. bardeen , a. duncan , e. eichten , n. isgur and h. thacker , phys . rev . d * 65 * ( 2002 ) 014509 .

we present results of a high - statistics study of scalar and pseudoscalar meson propagators in quenched qcd at two values of lattice spacing , @xmath0 and 5.9 , with clover - improved wilson fermions . the study of the chiral limit is facilitated by the pole - shifting ansatz of the modified quenched approximation . pseudoscalar masses and decay constants are determined as a function of quark mass and quenched chiral log effects are estimated . a study of the flavor singlet @xmath1 hairpin diagram yields a precise determination of the @xmath1 mass insertion . the corresponding value of the quenched chiral log parameter @xmath2 is compared with the observed qcl effects . removal of qcl effects from the scalar propagator allows a determination of the mass of the lowest lying isovector scalar @xmath3 meson .

the study of time - dependent solutions of the schrdinger equation for initial wave packets of gaussian form has a long history , dating back to the earliest days of the development of quantum theory . schrdinger himself @xcite and others @xcite - @xcite used such solutions to discuss the connections between the quantum and classical formulations of mechanics and found explicit wave packet solutions to many of the standard problems of classical mechanics , including the free - particle , case of uniform acceleration , the harmonic oscillator , and the particle in a uniform magnetic field . these examples first appeared in a number of textbooks @xcite - @xcite less than a decade later . in contrast , many treatments of introductory quantum mechanics today focus almost exclusively on time - independent , energy eigenstate ( or stationary state ) solutions of the schrdinger equation , although a large number of texts do include the explicit construction of a time - dependent gaussian wave packet solution for the free - particle case . this easily obtained , analytic result ( seemingly first obtained by darwin @xcite ) illustrates , in a closed form solution , many of the notions central to wave packet spreading and the essentially classical time - development of the expectation value as @xmath1 . in many modern software packages , these solutions are also visualized , helping to illustrate the correlated time - development of the real and imaginary parts , and especially their phase relationships , often using a popular color - coding scheme @xcite , @xcite , @xcite . an example of a static image representing such time - development is shown in fig . 1 . one can easily note the non - symmetric pattern of the ` wiggliness ' apparent for @xmath2 in both the real ( dotted ) and imaginary ( dashed ) parts of @xmath3 , which , as we will argue below , can be used as a qualitative measure of the local kinetic energy . the faster ( higher momentum , and hence ` wigglier ' ) components are more obviously associated with the ` front end ' of the spreading wave packet , while the ` back end ' exhibits much less rapid spatial variation , consistent with the slower ( low momentum ) components trailing behind . in this note , we attempt to extend and amplify upon this type of intuitive observation and try to answer such questions as _ where is the kinetic energy localized in a wave packet ? _ to this end , we begin with the familiar observation that the expectation value of the kinetic energy operator in any time - dependent position - space state , @xmath3 , can be written , using a simple integration - by - parts ( or ibp ) argument , in the form @xmath4 or @xmath5 where the ` boundary terms ' ( involving @xmath6 and @xmath7 at @xmath8 ) are assumed to vanish for any appropriately localized solution . this global identification suggests that we define a local _ kinetic energy density _ , @xmath9 , via @xmath10 this quantity , which is clearly locally real and positive - definite , can then be used to quantify the distribution of kinetic energy , and how it changes with time , for any time - dependent solution of the schrdinger equation . for gaussian solutions , we will be able to perform many of the desired integrals in closed form , leading to explicit analytic results for such related quantities as @xmath11 where the expectation value @xmath12 serves to define the ` center ' of the wave packet . these quantities can be respectively associated with the kinetic energy in the ` right ' ( or ` front ' , at least for packets moving generally to the right ) half and the ` left ' ( or ` back ' ) half of the wave packet . in what follows , we will present several explicit closed - form examples of the calculation of @xmath13 for gaussian wave packet solutions , starting in sec . [ sec_free_particle ] with free - particle wave packets for a standard gaussian momentum distribution , while in sec . [ sec_uniform_acceleration ] we illustrate similar results for gaussian solutions to the problem of a particle undergoing uniform acceleration , working initially in momentum - space . in sec . iv we use standard propagator techniques to examine gaussian wave packet solutions for the harmonic oscillator problem in this context , while in sec . v we extend these results to the case of an ` inverted oscillator ' , corresponding to a particle in unstable equilibrium . the time - dependent schrdinger equation for the one - dimensional free particle case can be written , and easily solved , in either position- or momentum - space in the equivalent forms @xmath14 since we will find the momentum - space approach more useful in sec . [ sec_uniform_acceleration ] for the case of uniform acceleration , we will also use that approach here and write @xmath15 where @xmath16 is the initial momentum distribution . using this very general form , and the appropriate operator form of @xmath17 , we recall that @xmath18 the position - space solution can be written , of course , using the fourier transform as @xmath19 the standard initial gaussian momentum - space distribution , which gives arbitrary initial momentum ( @xmath20 ) and position ( @xmath21 ) values , can be written in the form @xmath22 which gives @xmath23 the explicit form of the position - space wave function is given by the gaussian integral @xmath24 which can be evaluated in closed form ( using the change of variables @xmath25 and standard integrals ) to obtain @xmath26 where @xmath27 . ( this result is sometimes attributed to darwin @xcite . ) the corresponding probability density is easily shown to be @xmath28 where @xmath29 and the time - dependent expectation values of position are @xmath30 all of which are familiar results . turning now to the kinetic energy distribution defined in eqn . ( [ kinetic_energy_distribution ] ) , we find that the required spatial derivative is given by @xmath31 the kinetic energy density can therefore be written in the form @xmath32 \left[\frac{t / t_0}{(1+t^2/t_0 ^ 2)}\right ] + \frac{(x - x(t))^2}{(\alpha^2 \hbar)^2 ( 1+t^2/t_0 ^ 2)}\right ) \ , . \label{gaussian_case}\ ] ] the expectation value of the kinetic energy is correctly given by @xmath33 and receives non - zero contributions from the first and last terms in brackets in eqn . ( [ gaussian_case ] ) , since the middle term vanishes ( when integrated over all space ) for symmetry reasons . on the other hand , the individual values of @xmath13 in eqn . ( [ half_kinetic_energies ] ) can also be evaluated giving @xmath34 both of which are easily seen to be positive definite , as they must , due to the non - negativity of @xmath9 . the time - dependent fractions of the total kinetic energy contained in the @xmath35 ( right / left ) halves of this archetypical wave packet are given by @xmath36 which will clearly increase / decrease monotonically as @xmath37 . the limiting values are then @xmath38 which have the extremal values @xmath39 which occur when @xmath40 thus , as much as @xmath0 of the total kinetic energy of this gaussian wave packet solution can be in the ` front half ' of the wave at long times ( i.e. those for which @xmath41 . ) to illustrate this effect , we plot in the left column of fig . 2 , the modulus ( @xmath42 , solid ) , and real ( dotted ) and imaginary ( dashed ) parts of a typical solution corresponding to @xmath43 ( top ) , @xmath44 ( the extremal value , middle ) , and @xmath45 ( bottom ) for long times ( @xmath46 ) . the values are plotted in terms of the variable @xmath47 for easier comparison . for the @xmath48 case , it is clear that the larger momentum components ( in magnitude ) spread faster , but uniformly , in the opposite @xmath49 and @xmath50 directions , giving equal ` wiggliness ' on each side , while for large values of @xmath20 , the total kinetic energy is clearly increased ( many more ` wiggles ' everywhere ) , but the amount on each side of the expectation value peak at @xmath12 is roughly the same . for the extremal value of @xmath44 , there is the clearest distinction between the ` front ' and ` back ' halves , as the magnitudes of the momentum components in the ` back ' half are at a minimum , resulting in the greatest separation between the kinetic energy in the two halves . we can also compare the distribution of probability , described as usual by @xmath51 , to how the kinetic energy is localized . the kinetic energy density , @xmath9 , can be scaled to the total ( and possibly time - dependent ) value of @xmath52 via @xmath53 and so is normalized in the same way as the probability density . we plot both @xmath54 ( solid curve ) and @xmath55 ( dot - dashed curve ) in the right hand column of fig . 2 for the same three cases considered above and we note how the shape of @xmath55 is correlated with the obvious ` wiggliness ' shown in the real and imaginary parts of @xmath3 . the problem of a particle under the influence of a constant force is a staple in classical mechanics , and was considered early in the history of quantum mechanics @xcite , but is less often discussed in introductory treatments of the subject , especially in terms of time - dependent solutions . for that reason , we briefly review the most straightforward momentum - space approach to this problem . in this case , where the potential is given by @xmath56 , we can write the time - dependent schrdinger equation in momentum - space as @xmath57 \phi(p , t ) = i\hbar \frac{\partial \phi(p , t)}{\partial t } \label{pspace}\ ] ] or @xmath58 we note that the simple combination of derivatives guarantees that a function of the form @xmath59 will make the left - hand side vanish , so we assume a solution of the form @xmath60 , with @xmath61 arbitrary and @xmath62 to be determined . using this form , eqn . ( [ newp ] ) reduces to @xmath63 with the solution @xmath64 we can then write the general solution as @xmath65 or , using the arbitrariness of @xmath61 , as @xmath66 where now @xmath67 is some initial momentum distribution since @xmath68 . note that because the @xmath69 terms cancel in the exponential , we will be able to explicitly integrate gaussian type initial momentum - space waveforms . for any general initial @xmath70 we now have the time - dependent expectation values @xmath71 which also give the expectation value @xmath72 which , in turn , also agrees with a similar calculation of @xmath73 using @xmath74 , all of which are consistent with a particle undergoing uniform acceleration . using the standard initial gaussian momentum - space wavefunction in eqn . ( [ initial_gaussian ] ) as the @xmath75 in eqn . ( [ pacc ] ) , we can evaluate the position - space solution using eqn . ( [ fourier_transform ] ) to obtain @xmath76 \left ( \frac{1}{\sqrt{\sqrt{\pi}\alpha \hbar ( 1+it / t_0 ) } } \right ) \nonumber \\ & & \,\,\ , \times \ , e^{-(x-(x_0+p_0t / m+ft^2/2m))^2/2(\alpha \hbar)^2(1+it / t_0 ) } \ , . \label{accelerating_solution}\end{aligned}\ ] ] the corresponding probability density is given by @xmath77 where @xmath78 and @xmath79 so that this accelerating wave packet spreads in the same manner as the free - particle gaussian example . the calculation of the kinetic energy density proceeds exactly as in sec . [ sec_free_particle ] , with @xmath80 kinetic energies are then derived in the same way as before and are given by @xmath81 which is the same result as in eqn . ( [ left_and_right_kinetic_energies ] ) , with @xmath82 . this similarity in form implies that the maximum ( minimum ) values of @xmath13 are once again given by eqn . ( [ maximum_minimum_values ] ) ( provided that @xmath83 ) which now occurs when @xmath84 . in this more dynamic situation , if @xmath20 and @xmath85 have different signs ( so that the motion includes one ` back ' and one ` forth ' component ) , this situation can occur twice during a single trajectory , with the roles of @xmath86 and @xmath87 changing between the ` back ' and the ` forth ' traversal . the third model system which is easily shown to exhibit time - dependent gaussian wave packet solutions is also the most frequently studied of all classical and quantum mechanical examples , namely the simple harmonic oscillator , defined by the potential energy @xmath88 . in this case , the initial value problem is perhaps most easily solved , especially for gaussian wave packets , by propagator techniques @xcite @xcite . in this approach , one writes @xmath89 where the propagator can be derived in a variety of ways @xcite and can be written in the form @xmath90 \ , .\ ] ] for the initial state wavefunction we will use position - space version of eqn . ( [ initial_gaussian ] ) , but for notational and visualization simplicity , we will specialize to the case of @xmath91 , namely @xmath92 where @xmath93 and @xmath94 . in this state , the expectation value of the energy is @xmath95 the integral in eqn . ( [ propagator ] ) can be done in closed form for the initial gaussian in eqn . ( [ initial_position_gaussian ] ) with the result @xmath96 \frac{1}{\sqrt{a(t ) \sqrt{\pi } } } \exp \left [ -\frac{i m \omega \beta}{2\hbar \sin(\omega t ) } \frac{(x - x_s(t))^2}{a(t ) } \right ] \label{position_space_sho_solution}\ ] ] where @xmath97 ( we note that in the force - free limit , when @xmath98 , one can show that this solution reduces to the free - particle form in eqn . ( [ free_particle_position_solution ] ) , with @xmath91 , as expected . ) the time - dependent probability density is given by @xmath99 where @xmath100 and @xmath101 so , while the expectation value of the wave packet oscillates in a way which mimics the classical result , the time - dependent spatial width of the packet changes in time , consistent with very general expectations for the oscillator case @xcite , @xcite . this more general time - dependent gaussian solution is described in several textbooks @xcite , @xcite and special cases of it are often rediscovered @xcite , @xcite . we note that in the special case of the minimum uncertainty wavepacket where @xmath102 the time - dependent width of the gaussian simplifies to @xmath103 so the wave packet oscillates with no change in shape . this is the special result seen more standardly @xcite @xcite in textbooks and pedagogical articles , and is similar to the coherent - state like solution discussed by schrdinger @xcite in a famous paper . the time - dependent expectation value of the potential energy is easily found to be @xmath104 the time - dependent ( total ) kinetic energy then follows directly from this equation combined with eqn . ( [ total_sho_energy ] ) and is given by @xmath105 in order to determine the _ distribution _ of kinetic energy , however , we evaluate @xmath9 using @xmath106 and we find that @xmath107 \label{sho_fraction } \ , .\ ] ] the asymmetry ( @xmath108 ) in the kinetic energy distribution vanishes at half - integral multiples of the classical period @xmath109 , namely when @xmath110 , but also at the classical turning points , _ i.e. _ , at odd multiples of @xmath111 . there is also no asymmetry in the case when @xmath43 and the wave packet expands and contracts uniformly in both the @xmath49 and @xmath50 directions . finally , there is no asymmetry in the special case of the ` fixed width ' gaussian , when @xmath112 , and this property can perhaps help explain some of the special features of that minimum - uncertainty state . to better visualize the general result in eqn . ( [ sho_fraction ] ) , we plot in figs . 3 and 4 representations of both the wave function ( modulus , real , and imaginary parts ) as well as the probability ( @xmath54 ) and ( scaled ) kinetic energy ( @xmath55 ) distributions over the first quarter period . we note that for wave packets initially moving to the right ( @xmath113 ) as shown here , narrow packets , i.e. , ones with @xmath114 , typically have more kinetic energy in the ` front ' half of the packet ( fig . 3 , middle panels ) , while initially wider packets have the opposite behavior ( fig . 4 ) consistent with eqn . ( [ sho_fraction ] ) . while the time - dependence of @xmath115 ( defined in eqn . ( [ define_r_function ] ) ) is more varied than for the simpler , non - periodic , cases considered so far , as a specific example , we can examine the distribution of kinetic energy at times such that @xmath116 in that case we find @xmath117 this has extremal values of @xmath118\ ] ] when @xmath119 which are obvious generalizations of eqns . ( [ maximum_minimum_values ] ) and ( [ extreme_value ] ) . the final case we present for which time - dependent gaussian wave packet solutions are easily obtained is a generalization of the harmonic oscillator which has been described as the ` inverted oscillator ' @xcite or the ` unstable particle ' @xcite , corresponding to the classical motion of a particle at the top of potential hill , given by @xmath120 many of the results obtained using propagator techniques can be easily carried over to this problem with the simple identifications @xmath121 for example , the position - space probability density corresponding to the initial state in eqn . ( [ initial_position_gaussian ] ) is given by @xmath122 where @xmath123 and @xmath124 and the probability density exhibits a ` runaway ' ( exponential ) behavior . in this case , the long time behavior of the kinetic energy fractions is dictated by the limits @xmath125 which gives @xmath126 \sqrt{\beta_0 ^ 4/\beta^2 + \beta^2}\ ] ] with the same extremal values as in eqn . ( [ maximum_minimum_values ] ) , when @xmath20 satisfies the equivalent of eqn . ( [ sho_extreme_value ] ) with @xmath127 . in this note we have provided a detailed analysis of gaussian wave packet solutions of the schrdinger equation in several model systems , helping to elucidate some of the qualitative aspects of wave packet time - development and spreading often seen in standard visualizations , and their relationship to the distribution of kinetic energy . we have focused on closed - form analytic results , all of which have been obtainable due to the special nature of the integrals which arise for gaussian wave packets . an obvious and interesting extension would be to extend these results to free particle wave packets for arbitrary non - gaussian initial momentum distributions , performing the required integrals in eqn . ( [ fourier_transform ] ) numerically to see just how general the results in eqns . ( [ maximum_minimum_values ] ) and ( [ extreme_value ] ) are . for example , are there other initial wave packets and initial conditions for which the kinetic energy can be even more localized that the @xmath128 fraction seen here for gaussians ? another straightforward generalization is to consider two - dimensional extensions of all four systems discussed here ( then visualized as functions of both @xmath129 and @xmath130 ) where closed form expressions are also possible , as well as the related case of the charged particle ( confined to a plane ) subject to a uniform magnetic field . other studies of dynamical wave packet propagation have discussed aspects of the time - development of @xmath3 which arise from effects related to the differing behavior of various momentum - components . examples have included discussions of the average speed of wave packet components which are transmitted or reflected from a rectangular barrier @xcite ( due to the energy dependence of probabilities of transmission and reflection ) as well as the time - dependent shape of @xmath131 for a wave packet hitting an infinite wall @xcite or similar ` bouncing ' systems @xcite . the consideration of the distribution of kinetic energy distribution in many other such time - dependent wave packet solutions might prove useful in understanding their behavior . one such example certainly would be gaussian - like wave packet solutions in the infinite square well , where additional interesting features , such as exact wave packet revivals , are present . gaussian solutions such as those considered here can find use as examples of model systems in discussing other theoretical constructs , such as the wigner quasi - probability distribution @xcite - @xcite . in that case , one defines @xmath132 via @xmath133 which can then be evaluated explicitly in closed form , in either position- or momentum - space , for all of the solutions discussed above . in an accompanying paper @xcite we study the same four examples considered here , in yet a different context . 99 e. schrdinger , _ der stetige bergang von der mikro- zur makromechanik _ naturwissenschaften * 14 * 664 - 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211 ( 1995 ) . a. m. ozorio de almeida , _ the weyl representation in classical and quantum mechanics _ , phys . rep . * 296 * , 265 - 342 ( 1998 ) m. belloni , m. doncheski , and r. w. robinett , _ wigner quasi - probability distribution for the infinite square well : energy eigenstates and time - dependent wave packets _ ; e - print arxiv : quant - ph/0312086 . r. w. robinett and l. c. bassett , _ analytic results for gaussian wave packets in four model systems : ii . autocorrelation functions _ , submitted to found . phys . * time - development of a free - particle gaussian wave packet , illustrating the modulus ( @xmath42 , solid ) , the real ( @xmath134 , dotted ) and imaginary ( @xmath135 , dashed ) parts , as functions of time . * the figures in the left column show the modulus ( solid ) , and the real ( dotted ) and imaginary ( dashed ) parts of gaussian wave packets at long times ( @xmath46 ) plotted versus @xmath136 , illustrating the large fraction of kinetic energy ( note the pattern of ` wiggliness ' ) in the front of the wave for the @xmath44 case , as in eqn . ( [ maximum_minimum_values ] ) . in the right column , we plot the probability density ( @xmath137 , solid ) and the ( scaled ) kinetic energy density ( @xmath138 , dashed ) from eqn . ( [ scaled_kinetic_energy_density ] ) . * the figures in the top row show the modulus ( solid ) , and the real ( dotted ) and imaginary ( dashed ) parts of gaussian wave packet solutions of the harmonic oscillator corresponding to @xmath91 and @xmath139 with @xmath140 . times over the first quarter period are shown . in the bottom row , we plot the probability density ( @xmath137 , solid ) and the ( scaled ) kinetic energy density ( @xmath138 , dashed ) from eqn . ( [ scaled_kinetic_energy_density ] ) at the corresponding times . * same as fig . 3 , but for an initial gaussian wave packet solution of the harmonic oscillator with @xmath141 .

using gaussian wave packet solutions , we examine how the kinetic energy is distributed in time - dependent solutions of the schrdinger equation corresponding to the cases of a free particle , a particle undergoing uniform acceleration , a particle in a harmonic oscillator potential , and a system corresponding to an unstable equilibrium . we find , for specific choices of initial parameters , that as much as @xmath0 of the kinetic energy can be localized ( at least conceptually ) in the ` front half ' of such gaussian wave packets , and we visualize these effects .

beng - yen chen investigated the curves whose position vectors lie in their rectifying planes in the 3-dimensional euclidean space , @xcite . these curves are called the rectifying curves which are not plane curves . they can be equivalently defined as the twisted curves whose osculating planes are at the same distance from a fixed point . here we consider the curves whose all normal planes are at the same distance from a fixed point and obtain some characterizations of them in the 3-dimensional euclidean space . we first review the results on the rectifying curves in the 3-dimensional euclidean space , @xcite . let @xmath0 be a unit - speed curve with the position vector @xmath1 . therefore @xmath2 is the natural parameter of @xmath0 . frenet formulas follow as @xmath3 where@xmath4 and @xmath5 are frenet vectors and @xmath6 the curvature and @xmath7 the torsion of the curve.a rectifying curve can be defined by the equation @xmath8 by means of ( [ 11 ] ) we can write it as @xmath9 accordingly @xmath10 and so @xmath11integrating ( [ 14 ] ) we have @xmath12 where @xmath13is a constant . without loss the generality we can write ( [ 15 ] ) as @xmath14 where @xmath15 is a positive number.([12 ] ) , ( [ 14 ] ) and ( [ 16 ] ) imply that the position vector can be written as @xmath16differentiating ( [ 17 ] ) we have@xmath17and so@xmath18this implies that a rectifying curve is a twisted curve , that is @xmath19on the other hand , from ( [ 17 ] ) we have@xmath20so if a curve is a rectifying curve , then its osculating planes are at the same distance from the origin . conversely , let us assume that osculating planes of a twisted curve are at the same distance from the origin . differentiating ( [ 19 ] ) we get ( [ 12 ] ) and so it is a rectifying curve . therefore we have the following property : _ a twisted curve is a rectifying curve if and only if its osculating planes are at the same distance from a fixed point . _ on the other hand , if a unit spherical curve @xmath21 is defined by the equation @xmath22 where @xmath23 is the position vector of a rectifying curve @xmath24 with the natural parameter @xmath2 , it can be show that the position vector of a rectifying curve @xmath0 can be represented as @xmath25 where @xmath26is the natural parameter of the spherical curve @xmath27 then the tangent unit vector of the curve @xmath0 can be written as @xmath28 where @xmath29 is the tangent unit vector of the spherical curve @xmath30differentiating the last equation we have @xmath31 @xmath32 is the curvature of the spherical curve @xmath27from ( [ 23 ] ) we find the following relation between the curvatures of the curves @xmath33and @xmath21 associated with each other : @xmath34the last relation can be written , in terms of the natural parameter @xmath2 of the rectifying curve , as @xmath35 let us assume that all normal planes of a curve @xmath0 with the position vector@xmath36 * * * * are at the same distance from a fixed point where @xmath2 is the natural parameter of @xmath0 . we denote frenet vectors and the curvature and the torsion of @xmath0 by @xmath37 and @xmath5 and @xmath6 and @xmath7 respectively . we can choose the fixed point as the origin . therefore our condition becomes @xmath38 by the integration we find that @xmath39 where @xmath40 is a constant.since the case of @xmath41 corresponds to a spherical curve , we can assume that @xmath42 . then , without loss the generality , we can write ( [ 32 ] ) as @xmath43 where @xmath44 is a positive constant . then ( [ 31 ] ) reduces to @xmath45 ( [ 33 ] ) and ( [ 34 ] ) imply @xmath46differentiating ( [ 34 ] ) we have @xmath47 or @xmath48 where @xmath49 is the radius of curvature.from ( [ 34 ] ) and ( [ 37 ] ) we can write the position vector as @xmath50differentiating ( [ 38 ] ) we have @xmath51 and @xmath52 therefore we first have@xmath53so the curve is a plane curve if and only if @xmath54without the generality , for case of a plane curve we can write@xmath55then from ( [ 33 ] ) and ( [ 401 ] ) @xmath56accordingly the natural equations of the curve are @xmath57 we can now assume that our curve is a twisted curve , that is @xmath58 . then@xmath59where @xmath60is the radius of torsion . hence ( [ 38 ] ) can be written as@xmath61therefore , according to ( [ 33 ] ) we have@xmath62on the other hand , from ( [ 402 ] ) we can write@xmath63we can show that ( [ 311 ] ) and ( [ 312 ] ) are equivalent equations . in fact , the equation ( [ 312 ] ) can be written as @xmath64[rr^{\prime } -2\epsilon c^{2}]=0 \label{313}\ ] ] because of @xmath65 , corresponds to the plane curve . then we have @xmath66r^{\prime } t+4c^{4}\kappa t(\kappa t)^{\prime } -2\epsilon c^{2}\kappa t(r^{\prime } t)^{\prime } \]]and@xmath67\]]this implies that@xmath68^{\prime } \label{314}\]]therefore we have@xmath69it is obvious that using the equation ( [ 311 ] ) we obtain the equation ( [ 312 ] ) . let us define a unit spherical curve @xmath21 by the equation @xmath70 where @xmath71 is the position vector of a curve with normal planes at constant distance from a fixed point.so we have @xmath72 differentiating we get @xmath73 from the last equation we find @xmath74 for the speed of the spherical curve . and so the natural parameter of the unit spherical curve@xmath75 , from @xmath76 is obtained as @xmath77 since @xmath78 there exists a function @xmath79 which satisfies the equation ( [ 44 ] ) therefore ( [ 41 ] ) can be written as @xmath80 so for a given curve @xmath0 with normal planes at constant distance from a fixed point , whose position vector is@xmath81 , we have a unit spherical curve @xmath21 whose position vector @xmath82 is defined by ( [ 46 ] ) with the natural parameter @xmath83 . we call @xmath84the unit spherical curve associated with the curve @xmath0 with normal planes at constant distance from a fixed point . now let us consider a curve @xmath0 defined by @xmath85 where@xmath86 is the position vector of a unit spherical curve @xmath84with the natural parameter @xmath83 and @xmath79 is defined by ( [ 44 ] ) and @xmath44 is constant . let @xmath87 and @xmath32 and@xmath88 be frenet vectors and the curvature and the torsion of @xmath21 respectively.according to ( [ 44 ] ) @xmath89 differentiating ( [ 47 ] ) with respect to @xmath83 we have @xmath90 since @xmath91 and @xmath92 from ( [ 49 ] ) we have @xmath93 so the unit tangent vector of @xmath0 is found as @xmath94 since @xmath95and @xmath96 @xmath97 this means that the curve @xmath0 is a curve with normal planes at constant distance from a fixed point . we call @xmath33the curve with normal planes at constant distance from a fixed point associated with the unit spherical curve @xmath21.let us now differentiate ( [ 410 ] ) with respect to@xmath98 . we have @xmath99 using * * * * @xmath100 @xmath101 and @xmath102 , from ( [ 412 ] ) we obtain the following relation between curvatures of the curves @xmath0 and @xmath21 associated with each other : @xmath103 or@xmath104since @xmath105the curvature of the unit spherical curve is not smaller than @xmath106that is @xmath107 the equation ( [ 4131 ] ) implies that the curvature of the unit spherical curve associated with the plane curve given by the natural equations ( 404 ) is @xmath108 this means that the spherical curve is a great circle of the unit sphere . hence we can obtain the cartesian equations of the plane curve associated with a great circle of the unit sphere using the equation ( [ 42 ] ) . in fact , we can choose the equation of a great circle of the unit sphere as @xmath109since @xmath110@xmath111and@xmath112then from ( [ 42 ] ) we obtain the cartesian equations of the plane curve with normal planes at constant distance from the origin as@xmath113this curve is a plane curve with normal planes at a distance @xmath114 from the origin . this means that all normal lines of the curve are at a distance @xmath115 from the origin . so it is an involute of the circle of radius @xmath116 centered at the origin . since it is an involute of a plane curve , it is also a plane curve ( * ? ? ? in fact , the position vector of an involute @xmath0 of the curve @xmath117 whose position vector is@xmath118can be written as@xmath119where @xmath120is constant and @xmath121 is unit tangent vector to the curve @xmath117 , @xcite , @xcite , @xcite . then@xmath122where @xmath123 is the curvature of @xmath117 and @xmath124 the principal normal vector . therefore the unit tangent vector t to @xmath0 can be written as @xmath125since @xmath117 is a circle of radius @xmath114 centered at the origin,@xmath126@xmath127@xmath128so@xmath129 , @xmath130 . then from ( [ 3850 ] ) and ( [ 3851 ] ) we have @xmath131this means that the involute @xmath0 of the circle @xmath117 is a plane curve with normal planes at a distance @xmath114 from the origin . figure 1 illustrates the plane curve ( [ 385 ] ) with normal planes at a distance @xmath132 from the origin ( the solid line for @xmath133 , the dashed line for @xmath134 ) . it is an involute of the circle of radius @xmath135 centered at the origin.let us note the equations ( [ 385 ] ) can be also obtained using the equations ( [ 3850])-([3852 ] ) or using ( [ 385 ] ) from the equations of a plane curve given by@xmath136where @xmath2 is the natural parameter of the curve , @xcite , 3 , @xcite . in the following we give an example of a twisted curve with normal planes at constant distance from a fixed point . let us choose the unit spherical curve @xmath21 as the circle of radius @xmath137 given by the equation @xmath138 then the curve @xmath0 associated with @xmath21 is given by the equation@xmath139where @xmath140since the curvature of the circle @xmath21 is @xmath141(4131 ) implies that@xmath142figure @xmath143 illustrates the twisted curve ( [ 4141 ] ) with normal planes at a distance @xmath132 from the origin ( the solid line for @xmath144 , the dashed line for @xmath134 ) . if @xmath145constant , ( [ 403 ] ) reduce to@xmath146from ( [ 4151 ] ) we have @xmath147constant . this means that rectifying planes of a curve of constant curvature with normal planes at constant distance from a fixed point are also at constant distance from the same point .

we consider the curves whose all normal planes are at the same distance from a fixed point and obtain some characterizations of them in the 3-dimensional euclidean space . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ keywords : curve , plane curve , distance , spherical curve , involute . 2000 mathematics subject classification : 53a04 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

helioseismology has confirmed the importance of gravitational settling in the sun s external regions ( @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ) . turnoff stars in globular clusters are only slightly less massive than the sun and have convection zones that tend to be somewhat shallower . in solar type stars , radiative accelerations have been shown to become equal to gravity for some metals around the end of the main sequence lives @xcite . the question of the atomic diffusion of metals in pop ii stars then naturally arises . it is currently of special interest because large telescopes are now making possible the determination of the abundance of metals in the turnoff stars of globular clusters . in this paper evolutionary models that take into account the diffusion of he , libeb and metals in pop ii stars are presented for the first time . surface abundances may then be used as additional constraints in the determination of the age of globular clusters and of the universe ( @xcite , hereafter paper ii ) . given their old age , pop ii stars are those where the slow effect of atomic diffusion has the largest chance to play a role on the evolutionary properties . previously published evolutionary models of pop ii stars have included some of the effects of diffusion ( @xcite ; @xcite ; @xcite ; @xcite ; @xcite ) but never included the effects of the diffusion of metals with their radiative accelerations self consistently . determining constraints on stellar hydrodynamics from abundance observations requires knowing the original chemical composition of the star . in pop ii stars there are larger variations in original abundances than in pop i. for that reason it is essential to use globular clusters for abundance determinations of metals since only then does one have a handle on the original abundances from observations of cluster giants . however libeb have an origin partly different from that of most metals . coupled with their sensitivity to low temperature nuclear burning , their abundance determination in halo stars provides useful constraints on hydrodynamics even if determinations in cluster stars would be preferable . the observation by @xcite of a plateau in the li concentration over a relatively large @xmath6 interval of pop ii stars has now been confirmed by many observations . furthermore , the li concentration is constant while that of fe varies by more than a factor of 100 @xcite from @xmath7 = - 3.7 $ ] to @xmath8 . this shows the primordial origin of li . its preservation for such a time interval over such a wide @xmath6 interval seriously challenges our understanding of convection and other potential mixing processes in those stars @xcite . if there were no mixing process outside of convection zones , the surface li abundance would vary with @xmath6 : at small @xmath6 because of nuclear reactions ( @xmath9 ) and at large @xmath6 because of gravitational settling . extending convection zones by a simple turbulence model does not solve the problem since , if the extension is sufficient to reduce li settling enough in the hotter stars , it causes excessive li destruction in the cooler stars of the plateau @xcite . efforts were also made to link such an extension to differential rotation ( @xcite or , as parametrized in the yale models , @xcite ) but the small apparent dispersion in the plateau makes this model unlikely . @xcite even claim that the destruction may not be by more than 0.1 dex . several groups ( @xcite , @xcite , @xcite ) have also observed @xmath10 in halo stars . since @xmath10 is destroyed at a smaller @xmath6 than @xmath11 , its survival in old stars implies a strict and small upper limit on the amount of mixing in those stars . the stars where a detection has been made are all concentrated close to the turnoff @xcite . while a mechanism has been suggested to produce @xmath12 in the sun @xcite , it is not expected to significantly affect atmospheric values so that a star in which @xmath12 is seen in the atmosphere may not have destroyed a significant fraction of its original @xmath10 . @xcite determined be abundances in halo stars as well as a linear correlation with the fe abundance suggesting that be has not been destroyed in those stars . the chemical composition of globular clusters is attracting more attention as large telescopes make the determination of the surface chemical composition of turnoff stars possible . the determination of the abundance of metals makes it possible to put additional constraints on the hydrodynamics of pop ii stars . some observers have reported factors of 2 differences between fe abundances in red giants and subgiants @xcite . however @xcite and @xcite find no difference between the abundances in red giants and turnoff stars . @xcite find the same relative abundances as in the sun in the turnoff stars of ngc@xmath136397 . unexplained anticorrelations between the abundances of o and na have been observed by @xcite . in this paper , the surface abundances to be expected in pop ii stars are calculated under different assumptions for the internal stellar hydrodynamics . the first and simplest assumption is that there is no macroscopic motion outside of convection zones . there remains only atomic diffusion , including the effects of gravitational settling , thermal diffusion and radiative accelerations , as a transport process outside of convection zones . on the main sequence , such models have been shown to lead to larger abundance anomalies than are observed @xcite but also to the appearance of an additional convection zone caused by the accumulation of iron peak elements @xcite . a relatively simple parametrization of turbulence , corresponding to an extension of those fe convection zones by a factor of about 5 in mass , was shown to lead to a simple explanation of the amfm phenomenon @xcite . a similar parametrization is used here in pop ii stars and leads to our second series of models . finally , we use an additional parametrization that is chosen in order to minimize the reduction of the surface li concentration , so as to provide the best representation of the spite plateau . @xcite introduced a similar parametrization of turbulence to compete with he and li settling ( other parametrizations of turbulence were introduced for the sun by @xcite and @xcite ) . @xcite has shown that weak turbulence below the solar convection zone also improves agreement with the solar pulsation spectrum . for comparison purposes , series of models are also calculated without diffusion and one model is calculated with gravitational settling but without @xmath4 ( see subsubsection [ sec : structure ] ) . the models were calculated as described in @xcite . the radiative accelerations are from @xcite with the correction for redistribution from @xcite and @xcite . the atomic diffusion coefficients were taken from @xcite . the uncertainties in the atomic and thermal diffusion coefficients have been discussed by @xcite , and by @xcite in particular for the temperature density domain of the interior of pop ii stars and the sun . turbulent transport is included as in @xcite and @xcite where the parameters specifying turbulent transport coefficients are indicated in the name assigned to the model . for instance , in the t5.5d400 - 3 model , the turbulent diffusion coefficient , @xmath14 , is 400 times larger than the he atomic diffusion coefficient at @xmath15 and varying as @xmath16 . to simplify writing , t5.5 will also be used instead of t5.5d400 - 3 since all models discussed in this paper have the d400 - 3 parametrization . the @xmath16 dependence is suggested ( see @xcite ) by observations that the be solar abundance today is hardly smaller than the original be abundance ( see for instance @xcite ) all models considered here were assumed to be chemically homogeneous on the pre - main sequence with the abundance mix appropriate for pop ii stars . the relative concentrations used are defined in table [ tab : xinit ] . the relative concentrations of the _ alpha _ elements are increased compared to the solar mix as is believed to be appropriate in pop ii stars @xcite . in @xcite , the solar luminosity and radius at the solar age were used to determine the value of @xmath17 , the ratio of the mixing length to pressure scale height , and of the he concentration in the zero age sun . they calibrated @xmath17 both in models using krishna - swamy s @xmath18 relation @xcite ( both with and without atomic diffusion ) and in models using eddington s @xmath18 relation ( only with diffusion ) . the he concentration mainly affects the luminosity while @xmath17 mainly determines the radius , through the depth of the surface convection zone . the required value of @xmath17 was found to be slightly larger in the diffusive than in the non diffusive models because an increased value of @xmath17 is needed to compensate for he and metals settling from the surface convection zone . the increased @xmath17 in the diffusion models of the sun is then determined by the settling occurring immediately below the solar surface convection zone . see @xcite for a discussion of uncertainties related with the use of the mixing length in pop ii stars . some of the models presented in this paper were calculated with turbulence below the convection zone . the appropriate value of @xmath17 to use then depends on the depth of turbulence . if turbulence is large enough to eliminate settling at the depth of solar models , the most appropriate @xmath17 to use is that determined for non diffusing solar models . if the adopted turbulence is more superficial than the solar convection zone , the appropriate @xmath17 to use is that for the solar models with diffusion . given the uncertainty , it was chosen to use the same value of @xmath19 for all models , both with and without diffusion , calculated with eddington s @xmath18 relation . in order to give an estimate of the uncertainty related to @xmath17 , two values of @xmath17 ( one determined by the solar model without diffusion and one by the solar model with diffusion ) will be used for one series of models with diffusion ( see subsection [ sec : alternate_models ] ) . the calculated series of models are identified in table [ tab : parameters ] . the series labeled ks@xmath17 used a value of @xmath17 determined from a solar model with diffusion while the series labeled ks used a value of @xmath17 determined from a solar model without diffusion . four series of evolutionary models were calculated . in the first subsection , the models with atomic diffusion are presented . in the second one , the models with turbulence and those without diffusion are introduced and compared to the models with atomic diffusion . two series of models with turbulence are discussed in some detail : a series that minimizes li underabundance and one that contains a level of turbulence similar to that needed to reproduce the observations of amfm stars . in the last subsection , the effect of changing boundary conditions is analyzed . the @xmath6 and @xmath20 as a function of age as well as the luminosity as a function of @xmath6 are shown in figure [ fig : historic_noturb ] for evolutionary models calculated including atomic diffusion and only those physical processes that can currently be evaluated properly from first principles . that figure also contains the time variation of the depth of the surface convection zone , of the temperature at its bottom and of central h concentration . models were calculated for 0.5 to 1.0 @xmath3 stars from the pre - main sequence to the bottom of the giant branch except for the 0.85 , 0.9 and 1.0 @xmath3 stars which were stopped earlier because of numerical instabilities . all models shown have @xmath21 . the mass interval was chosen to minimize the possibility of interpolation errors in the construction of isochrones ( see paper ii ) . the variation of the mass in the surface convection zone is large enough to play a major role on surface abundance evolution . in the next two subsections , the effect of radiative accelerations on surface abundances are related to the regression of the surface convection zone during evolution . the radiative accelerations below the surface convection zone play the most important role in determining the surface abundance variations ( see fig . [ fig : grad ] ) . as the evolution of a 0.8 @xmath3 star proceeds during the main sequence phase , the surface convection zone becomes progressively thinner : between 6 and 11 gyr , the mass in the convection zone is reduced by a factor of about 20 , from @xmath22 to @xmath23 ( see fig . [ fig : historic_noturb ] ) . the effect of this regression of the surface convection zone may be seen in the surface abundances through the dependence of @xmath4 on nuclear charge . the @xmath4 on li has a maximum at @xmath24 where it is in the hydrogenic configuration . it is completely ionized deeper in the star so that @xmath4(li ) is progressively reduced deeper in . the chemical species with a larger nuclear charge become ionized deeper in so that they are in an hydrogenic configuration deeper in and have the related maximum of their @xmath4 at greater depths . for , it is at @xmath25 , for at @xmath26 and for at @xmath27 . another maximum in the @xmath4 appears when a species is in between the atomic configurations of and . this maximum occurs at @xmath25 for p , and @xmath28 for . these maxima may be followed on the figure for other species . as may be seen from figure [ fig : grad ] , when the atomic species is in the hydrogenic configuration , the @xmath4 is rarely larger than gravity . it is larger only for be , b and c. at the maximum of @xmath4 between the -like and -like configurations , however , @xmath4 is usually larger than gravity . contrary to what happens in pop i stars @xcite , the @xmath4 do not depend on the abundance of the species since it is too small to cause flux saturation . the time variation of surface abundances in pop ii stars with atomic diffusion is shown in figure [ fig : surf_abundance_diff ] . the surface abundances reflect the variation of the @xmath4 below the surface convection zone as it moves toward the surface . from figure [ fig : historic_noturb ] , one sees that the depth of the surface convection zone gets progressively smaller during evolution until hydrogen is exhausted from the center . in the 0.8 @xmath3 model , it starts at a depth of @xmath26 and is at @xmath29 after 11 gyr . one sees in figure [ fig : grad ] that @xmath4(fe ) is smaller than gravity when the convection zone is deep but larger than gravity when it is more superficial . in the early evolution , fe settles gravitationally though _ less rapidly _ than he or c , largely because @xmath4(fe ) is not much smaller than gravity . around 9 gyr , @xmath4(fe ) becomes larger than gravity below the surface convection zone . one sees the effect in the surface abundances ( figure [ fig : surf_abundance_diff ] ) where the surface fe concentration starts increasing around 9 gyr , becoming larger than the original concentration around 10 gyr and reaching an overabundance by a factor of about 5 just before the star becomes a subgiant , at which point the fe abundance goes back to very nearly its original value as the convection zone becomes more massive . similar remarks apply to other chemical species . the depth dependence of the abundance variations is shown for a 0.8 @xmath3 model in figure [ fig : intern_abundances ] . one first notices that the abundance of @xmath10 is modified by nuclear reactions over the inner 98% of the mass . atomic diffusion modifies the abundances over the outer 1% of the mass . this leaves hardly any intermediate zone . at 6.1 gyr , the surface convection zone extends down to @xmath30 , where the @xmath4 are smaller than gravity for most species . only k , ca and ti start to be supported : their abundances in the surface convection zone are larger than immediately below . as evolution proceeds , the convection zone recedes . after 9.1 gyr , atomic species between na and ni are supported as the @xmath4 is larger than gravity below the bottom of the convection zone . this continues as the evolution proceeds . after 11.4 gyr , the fe abundance , for instance , is 3 times larger than the original abundance in the surface convection zone . that increased concentration of fe is caused by the migration of fe above @xmath31 where its @xmath4 becomes larger than gravity . between @xmath29 and @xmath28 , the concentration of fe decreases since it is either pushed into the surface convection zone by radiative accelerations ( for @xmath32 ) or settles gravitationally below . on the scale of that figure , the concentration variations of fe are not seen for @xmath33 . it sinks toward the center of the star where a small fe overabundance appears . the diffusion time scales are long enough that the effects amount to nearly a 10% increase at the center after 11.4 gyr ( see fig . [ fig : center_zoom ] ) . this figure may be compared to figures 15 and 16 of @xcite where the increase in fe concentration at the center is by about 3% at the age of the sun . transformation of c and o into n also modifies the value of @xmath34 so that it increases by only 4% at the center and has a maximum outside of the region where o is transformed into n. in stars of other masses , surface abundances are also mainly determined by the depth reached by the surface convection zone . in the 0.7 @xmath3 model , the surface convection zone never gets thinner than @xmath35 so that no overabundances appear during evolution and underabundances are limited to a factor of about 2 . in the 0.9@xmath3 model on the other hand , the surface fe abundance starts increasing around 2 gyr , when the surface convection zone becomes thinner than @xmath25 , where , as seen above , @xmath4(fe ) becomes larger than gravity . in this star , fe becomes overabundant around 3 gyr . in pop ii stars of larger mass , the overabundances are larger because the surface convection zones are thinner . from figure [ fig : historic_noturb ] , one sees that even in one single model , say the 0.8 @xmath3 one , the mass in the convection zone varies from about @xmath36 to @xmath37 during the main sequence evolution . linking turbulence to the position of the convection zone could never produce a small variation of li : if one increased the depth of the convection zone by a fixed factor , large enough to eliminate settling around 11 gyr for instance ( a factor of 100 or so is needed ) , this would lead to complete destruction of li in the early evolution . to minimize li abundance reduction , it is essential , if one uses time independent parametrization , to link turbulence to a fixed @xmath38 and not to the bottom of convection zones . the turbulent transport coefficients used in these calculations are shown in figure [ fig : coefficients ] . we chose to introduce the turbulence that minimizes the reduction of the surface li abundance . we consequently defined the turbulent diffusion coefficient as a function of @xmath38 in order to adjust it most closely to the profile that limits gravitational settling of li while not burning li . the nuclear reaction @xmath9 is highly @xmath38 sensitive and the li burning occurs at @xmath39 ( see @xcite for a detailed discussion ) . turbulent diffusion was adjusted to be smaller than atomic diffusion slightly below that @xmath38 so that turbulence would reduce settling as much as is possible in surface layers without forcing li to diffuse by turbulence to @xmath39 . a few series of evolutionary models were run in order to optimize turbulence parameters . examples of the resultant interior li profiles are shown in figure [ fig : coefficients ] . the li concentration always goes down rapidly as a function of increasing @xmath38 for @xmath39 . in the absence of turbulence , there is a peak in the li concentration at @xmath40 . this peak disappears as the strength of turbulence is increased . turbulence in the t5.5d400 - 3 model is too weak to influence li concentration in the @xmath38 interval shown . turbulence in the t6.0d400 - 3 model eliminates the li abundance gradient caused by gravitational settling down to @xmath41 while causing little reduction on the largest li abundance at @xmath40 . increasing turbulence further reduces the li peak , and in the t6.2 model the surface li abundance is smaller than in the t6.13 model , showing that one has passed the optimal value of turbulence . it is interesting to note that the turbulent diffusion coefficient of the t6.09d400 - 3 model of 0.8 @xmath3 is a factor of about 10 smaller than the turbulent diffusion coefficient found necessary to reproduce the solar li surface abundance by @xcite . the latter corresponds approximately to the t6.2d400 - 3 turbulent diffusion coefficient profile . on the other hand the t5.5d400 - 3 turbulent diffusion parametrization corresponds approximately , as a function of @xmath38 , to that found by @xcite to lead to the abundance anomalies observed in amfm stars . a comparison of the 0.8 @xmath3 models with and without diffusion is shown in figure [ fig : with_without ] . first compare the hertzsprung - russell diagrams . all models with diffusion ( both with and without turbulence ) nearly have the same evolutionary tracks ( those with turbulence are somewhat hotter ) . however the one without atomic diffusion goes to significantly higher @xmath6 than all models with diffusion . we have also verified that the turnoff temperature is reached 6 to 7% earlier in the models with diffusion ( with or without turbulence ) than in the model without diffusion . these differences are reflected in the isochrones based on these tracks and therefore on globular cluster age determinations ( see paper ii ) . they partially come from the structural effects of the global redistribution of he but also from the 4 to 5% difference in the age at which the central h abundance is exhausted . this in turn is reflected in the shorter age ( also 4 to 5% ) at which all diffusion models become subgiants , as seen in the rapid decrease of @xmath6 but the rapid increases of the mass and @xmath38 at the bottom of the surface convection zone . while there are significant differences between the model without diffusion and all those with diffusion , the effect of the different turbulent transport models on the depth and mass at the bottom of the surface convection zone , and the time variation of the central h concentration are seen from figure [ fig : with_without ] to be practically negligible on the scale of that figure . of the macroscopic properties defining the evolution of a 0.8@xmath3 star , only the @xmath6 and @xmath42 appear to be modified by the turbulence models that were introduced . while it is not shown for other masses , it is also true for them . however it is shown in paper ii that the differences in @xmath6 that seem small on the scale of figure [ fig : with_without ] are important in the comparison of calculated and observed isochrones . given that there are , in the literature , calculations of pop ii evolutionary tracks with gravitational settling but no @xmath4 , it is worth noting that the track ( @xmath43 relation ; upper left hand part of fig . [ fig : with_without ] ) of such a model is , around turnoff , slightly cooler than the track of the model with diffusion and @xmath4 but no turbulence . furthermore the abundance is very different , by a factor of about 1000 , at 12 gyr . the @xmath44 is smaller by 0.1 dex mainly because of the reduction of opacity caused by the reduction of the abundance . _ it is a better approximation not to let diffuse at all in a 0.8@xmath3 star than to calculate its gravitational settling without including the effects of @xmath4(fe)_. the abundance is smaller by a factor of 2 in this case both because of the reduction of the @xmath44 and because @xmath4(li ) has some effect in the 0.8 @xmath3 model . the surface abundance anomalies caused by atomic diffusion are much more affected by turbulence than the macroscopic properties . in the t5.5 model , turbulence reduces surface abundance anomalies but only in the time interval from 9 to 12 gyr . before 9 gyr , the surface convection zone is deeper than @xmath45 while in the t5.5d400 - 3 model , the turbulent diffusion coefficient is larger than atomic diffusion only for temperatures up to @xmath46 . only when the surface convection zone has retracted sufficiently for turbulent diffusion to be larger than atomic diffusion below the convection zone , does turbulence have any effect . in this model , the turbulent diffusion coefficient does not suppress completely the effects of atomic diffusion ( see fig . [ fig : with_without ] ) . it reduces them by an amount dependent on the chemical species . an fe overabundance appears in the absence of turbulence but it is eliminated in the t5.5 model while the he underabundance is much less affected . the t6.0 model has the same turbulence profile as the t5.5d400 - 3 model except that the profile is shifted towards higher temperatures by a factor of 3 in @xmath38 ( see fig . [ fig : coefficients ] ) ; this leads to a much larger effect on abundance anomalies . it limits the effect of atomic diffusion on the surface abundances of he and li to a factor of @xmath47 . the effect of turbulent transport on the interior concentrations in a 0.8 @xmath0 star is shown , for two values of turbulence , in figure [ fig : abtot_dif_6.09 ] . the no turbulence case is also shown for comparative purposes . the internal concentration profiles are shown at nearly the same age in the three models . such a star today would be close to the turnoff . it is also the evolutionary epoch when the effect of radiative accelerations and diffusion are largest and when the effect of adding a turbulent diffusion coefficient is greatest . in the case , the @xmath4 never play a large role since turbulence mixes down to @xmath48 and the @xmath4 are not much greater than gravity at that depth . in fact they are greater than gravity only for ti , cr , mn and fe and only around the end of evolution when @xmath4 causes settling to slow down in the exterior region . a minimum in abundances appears at @xmath49 because of that slow down in the settling from above while settling continues unabated below . in the star with the t5.5d400 - 3 parametrization , there is mixing from the surface down to @xmath50 . deeper in the star , radiative accelerations ( see fig . [ fig : grad ] ) are greater than gravity over a sufficient mass interval to cause overabundances in the surface regions for chemical species between al and ca and for ni . the @xmath4 for mg , ti , cr , mn and fe are just sufficient to bring them back to their original abundance while most of the chemical species between he and na are less supported and remain underabundant in the atmosphere . the exceptions are b and c which are very nearly normal . this behavior is to be contrasted to that in the star with no turbulence , where mixing by the surface convection zone extends from the surface down to @xmath51 ; the concentration gradients below the convection zone are very steep because of the absence of turbulence ; this leads to larger overabundances , in particular a factor of 10 for si and s but of 3 for most fe peak elements ; while c becomes underabundant , b becomes overabundant . turbulence changes the position in the star where atomic diffusion dominates and so where the sign of @xmath52 matters . however , turbulent transport also modifies the steepness of concentration gradients . together with evolutionary time scales , these determine over vs under abundances . this shows the need for complete evolutionary models in order to determine surface abundances . changing the surface boundary conditions has a significant effect on the depth of the surface convection zone ( see section [ subsec : alpha ] and fig . [ fig : m_tbzc ] ) . two boundary conditions were used . both are often used for popii models . in their solar models , @xcite used mainly krishna swamy s @xmath18 relation . it is used here for some series of models . @xmath18 relation is also used : it was used for a and f pop i stars by @xcite and @xcite . in @xcite , the sun is used to determine the appropriate value of @xmath19 to use with each of these boundary conditions . one may view the difference in convection zone mass between the models ( fig . [ fig : m_tbzc ] ) as a reasonable estimate of the uncertainty . during evolution , the uncertainty varies from a factor of 1.3 to 1.6 in mass . this is sufficient to modify the fits to globular cluster isochrones ( see paperii ) . it has however a less pronounced effect on surface abundances than the uncertainty on turbulence . it modifies significantly surface abundances only when no turbulence is included in the models . the surface concentrations of the 28 species impose constraints on stellar models . they will be shown for three series of models , the atomic diffusion models with no turbulence , those that mimic the turbulence of amfm stars and those that minimize the effects of transport on surface li concentration . surface abundance variations as a function of time have been discussed above ( see subsections [ sec : with diffusion ] and [ sec : turbulent diffusion ] ) . they are discussed below as a function of atomic number and of @xmath53 in order to facilitate comparison to observations . in the first subsection , we present the results for all calculated species for individual stars . in the second subsection , we present results at a given age for stars of various masses as a function of @xmath53 . they are then compared to observations in the last subsection . the surface concentrations at 10 and 12 gyr are shown as a function atomic number in figure [ fig : ab_z10 ] for the evolutionary models with atomic diffusion and no turbulence ( see also subsection [ sec : with diffusion ] ) . the 0.7 @xmath3 model is the one with the smallest mass shown . in it , gravitational settling causes a general reduction of surface abundances by close to 0.2 dex . as may be seen from figure [ fig : historic_noturb ] , the convection zone is always at least 1% of the mass in a 0.7 @xmath3 star so that radiative accelerations play a modest role ( see fig . [ fig : grad ] ) . in the 0.6 and 0.5 @xmath3 models ( not shown ) , the mass in the surface convection zone is always larger than about 20% of the stellar mass so that gravitational settling is negligible even after 10 or 12 gyr . in stars of 0.75 @xmath3 or more , the surface convection zone occupies less than 0.1% of the stellar mass for part of the evolution . the @xmath4 then play the major role for that part of the evolution . at 10 gyr , the 0.8 @xmath3 model is significantly affected by @xmath4 ( see fig . [ fig : ab_z10 ] ) . the atomic species most affected are those between al and ca . they have overabundances by factors of up to 5 . the fe peak elements are hardly affected . in the 0.84 @xmath3 model , ni is overabundant by a factor of 70 , while fe is overabundant by a factor of 30 . large underabundances are expected for cno and li while be and b are supported at least partly by their @xmath4 . since the original metallic abundances are small , the @xmath4 are little affected by line saturation and the abundance anomalies reflect essentially the atomic configuration of the dominant ionization state of each chemical species . for instance , li is completely stripped of its electrons while c , n and o are mainly in he - like configurations ( see also @xcite ) . at 12 gyr , the 0.84 @xmath3 model has already evolved to the giant phase . the most massive star around the turnoff has 0.81 @xmath3 . it is already 300 k cooler than the turnoff so that its surface convection zone has already started to expand and its abundance anomalies to decrease . while its fe peak shows approximately the original abundances , the species between al and ca have overabundances by up to a factor of 5 . note again that be and b are supported by their @xmath4 while li has sunk . at that age , the 0.8 @xmath3 star is the star with the largest anomalies . ni and species between na and cl are about a factor of 15 overabundant . he , li and cno are underabundant by a factor of about 10 . at 12 gyr , the 0.80 @xmath3 is at the peak of its abundance anomalies : its surface convection zone is just about to start getting deeper ( see fig . [ fig : historic_noturb ] ) . the 0.7 @xmath3 star also has larger anomalies than at 10 gyr . its surface convection zone is still getting smaller ( see fig . [ fig : historic_noturb ] ) while the 20% longer life means more time for gravitational settling . at 13.5 gyr ( the age of the oldest clusters as determined in paper ii ) the surface abundances are shown in figure [ fig : ab_z13.5 ] for stars with no turbulence . the lower mass star , 0.7 @xmath3 , is at @xmath54 k and has seen its original abundances reduced by approximately 0.3 dex . this applies , in particular , to li . the slightly warmer ( 6300 k ) 0.75 @xmath3 star has a larger li underabundance ( @xmath550.5 dex ) , marginally larger fe peak underabundances ( @xmath550.39 dex ) but the ca abundance goes back to its original value because it is supported by its @xmath4 . chemical species between al and ti are at least partially supported by their @xmath4 . the slightly more evolved 0.77 and 0.78 @xmath3 stars have nearly the same @xmath6 but slightly lower gravities , less massive surface convection zones ( see fig . [ fig : historic_noturb ] ) and larger abundance anomalies than the 0.75 @xmath3 star . the two have very similar abundance anomalies . fe has about its original abundance while species between al and ca have larger surface abundances than their original ones . li is underabundant by a factor of 7.3 . a comparison to figure [ fig : ab_z10 ] shows that , for instance , ni and al abundance anomalies are reduced by a factor of 10 as one goes from clusters of 10 gyr to clusters of 12 gyr and then by a factor of 5 as one goes from clusters of 12 gyr to clusters of 13.5 gyr . as clusters age , the abundance anomalies caused by atomic diffusion in turnoff stars get progressively smaller . the effect of turbulence strength on surface composition is shown for the 0.8 @xmath3 model in figure [ fig : ab_z12_08 ] at 10 and 12 gyr . as a general rule , turbulence reduces abundance anomalies . however , in a 0.8 @xmath3 star , the effect of the t5.5d400 - 3 turbulence is more complex . mg is mildly overabundant in the absence of turbulence but normal for the turbulence used , while na goes from slightly over to slightly underabundant and the overabundance of ca is increased by the increased turbulence . the anomalies between al and cl are more robust . these differences in behavior may be related to the differences in the internal distribution of the abundance variations caused by diffusion for various species ( see fig . [ fig : abtot_dif_6.09 ] ) . at 11.7 gyr , species from na to cl have their maximum concentrations in the surface convection zone in the absence of turbulence . they are reduced by the extra mixing brought about by turbulence . however , ar , k , ca and ti have their maximum concentrations below the convection zone so that some extra mixing increases their surface abundances . as a comparison with figure [ fig : ab_z10 ] shows , the effect of adding turbulence is similar to that of considering a slightly more evolved , slightly more massive model ( compare the 0.8 @xmath3 star with t5.5 of fig . [ fig : ab_z12_08 ] with the 0.81 @xmath3 of fig . [ fig : ab_z10 ] ) , in that fe peak anomalies are nearly eliminated in both while those between c and ca are mildly reduced in both . the 0.78 @xmath3 no - turbulence model ( not shown ) has very similar surface concentrations as both the 0.81 @xmath3 model without turbulence and the 0.8 @xmath3 star with t5.5 turbulence . it differentiates itself by slightly larger n , o and f underabundances . however , the mixing that is strong enough to minimize li abundance variations ( the t6.09 model ) goes deeper than the region where @xmath4 play a substantial role . turbulence then reduces the abundance anomalies of all metals to 0.1@xmath550.2 dex . all metals are underabundant if mixing is that strong . the 12 gyr surface abundance isochrones are shown as a function of @xmath53 for a number of species ( @xmath56li , @xmath57li , be , b , c , o , na , mg , al , si , s , cl , ca , cr , fe and ni ) in figure [ fig : ab_teff12 ] for the cases of atomic diffusion with no turbulence as well as with two turbulence strengths ( t5.5 and t6.09 ) . one first notes that , at the cool end , below 5000 k , there are only very small underabundances ( @xmath58 dex ) for all species and all turbulence strengths , except for li which is destroyed by nuclear reactions and , to a lesser extent , be . as one considers hotter stars , all underabundance factors increase until one reaches 6000 k. there the @xmath4 start to play a role for some species . furthermore , in the vicinity of the turnoff , stars of similar @xmath6 have very different evolution states . in the absence of turbulence , some turnoff stars can have overabundances of fe by up to 0.7 dex . around the turnoff the fe abundance can vary between @xmath550.3 dex underabundance and 0.7 dex overabundance . the presence of even a very small amount of turbulence reduces this range by more than 0.7 dex ( compare the continuous and dotted lines ) . the maximum abundance is reduced to the original fe abundance while the underabundance factor is only slightly reduced to 0.25 dex . b , mg , cr , mn and ni are similarly affected by the same turbulence model though by slightly smaller factors . however the overabundances between p ( not shown ) and ca are more robust . the range of cl anomalies is merely reduced from a factor of 20 to a factor of about 7 . similarly , the large underabundances ( he , li , n and o ; he and n are not shown ) are not much reduced by the introduction of a small amount of turbulence . in a given cluster , the range of chemical species calculated then offers the possibility of testing turbulence models . the surface abundance isochrones at the age determined for m92 in paper ii are shown in figure [ fig : ab_teff13.5 ] . one first notes , by comparing to figure [ fig : ab_teff12 ] , the strong reduction in all abundance variations around the turnoff but the general slight increase in all underabundances in stars with @xmath59 k. the increased underabundances reflect the increased time available for gravitational settling and the slight reduction in the depth of the surface convection zone between 12 and 13.5 gyr ( see fig . [ fig : historic_noturb ] ) . at 13.5 gyr , one then expects , in the absence of turbulence , underabundances of 0.05 dex at @xmath60 k increasing to 0.1 dex around 5500 k. above that @xmath6 , underabundances remain approximately constant for stars with the t6.09 turbulence model . if there is no turbulence , underabundances increase to 0.3 dex at 6000 k and above that @xmath6 both over and underabundances are present as @xmath4 play a significant role . for the fe peak elements , abundances are between the original and 0.3 dex underabundance . only ni could have an overabundance . between c and na all species are underabundant by a factor of order 2 while species between al and ca are generally overabundant . the presence of the t5.5 turbulence model mainly affects al , si , fe and ni overabundances which it reduces to or below the original abundance . its presence is sufficient to reduce the expected range of underabundances of fe to @xmath61 dex . since in this paper results are presented only for stars with @xmath21 , we will compare with the abundance determinations in m92 ( see @xcite and @xcite ) . in so far as one may assume all cluster stars to share the same original composition , giant stars give a handle on the original turnoff star composition . the observations of clusters with larger @xmath34 values will be discussed in a forthcoming paper where the evolutionary calculations for various values of @xmath34 will be presented . we will use li abundance in field halo stars with @xmath62 to complement the m92 observations . since li is believed to be of cosmological origin in these stars , one expects that the original abundance is the same for all . the @xmath11 observations of @xcite and @xcite , who have established the presence of a li abundance plateau , are compared in figure [ fig : li_teff ] to @xmath11 abundance isochrones at the age of m92 ( 13.5 gyr ) and at 12 gyr . the observations of @xcite , are also included since they further constrain the high @xmath38 part of the plateau . given these results , it appears difficult to question the constraints imposed by the li plateau , as done by @xcite . results for 3 series of models are shown : those with atomic diffusion without turbulence and those with the t5.5 and t6.09 turbulent transport models . the stars calculated with the t6.09 model clearly fit the @xmath11 observations much better . both at 12 and 13.5 gyr , there is a uniform @xmath11 abundance extending from @xmath63 to 6000 k , in agreement with observations . it corresponds to an underabundance by 0.17 dex from the original li abundance . note also that @xmath10 is reduced by less than a factor of 2 in turnoff stars ( see fig . [ fig : ab_teff12 ] and fig . [ fig : ab_teff13.5 ] ) , which is compatible with the observations of the presence of @xmath10 in those pop ii objects ( see @xcite ) . the series of models calculated without turbulence and that calculated with the t5.5 model do not reproduce the observations for @xmath64 k. given the observational error bars , the t6.09 model leads to a @xmath6 dependence of @xmath65(li ) which is perhaps more constant than required . however , at 12 gyr , calculations with the t6.0 model ( not shown ) give a li abundance 0.15 dex smaller at 6500 k than at 5600 k. while perhaps compatible with the error bars , this would lead to a significantly poorer fit and may be considered the lower limit to the turbulent transport required by observations . the lithium abundances observed in the m92 turnoff stars by @xcite are also shown in the 13.5 gyr part of the figure . the error bars are much larger than for field stars mainly because of the much smaller signal to noise ratio . the determined li abundances suggest a slightly larger original value for the li abundance in m92 than in halo stars but , given the size of the error bars , this could be considered uncertain . one star has a li abundance higher than the others by a factor larger than the error bars . other stars at the same @xmath6 have a clearly smaller equivalent width of the li doublet . the authors used this to argue for li destruction by a large factor in pop ii stars . this appears premature to us . @xcite have observed 7 metals mainly in three subgiant stars of m92 with @xmath66 k ( see their discussion of @xmath6 in their section 4.3 ) , which is slightly cooler than our 6300 k turnoff @xmath6 at 13.5 gyr . their determination of the fe abundance is based on more lines and appears more reliable than that of other metals . it is the only metal we discuss . their main conclusion is that the subgiants of m92 have a 0.26 dex lower fe abundance than the giants of m92 as determined by @xcite . they carefully discuss all sources of error and can not completely exclude that this 0.26 dex difference be reduced even possibly to zero . if the t6.09 turbulent transport model applies , all 3 stars should have the same fe abundance and it should be approximately 0.1 dex smaller than the original fe abundance according to our figure [ fig : ab_teff13.5 ] . the giants should have the original fe abundance because of their large surface convection zones . this seems quite compatible with the observations of fe given the uncertainties . @xcite also find ( their section 4.2 ) that star 21 has a 0.15 dex larger fe abundance than the other two ( stars 18 and 46 ) . furthermore @xcite observed that the @xmath57li abundance is a factor of 3 smaller in star 21 than in star 18 . this is to be compared to our results in figure [ fig : ab_teff13.5 ] . at the turnoff , the li and fe abundances are sensitive functions of both the exact evolutionary state and turbulence . at 13.5 gyr , in 0.78 @xmath3 models without turbulence , the fe abundance is @xmath550.05 dex from normal . introducing the t5.5 turbulence model reduces it by 0.25 dex while the t6.09 model brings it back to @xmath550.1 dex from normal . as may be seen in figure [ fig : life_teff ] , the fe abundance as a function of @xmath6 has a complex loop structure around the turnoff in both 0.8 and 0.78 @xmath3 stars . small changes in evolutionary state and/or turbulence can change the fe abundance by 0.2@xmath550.3 dex and the @xmath57li abundance by 0.5@xmath550.6 dex . at the turnoff , the different li and fe abundances would then be explained by a different turbulence between the stars , resolving the difficulty described in section 4.2 of @xcite . they would become the pop ii analogue of amfm stars . however all three of those stars appear to be past the turnoff . as figure [ fig : life_teff ] shows , the @xmath4 have significant effects on surface abundances in stars of mass 0.77 @xmath3 and more but , for fe , these effects are limited to @xmath64 k. as seen in figure [ fig : ab_z_teff ] , the abundance variations at the turnoff are not expected in subgiants when they reach 6020 k. at this temperature , reducing turbulence leads to more gravitational settling for both li and fe which can not explain the differences between stars 18 and 21 . while turnoff stars of a cluster could have abundance variations as observed for stars 18 , 21 and 46 , these are not expected in stars with exactly the parameters determined for them and this is a difficulty even if there appears to be considerable uncertainty in the @xmath6 of stars in this cluster ( compare the @xmath6 of @xcite to those of @xcite and the discussion in section 4.3 of @xcite ) . note that in the comparison to our evolutionary models , it is the absolute @xmath6 scale that matters and not just the @xmath6 with respect to the turnoff @xmath6 . if anomalies are confirmed , agreement with our model would require the higher @xmath6 scale for m92 . it has been clearly shown that , contrary to the belief expressed in many papers and recently by @xcite , atomic diffusion does not necessarily lead to underabundances of metals in pop ii stars . differential radiative accelerations lead to overabundances of fe and some other chemical elements in some turnoff stars . consider the evolution of pop ii stars with no tubulence . as one may see by considering the solid curves in figures [ fig : ab_teff12 ] and [ fig : ab_teff13.5 ] , generalized underabundances by 0.1 dex are expected in the @xmath6 interval from 4600 to 5500 . between 5500 and 6000 , the underabundances are still generalized and increase to 0.3 dex for some species such as . in 12 gyr turnoff stars however ( @xmath67 ) , overabundances by a factor of up to 10 are possible ( e.g. al and ni ) . all calculated species heavier than na may have overabundances . at a given @xmath6 , variations are expected from star to star . at 13.5 gyr , similar but smaller anomalies are expected . the overabundances are sensitive to any left over turbulence below the convection zone . in this paper , the evolution of stars with @xmath21 has been described both with and without turbulence . a 0.1 dex underabundance of metals in turnoff stars as compared to giants has been shown to be the smallest anomaly to be expected ( section [ sec : ab_teff ] ) . star to star variations were seen to be possible around the turnoff , if turbulence is small enough . observations ( see section [ sec : observations ] ) suggest the presence of abundance variations similar to those expected . the comparison to observations is , however , sensitive to the @xmath6 scale . as @xcite concluded at the end of their section 4.2 , higher quality data is probably required to establish the reality of fe abundance variations within m92 . the accurate determination of the abundance of more species is also needed . this may well have implications not only for intrinsic abundance variations in clusters but for internal stellar structure . the effect of varying @xmath34 on the evolution of pop ii stars will be investigated in a forthcoming paper , ( paper iii ) , where comparisons to higher @xmath34 clusters will be undertaken . increasing @xmath34 in pop ii stars will be shown to reduce considerably the expected abundance anomalies . note also that paper ii shows that the present models for @xmath7_0 = -1.31 $ ] accurately reproduce the cmd locations of local population ii subdwarfs having hipparcos parallaxes and metallicities within + /@xmath55 0.2 dex of @xmath7 = -1.3 $ ] . in a number of clusters with higher @xmath34 than m92 , observations suggest that only small variations , if any , are present in turnoff stars ( see for instance @xcite ; @xcite and @xcite ) . furthermore in field halo stars , the li abundance puts strict constraints on any chemical separation . in the companion paper ( paper ii ) we therefore took the cautious approach to use mainly evolutionary models that minimize the effect of atomic diffusion . it has been shown that the use , in complete stellar evolution models , of a relatively simple parametrization of turbulent transport leads to li surface abundances compatible with the li plateau observed in field halo stars ( with a 0.17 dex reduction from the original li abundance ) and small variations in the surface abundances of metals ( a 0.1 dex reduction of the metal abundance in turnoff stars as compared to that in giants in clusters with @xmath21 ) . at the same time , the gravitational settling of he leads to a reduction in the age of globular clusters by some 10% ( see paper ii ) . however simple the parametrization of turbulent transport , it is not understood from first principles . the high level of constancy of li abundance as a function of @xmath6 requires that turbulence mixes to very nearly the same @xmath38 throughout the star evolution and in stars covering a mass interval of approximately 0.6 to 0.8 @xmath3 . as already noted by @xcite , this is not expected in standard stellar models . no convincing hydrodynamic model has been proposed that explains this property . pop ii stars appear to tell us that this is the case , however . mass loss is another physical process that could compete with atomic diffusion and maintain a constant value of li as a function of @xmath6 @xcite . whether , in the absence of turbulent transport , it could be made consistent with the observations of metals in globular cluster turnoff stars is a question that requires further calculations . these may lead to observational tests of the relative importance of mass loss and turbulence in these objects . the number of chemical species that are now included in these calculations and that can be observed makes such tests possible . we thank an anonymous referee for constructive comments that led to significant improvements of this paper . this research was partially supported at the university of victoria and the universit de montral by nserc . we thank the rseau qubcois de calcul de haute performance ( rqchp ) for providing us with the computational resources required for this work . extends his warm thanks to the service dastrophysique at cea - saclay for an enjoyable and productive stay during which part of this research was performed . this work was performed in part under the auspices of the u.s . department of energy , national nuclear security administration , by the university of california , lawrence livermore national laboratory under contract no.w-7405-eng-48 . , r. g. , bonifacio , p. , bragaglia , a. , carretta , e. , castellani , v. , centurion , m. , chieffi , a. , claudi , r. , clementini , g. , dantona , f. , desidera , s. , franois , p. , grundahl , f. , lucatello , s. , molaro , p. , pasquini , l. , sneden , c. , spite , f. , & straniero , o. 2001 , a&a , 369 , 87 , @xmath42 , temperature at the base of the surface convection zone ( @xmath44 ) , mass above the base of the surface convection zone @xmath68 and mass fraction of hydrogen ( @xmath69 ) at the center of stars of 0.5 to 1.0 @xmath3 with @xmath34 = 0.00017 , or @xmath70=-2.31.$ ] all models were calculated with atomic diffusion and radiative accelerations but no turbulent transport . in the black and white figure , are shown 6 of the 15 models that may be seen in the color figure available in the electronic version of the paper . ] with @xmath7=-2.31 $ ] at four epochs identified in the upper part of the figure . the vertical lines give the position of the bottom of the surface convection zone . for other stellar masses , it is the position of the bottom of the convection zone that is most different . the @xmath4 of the various species varies from star to star but much less than the depth of convection zones . gravity is shown in the lower right hand corner and repeated in each panel of the figure . ] with @xmath7=-2.31 $ ] . all models were calculated with atomic diffusion and radiative accelerations but no turbulent transport . in the black and white figure , are shown 6 of the 15 models that may be seen in the color figure available in the electronic version of the paper . as is evident in the color version , a vertical line drawn in each panel of the figure , corresponding to a fixed age , permits one to evaluate the range of surface abundances of a species at that age ( in , e. g. , a globular cluster ) . ] with @xmath7=-2.31 $ ] . the profiles are shown at four different ages , 6.1 , 9.1 , 10.3 and 11.4 gyr . the last is shortly before the star moves to the giant branch . calculations included atomic diffusion and radiative accelerations but no turbulent transport . ] with @xmath7=-2.31 $ ] over the inner 2/3 of the mass of the star at an age of 11.8 gyr . calculations included atomic diffusion and radiative accelerations but no turbulent transport . the zero abundance change occurs around 1/2 the mass of the star at @xmath71 . interior to that point , the abundance of metals is generally larger than the original abundance . the variations of cno are due to their transformation by the cno chain which would lead to a decrease of z at the center if it were not for the settling of the other metals . only mg , fe and ni are shown from the 16 species between ne and ni included in the calculations but all others have very similar behavior . ] models . the nearly horizontal line is the he atomic diffusion coefficient . the other lines are various parametrizations of turbulence . in the upper part of the figure , the corresponding li concentrations are shown , at an age of 10.2 gyr , with the same line identification to give the link between turbulent transport and li burning at @xmath39 . the parameters specifying turbulent transport coefficients are indicated in the name assigned to the model . for instance , in the t5.5d400 - 3 model , the turbulent diffusion coefficient , @xmath14 , is 400 times larger than the he atomic diffusion coefficient at @xmath15 and varying as @xmath16 . ] , @xmath42 , temperature at the base of the surface convection zone ( @xmath44 ) , mass at the base of the surface convection zone ( @xmath68 ) and mass fraction of hydrogen ( @xmath69 ) at the center and of the he , fe and li abundances in 0.8 @xmath3 models both with and without diffusion . see figure [ fig : coefficients ] and table [ tab : parameters ] for a definition of the notation used for models with turbulent transport . ] with @xmath7=-2.31 $ ] and turbulence parametrized by t6.09d400 - 3 and t5.5d400 - 3 . the model with no turbulence is also shown for comparison purposes . the profiles are shown at 11.7 gyr , shortly before the star moves to the giant branch . the same scale is used as for figure [ fig : intern_abundances ] . see figure [ fig : coefficients ] and table [ tab : parameters ] for a definition of the notation used for models with turbulent transport . ] model . the models differ by the boundary conditions and by the value of @xmath19 used in the calculations . see figure [ fig : coefficients ] and table [ tab : parameters ] for a definition of the notation used for models with turbulent transport . ] after 10 gyr of evolution . in the right hand panel , the 0.84 @xmath3 model is replaced by the 0.81 @xmath3 one and it is after 12 gyr of evolution.,title="fig:",scaledwidth=45.0% ] after 10 gyr of evolution . in the right hand panel , the 0.84 @xmath3 model is replaced by the 0.81 @xmath3 one and it is after 12 gyr of evolution.,title="fig:",scaledwidth=45.0% ] star , after 10/12 gyr of evolution for 3 turbulence strengths : atomic diffusion only , the t5.5 and t6.09 turbulence models . , title="fig:",scaledwidth=45.0% ] star , after 10/12 gyr of evolution for 3 turbulence strengths : atomic diffusion only , the t5.5 and t6.09 turbulence models . , title="fig:",scaledwidth=45.0% ] . the continuous line segments link models calculated with atomic diffusion . the dashed and dotted line segments link models calculated with atomic diffusion and , respectively , the t6.09 and t5.5 turbulence models . ] . the continuous line segments link models calculated with atomic diffusion . the dashed and dotted line segments link models calculated with atomic diffusion and , respectively , the t6.09 and t5.5 turbulence models . ] 6000 k , one should remember that , at a given mass and age , a star with the t6.09 turbulence model has a higher @xmath6 ( by some 100 ) than one with atomic diffusion only ( see fig . [ fig : with_without ] ) . references for the observed li abundances are identified on the figure . ] throughout the main sequence and subgiant evolution of 0.84 , 0.8 , 0.77 and 0.75 @xmath3 stars when atomic diffusion is included in the calculations . turbulence is assumed negligible . for the 0.84 @xmath3 star the main sequence starts at @xmath72k where @xmath73 . as evolution proceeds , the fe abundance decreases until an underabundance of @xmath550.4 dex is reached . at that point , the surface convection zone has retracted sufficiently for @xmath4 ( ) to be greater than gravity and the fe abundance increases up to a 1.7 dex overabundance . this occurs at the end of the main sequence evolution ( around the turnoff ) at @xmath74k . as evolution proceeds on the subgiant branch the surface convection zone gets deeper and the fe abundance decreases to 0.1 dex below the original abundance . the fe abundance goes back to its original value when a @xmath6 of 5300@xmath555400 k is reached . similar evolution occurs for the other stars except that it starts at different @xmath6 for each . the li abundance in the 0.84 @xmath3 star has a similar evolution . the li abundance decreases until an underabundance of @xmath551.2 dex is reached . at that point , the surface convection zone has retracted sufficiently for @xmath4 ( ) to be greater than gravity . however @xmath4 ( ) is larger than gravity over a smaller interval than @xmath4 ( ) . it leads to an increase of the li abundance at @xmath75k but to no overabundance . as the subgiant evolution starts , li gets below its main sequence abundance at @xmath76k , creating a loop in the li abundance curve . the li abundance never reaches its original abundance since , before this occurs , the surface convection zone approaches the region of li burning and the li abundance decreases rapidly below 5800 k . ] k for a 0.8 and a 0.78 @xmath3 star with atomic diffusion only . their ages bracket the age of m92 , 13.5 gyr . both stars have very closely the same composition showing that surface abundances are not a sensitive function of mass at @xmath77k , when the star is on the subgiant branch . a 0.78 @xmath3 star with the t6.09 turbulence model is also shown at the same @xmath6 . finally the 0.78 @xmath3 star with no turbulence is shown at 13.5 gyr when it is a turnoff star ( @xmath78k ) . , scaledwidth=60.0% ] lccc h & & 7.646@xmath79 & 7.613@xmath79 + @xmath80he & & 2.352@xmath79 & 2.370@xmath79 + @xmath81c & & 1.727@xmath82 & 1.727@xmath83 + n & & 5.294@xmath84 & 5.294@xmath82 + o & & 9.612@xmath82 & 9.612@xmath83 + ne & & 1.966@xmath82 & 1.966@xmath83 + na & & 3.986@xmath85 & 3.986@xmath84 + mg & & 7.484@xmath84 & 7.484@xmath82 + al & & 1.627@xmath85 & 1.627@xmath84 + si & & 8.072@xmath84 & 8.072@xmath82 + p & & 6.976@xmath86 & 6.976@xmath85 + s & & 4.215@xmath84 & 4.215@xmath82 + cl & & 8.969@xmath86 & 8.969@xmath85 + ar & & 1.076@xmath84 & 1.076@xmath82 + k & & 1.998@xmath86 & 1.998@xmath85 + ca & & 7.474@xmath85 & 7.474@xmath84 + ti & & 3.986@xmath86 & 3.986@xmath85 + cr & & 9.989@xmath86 & 9.989@xmath85 + mn & & 3.890@xmath86 & 3.890@xmath85 + fe & & 7.172@xmath84 & 7.172@xmath82 + ni & & 4.445@xmath85 & 4.445@xmath84 + z & & 1.675@xmath83 & 1.675@xmath87 [ tab : xinit ] lccccc non - diffusive & eddington & 1.687 & @xmath88 & no & no + diffusive & eddington & 1.687 & @xmath88 & yes & no + t5.5 & eddington & 1.687 & @xmath88 & yes & t5.5d400 - 3 + t6.0 & eddington & 1.687 & @xmath88 & yes & t6.0d400 - 3 + t6.09 & eddington & 1.687 & @xmath88 & yes & t6.09d400 - 3 + t6.13 & eddington & 1.687 & @xmath88 & yes & t6.13d400 - 3 + t6.09ks & krishna - swamy & 1.869 & @xmath88 & yes & t6.09d400 - 3 + t6.09ks@xmath19 & krishna - swamy & 2.017 & @xmath88 & yes & t6.09d400 - 3 + non - diffusive@xmath89 & eddington & 1.687 & @xmath89 & no & no + diffusive@xmath89 & eddington & 1.687 & @xmath89 & yes & no + t6.0@xmath89 & eddington & 1.687 & @xmath89 & yes & t6.00d400 - 3 + t6.09@xmath89 & eddington & 1.687 & @xmath89 & yes & t6.09d400 - 3 + [ tab : parameters ]

evolutionary models have been calculated for pop ii stars of 0.5 to 1.0@xmath0 from the pre - main sequence to the lower part of the giant branch . rosseland opacities and radiative accelerations were calculated taking into account the concentration variations of 28 chemical species , including all species contributing to rosseland opacities in the opal tables . the effects of radiative accelerations , thermal diffusion and gravitational settling are included . while models were calculated both for @xmath1 and @xmath2 , we concentrate on models with @xmath1 in this paper . these are the first pop ii models calculated taking radiative acceleration into account . it is shown that , at least in a 0.8 @xmath3 star , it is a better approximation not to let diffuse than to calculate its gravitational settling without including the effects of @xmath4(fe ) . in the absence of any turbulence outside of convection zones , the effects of atomic diffusion are large mainly for stars more massive than 0.7@xmath0 . overabundances are expected in _ some _ stars with @xmath5 . most chemical species heavier than cno are affected . at 12 gyr , overabundance factors may reach 10 in some cases ( e.g. for al or ni ) while others are limited to 3 ( e.g. for fe ) . the calculated surface abundances are compared to recent observations of abundances in globular clusters as well as to observations of li in halo stars . it is shown that , as in the case of pop i stars , additional turbulence appears to be present . series of models with different assumptions about the strength of turbulence were then calculated . one series minimizes the spread on the li plateau while another was chosen with turbulence similar to that present in amfm stars of pop i. even when turbulence is adjusted to minimize the reduction of li abundance , there remains a reduction by a factor of at least 1.6 from the original li abundance . independent of the degree of turbulence in the outer regions , gravitational settling of he in the central region reduces the lifetime of pop ii stars by 4 to 7 % depending on the criterion used . the effect on the age of the oldest clusters is discussed in an accompanying paper . just as in pop i stars where only a fraction of stars , such as amfm stars , have abundance anomalies , one should look for the possibility of abundance anomalies of metals in some pop ii turnoff stars , and not necessarily in all . expected abundance anomalies are calculated for 28 species and compared to observations of m92 as well as to li observations in halo field stars .

the process of elastic vector meson electroproduction on a nucleon , @xmath1 where @xmath2 , was studied in many fix target and in hera collider experiments . on the theoretical side , the large negative virtuality of the photon , @xmath3 , provides a hard scale for the process which justifies the application of qcd factorization methods that allow to separate the contributions to the amplitude coming from different scales . the factorization theorem @xcite states that in a scaling limit , @xmath4 and @xmath5 fixed , a vector meson is produced in the longitudinally polarized state by the longitudinally polarized photon and that the amplitude of the process ( [ process ] ) is given by a convolution of the perturbatively calculable hard - scattering amplitudes @xmath6 , the nonperturbative meson distribution amplitude ( da ) @xmath7 , and the generalized parton densities ( gpds ) @xmath8 . @xmath9 where @xmath10 is the skewness variable , @xmath11 and @xmath12 is a factorization scale . gpds encode important information on hadron structure , including aspects that can not be deduced directly from experiment , like the transverse spatial distribution of partons and their orbital angular momentum , for more details see @xcite . deeply virtual compton process ( dvcs ) provides the theoretically cleanest access to gpds . recently two - loop effects were incorporated into the analysis of dvcs @xcite . a theoretical description of exclusive meson production is more involved since it includes an additional nonperturbative quantity , a meson da . the primary motivation for the strong interest in this process ( and in the similar process of heavy quarkonium production ) is that it can serve to constrain the gluon density in a nucleon . indeed , in vector meson production case the gluon gpd enters the description already at the leading order ( lo ) in the strong coupling @xmath13 , whereas in dvcs it appears first at nlo , and , like in inclusive dis , is accessible only through scaling violation . the hard - scattering amplitudes for process ( [ process ] ) were calculated at nlo in @xcite , and for exclusive heavy quarkonium photoproduction in @xcite . the analysis of nlo effects showed that in kinematics typical for the hera collider experiments , @xmath14 , the nlo corrections are huge even for really large values of hard scales @xmath15 . if the factorization scale is chosen close to the value of a hard scale , @xmath16 , the corrections have opposite signs in comparison to the born term . which may lead to the change of signs of the imaginary and the real parts of the amplitude within phenomenologically relevant interval of @xmath17 . besides , the factorization and renormalization scale uncertainties were found being very large . recently these findings were confirmed in @xcite , where very detailed analysis of the cross sections and the transverse target polarization asymmetries in exclusive meson production was performed both for small and larger values of @xmath17 , typical for fixed - target experiments . for the fixed target kinematics it seems that nlo corrections start to be under control , though their values are still large at presently available values of @xmath18 . for the transverse target polarization asymmetries the situation is better , in some cases . going back to small @xmath17 , why nlo corrections are large in this case ? the inspection of nlo hard - scattering amplitudes shows that the imaginary part of the amplitude dominates and that the leading contribution to the nlo correction originates from the broad integration region @xmath19 , where the gluonic part approximates ( @xmath20 is a number of colors ) @xmath21 \ , . \label{appr}\ ] ] given the behavior of the gluon gpd at small @xmath22 , @xmath23 , we see that nlo correction is parametrically large , @xmath24 , and negative unless one chooses the value of the factorization scale sufficiently lower than the kinematic scale . for the asymptotic form of meson da , @xmath25 , the last term in ( [ appr ] ) changes the sign at @xmath26 , for the da with a more broad shape this happens at even lower values of @xmath12 . similar , @xmath27 enhanced , contribution appears also in the quark singlet channel . the partonic momentum fraction @xmath22 is related to the mandelstam energy variable @xmath28 of the partonic subprocess @xmath29 . the leading part of nlo partonic amplitude ( proper normalized ) grows as the first power of energy , @xmath30 , whereas at lo partonic amplitude behaves like a constant at large @xmath28 . the reason for this difference is the appearance , starting from nlo , of partonic diagrams with the gluon exchange in the @xmath31 channel , see fig . 1 . at lo one has only diagrams with the quark exchange , both for the gluon and quark channels . at higher orders the diagrams with gluon @xmath31 channel exchange give contributions to the amplitudes of partonic subprocesses enhanced , for @xmath32 loops , as @xmath33 . in its turn , these terms inserted in the factorization formula will produce large contributions @xmath34 to the process amplitude , where each power of the strong coupling is compensated by the same power of a large logarithm of energy . it is a natural idea to resum these enhanced at small @xmath17 contributions using the bfkl approach @xcite . the central point in this high energy resummation is to perform it consistently , without spoiling the all - order factorization of collinear singularities . a care should be taken of the factorization scheme used at the factorization of the process amplitude ( [ factor ] ) in terms of gpds and hard - scattering amplitudes . the higher order terms of the hard - scattering amplitudes derived within the high energy approximation ( bfkl approach ) can be supplemented by the knowledge of hard - scattering amplitudes calculated exactly at fixed order . then , one can use them together in factorization formula ( [ factor ] ) without double counting . for inclusive hard processes , heavy quark production and dis , the method of such high energy resummation was elaborated in @xcite . it is based on curci , furmanski and petronzio approach @xcite to separation of collinear singularities . the amplitudes on a parton ( quark , gluon ) target are considered in @xmath35 non - integer dimentions . that separates automatically the leading twist . collinear singularities appear in this approach as @xmath36 poles , these poles are absorbed into a definition of parton densities . another essential ingredients of @xcite method is an analysis of mellin moments , high - energy terms in mellin moment space @xmath37 look like singularities @xmath38 at @xmath39 , and a consideration of bfkl equation in @xmath35 dimensions . we found that this technique may be directly generalized on the analysis of exclusive non - forward reactions . below we present the first results of this study . like in dis the imaginary part of the a amplitude is given by the sum of quark singlet and gluon contributions @xmath40 \ , .\ ] ] @xmath41 and @xmath42 are the imaginary parts of the quark and gluon hard - scattering amplitudes . in difference to forward dis case the parton densities in ( [ ampl ] ) depend on both longitudinal momentum fractions . due to that the mellin moments of the amplitude do not factorize into the product of the moments @xmath43 \ , . & \nonumber\end{aligned}\ ] ] using polynomiality property of gpds , in particular for the gluon case @xmath44 one can show that for the integer odd @xmath37 @xmath45 \ , .\ ] ] which is a sum of moment products ( not just a product , as in dis case ) . one can analytically continue ( [ dn ] ) from the integer odd @xmath37 into entire complex @xmath37 plane . the high energy asymptotic of the amplitude is related with the behavior of @xmath46 near unphysical point @xmath39 . one can split ( [ dn ] ) into a sum of the singular and the regular at @xmath39 parts @xmath47 the singularities of the sum ( [ dn ] ) at @xmath39 are due to @xmath48 term only . therefore @xmath49 note that at @xmath50 , @xmath51 @xmath52 reduce to the moments of usual parton densities @xmath53 this consideration shows that a non - forward nature of hard exclusive reactions does not complicate much their analysis in the high energy limit . therefore the method used in dis @xcite may be applied here . the difference between dis and our case is in the different form of @xmath54 dependent amplitudes for corresponding partonic subprocesses . below i will concentrate on the dominant at high energy gluon contribution . the results will be presented for the process ( [ process ] ) ( assuming for simplicity the asymptotic form of meson da ) and for the process of heavy quarkonium electroproduction ( where the formation of quarkonium is treated in nrqcd ) . the amplitude is presented as follows @xmath55 here @xmath56 , we omitted normalization factors irrelevant for the subsequent discussion , in the r.h.s @xmath57 represents the born contribution and the sum stands for the high energy terms . @xmath58 are the polynomials of variable @xmath59 which we need to calculate . note that the born term belongs to the regular part ( in terms of ( [ dn ] ) ) , whereas the high energy terms behave as @xmath60 at @xmath39 . therefore in the high energy terms one can replace gluon gpd in ( [ forcalc ] ) by its forward limit , @xmath61 , but in the born contribution @xmath57 should be kept different from @xmath62 . omitting all details of the derivation we just present the results . we work in @xmath63 scheme . we define ( properly normalized ) @xmath54 dependent amplitude of the gluon subprocess @xmath64 then we calculate its mellin transform @xmath65\gamma[1-\gamma]}{\gamma[2 + 2\gamma]}\ , .\ ] ] the high energy terms are defined from the expression @xmath66 the gluon anomalous dimension is determined by the solution of equation @xmath67 , where @xmath68 is the bfkl eigenfunction , function @xmath69 depends on @xmath70 and is defined in @xcite . expanding @xmath71 in the series of variable @xmath72 one can obtain analytical expressions for the polynomials @xmath58 . below we illustrate the values of these polynomials for the case @xmath73 @xmath74 here the first two lines represent results for the exclusive light vector meson and quarkonium production respectively , in the third line we show for comparison the results for longitudinal dis structure function @xcite . we see that numerical values of @xmath75 are negative in all case , but for the exclusive reactions its absolute values are about @xmath76 times larger then in the case of @xmath77 , explaining very large negative nlo corrections found for exclusive meson production . on the other hand , the values of the second polynomial are positive and large , @xmath78 for the light vector meson production . this gives a hope that inclusion of these high energy terms in the analysis may stabilize predictions for exclusive meson production . to investigate this possibility we perform the following numerical study . we calculate the amplitude of light vector meson production with ( [ forcalc ] ) , where in the high energy terms we use very simple input for the gluon density @xmath79 , for the born term we take @xmath80 . definitely , more realistic input for gluon gpd should be used ( especially for @xmath57 ) , but at the present stage we just want to clarify the qualitative role of the high energy terms . in fig . 2 we present the energy dependence of the amplitude ( in arbitrary units ) calculated for two values of photon virtuality @xmath81 and @xmath82 , for the running coupling we use @xmath83 and @xmath84 , and for the factorization scale @xmath85 . the solid line on fig . 2 represents the born contribution , the dashed line the born @xmath86 the first high energy term , the dotted line the born @xmath86 2 first high energy terms , the dashed - dotted the born @xmath86 6 first high energy terms . [ cols="^,^ " , ] the dependence of the amplitude on the choice of factorization scale is shown in fig . , the dashed lines correspond to the born @xmath86 the first high energy term , the dashed - dotted lines the born @xmath86 6 first high energy terms . the upper dashed and dashed - dotted lines are for @xmath87 , the lower dashed and dashed - dotted lines are for @xmath85 . we observe sizable reduction of the factorization scale dependence if the high energy terms are resummed in comparison to the case when only the first of these terms is taken into account . large nlo corrections are found for hard exclusive vector meson production . at intermediate to larger values of @xmath17 , typical for fixed - target experiments , it seems that nlo corrections start to be under control for the large values of @xmath18 , say above @xmath88 . however , the situation is much worse for the region of small @xmath17 , typical for the hera collider experiments . here nlo corrections are not under control even for such large values of hard scales as @xmath89 , which prevents the interpretation of the precise hera data in terms of gpds . the problem is related to appearance of bfkl type logarithms in the hard - scattering amplitudes , that calls for a resummation of these effects at higher orders . here we present the first results for such study . the methods used earlier for forward dis process may be generalized to the case of nonforward hard exclusive reactions . we obtained analitical results for the corresponding high energy terms in ( [ forcalc ] ) . the first numerical calculation incorporating the high energy resummation is encouraging . _ i am very grateful to the organizers and to alexander von humboldt foundation for the support of my participation in eds07 conference . this work is also sponsored in part by grants rfbr-06 - 02 - 16064 and nsh 5362.2006.2 . _

we discuss exclusive vector meson electroproduction within the qcd collinear factorization framework . in bjorken kinematics the amplitude factorizes in a convolution of the nonperturbative meson distribution amplitude and the generalized parton densities with the perturbatively calculable hard - scattering amplitudes , which are presently known to next - to - leading order ( nlo ) . at small @xmath0 nlo corrections are very large . it is related to appearance of bfkl type logarithms in the hard - scattering amplitudes , that calls for a resummation of these effects at higher orders . here we report the first results of such resummation .

the nonstandard polymer quantization @xcite is a quantization procedure for mechanical models which implements some of the techniques of the loop quantization program @xcite . its main feature is a singular representation of the weyl algebra @xcite in a non - separable hilbert space and as a result , the canonical commutation relations @xcite can not be recovered by using the stone - von neumann theorems @xcite . polymer quantization can be applied to a broad range of mechanical systems related with atomic physics @xcite as well as in cosmology @xcite or in quantum field theory @xcite and statistical mechanics @xcite just to mention some examples . each of them serves as a toy model which provides specific hints about the procedure required to compare the non - regular and regular quantizations in the much more complicated context of quantum space - time . nevertheless , on its own it represents an interesting model to study from a mathematical physics perspective @xcite . the non - regularity of the weyl algebra representation avoids the definition of the momentum operator and makes impossible the usual dynamical characterization of the system . this obstacle is circumvented by proposing an operator that mimics the role of the absent operator @xcite . this artificial operator is formed as a combination of holonomies @xmath3 and depends on a length parameter @xmath4 . for the case of the polymer harmonic oscillator , the standard kinematical term @xmath5 is replaced by the operator @xmath6 , \label{kterm}\ ] ] where @xmath3 is the ` holonomy operator ' with a fixed value for the parameter @xmath4 . in the loop quantum gravity ( lqg ) scenario , the analogous of the parameter @xmath4 possesses a fixed value given by the planck length . for mechanical models , the nature of the parameter is ambiguous due to its non - fundamental characterization . however , its value is fixed although undetermined and can be seen as providing a measure of discretization of space . since no discreteness of space has been detected , the deviations of the polymer physical quantities from those observed within the regular quantization can not be experimentally tested . on the other hand , the physical quantities involved in polymer systems acquire corrections which are proportional to ( powers of ) @xmath4 . as a consequence , in mechanical systems @xmath4 must be much smaller than the natural length parameters associated with the system . the estimated value of the parameter @xmath4 for the free particle and the harmonic oscillator is about @xmath7 @xcite . the quantum corrections of these systems become significant in regimes where quantum mechanics would be inapplicable . hamiltonian commutation relations between the operator @xmath8 and an arbitrary holonomy @xmath9 are given by @xmath10 = - \alpha \widehat{v}_\alpha$ ] . they are irreducibly represented on the hilbert space @xmath11 which is given by @xmath12 . here , @xmath13 is the bohr compactification of the real line and @xmath14 is a regular haar measure on it @xcite . this hilbert space was built by mimicking the loop quantization program as was mentioned before @xcite . in this case , the representation of the operators @xmath8 and @xmath9 is @xmath15 where @xmath16 is an element of the uncountable basis on @xmath11 , i.e. , @xmath17 . a particular effect of the polymer construction coming as a result of the introduction of the aforementioned parameter is the modification of the dynamical equation of the quantum system . the hamiltonian of the system when using the kinetic term ( [ kterm ] ) takes the form @xmath18 + \frac{1}{2 } m \omega^2 \widehat{x}^2 . \label{phamiltonian}\ ] ] hamiltonian ( [ phamiltonian ] ) splits the full polymer hilbert space @xmath11 as an ( uncountable ) direct sum of separable hilbert spaces @xmath19 of the form @xmath20 or in the notation used in @xcite , @xmath21 where @xmath22 is the countable measure on the interval @xmath23 . consequently , the non - separability of the hilbert space @xmath11 renders discrete the nature of the parameter @xmath2 . each @xmath19 is given by @xmath24 and their observables algebra is now generated by @xmath8 and @xmath25 where @xmath26 . as a result , these hilbert spaces @xmath19 , each of them labeled by @xmath2 , are superselected . notice that the spatial scale given by @xmath4 naturally induces a scale with momentum dimensions given by @xmath27 the schrdinger equation for a given state @xmath28 takes the form @xmath29 \psi^{(\lambda)}(p ) = e^{(\lambda ) } \psi^{(\lambda)}(p ) , \label{schrp}\ ] ] with the additional condition @xmath30 in order to properly generate states in the entire polymer hilbert space using states of each superselected sector . observe that when @xmath31 in ( [ bcondition ] ) , then the wave functions @xmath32 are @xmath33 periodic functions . on the other hand , the parameter @xmath2 in ( [ bcondition ] ) can be seen as a fixed dual bloch wave vector . the equation ( [ schrp ] ) can be written into the more familiar form of the mathieu equation @xcite by using a dimensionless variable @xmath34 within the interval @xmath35 as @xmath36 where the parameters @xmath37 and @xmath38 in ( [ mequation ] ) are given by @xmath39 the spectrum of the eq . ( [ mequation ] ) is given by a series derived with a recursive method and the appropriate boundary conditions @xcite . consequently , the expression for any of the eigenvalues of the mathieu equation are very difficult to handle at analytical level , and hence , only its approximate expressions can be used . if we consider periodic solutions ( i.e. , @xmath40 ) then the spectrum of the mathieu equation @xcite given by ( [ mequation ] ) tends to the spectrum of the quantum harmonic oscillator in the formal limit @xmath41 ( figure [ eigenvalues ] ) . in this limit , the known results of the standard theory of the quantum harmonic oscillator are recovered as we already mentioned . for cases in which @xmath42 , the equation can be solved by using floquet theory for non - periodic mathieu functions @xcite . in terms of the quotient @xmath43.,width=302 ] as can be seen , an interesting result of the replacement of the kinetic term given in ( [ kterm ] ) is that in the momentum representation , this term turns out to be a periodic potential , in sharp contrast to the quantum harmonic oscillator in which this property is absent . nevertheless , it also renders periodic the semiclassical description , via path integral formulation @xcite which is , in this sense , a desired result . this feature results from the fact that ( [ kterm ] ) is a particular case of an almost periodic schrdinger operator @xcite i.e. , operators formed by a finite sum of holonomies . recently , barbero et al . @xcite showed that the energy spectrum of ( [ phamiltonian ] ) consists of an uncountable number of eigenvalues grouped in bands , very much like those appearing in the study of periodic potentials in standard quantum mechanics . the main difference is that the polymer energy spectrum is entirely discrete although grouped in bands whereas in the standard periodic potentials is continuum and grouped in bands . this peculiarity of the spectrum creates some difficulties in the development of the formalism both to define unitary evolution and to build suitable statistical ensembles . it also emerges both , at the full loop theory and , in its symmetry reduced quantum mechanical versions @xcite . however , some proposals for constructing a separable hilbert space are recently given for the case of quantum mechanical models @xcite . they essentially propose replacing the countable measure @xmath22 by a lebesgue measure @xmath44 on the same interval . a key point in barbero s results is the burnat - shubin - herczyski theorem @xcite . this theorem establishes that the spectrum of an almost periodic operator is the same no matter if the representation is regular or singular . representations only modify the nature of the spectrum of almost periodic schrdinger operators in that if it is discrete or continuous . in the case of the hamiltonian ( [ phamiltonian ] ) , its spectrum is identical to that of the quantum pendulum in the standard representation @xcite . periodic potential systems are very familiar in standard quantum mechanics , particularly in solid state physics @xcite . among other features , these potentials have an infinite set of oscillators - like eigenstates only when barrier penetration is ignored . otherwise , tunneling effect between each of the ` classical vacuums ' gives rise to the band structure of the spectrum . in standard quantum mechanics , the tunneling effect can be realized by instanton and anti - instanton pseudo - particles in the long imaginary time regime @xcite although it may be obtained within the deep quantum regime @xcite . instanton methods are a non - perturbative technique developed for handling phenomena in quantum field theory , and in particular in quantum chromodynamics , analogous to barrier penetration in quantum particle mechanics . in the polymer quantization , the tunnel effect can also yields the band structure of the spectrum indicated by barbero et al . in this scenario , the long time behavior of the penetration amplitude of the polymer harmonic oscillator by using instanton methods can be considered . in this case , each well at @xmath45 entails a classical vacuum from which the tunneling effect can occur in analogy to the result in the standard quantum pendulum @xcite . having said this , our goal here is to determine the penetration amplitude of the tunneling effect on the polymer harmonic oscillator between the minima in the entire polymer hilbert space and to obtain the band structure of the minimum of the spectrum . for this purpose , we are going to apply instanton methods to the polymer harmonic oscillator . it is worth mentioning that the mathematical structure of the polymer harmonic oscillator ( and other polymer quantum systems ) is very much similar to that of the lattice quantization of such systems @xcite . notice for example that the super - selected hilbert space @xmath19 is the same hilbert space used in the lattice quantization of the harmonic oscillator @xcite . however , although they coincides in this hilbert space , in both schemes there are conceptual aspects which makes them considerably different . for example , the polymer quantum mechanics ( pqm ) , as mimicking the lqg procedure , can be seen as a result of the gns - construction of the weyl algebra with a non - regular positive linear functional @xcite . the discreteness of the space in pqm is a result of such positive linear functional ( see @xcite for details ) and there is not a preferred lattice or so involved at this point . the lattice - type similarity emerges when the fictitious operator @xmath46 given in ( [ kterm ] ) is invoked . as a result , the fundamental discreteness of the space is unveiled by the parameter @xmath4 . for this reason , the limit @xmath41 can be seen as a formal mathematical trick in order to compare the physical quantities within the polymer description with their similar in the standard quantization . on the other hand , lattice quantum mechanics is a generalization of the standard quantization techniques on a discrete space . its powerful cousin , the lattice quantum field theory , provides remarkable results in removing the awkward divergences of the quantum field theory within a continuous space . such results has proven to be particularly useful in quantum chromodynamics @xcite . however , in these cases , the lattice structure together with the discreteness of the space are fixed by hand , that is to say , the lattice is not a fundamental description of the space . in order to recover the physical description of the system , the limit when the lattice parameter tends to zero is the ultimate step . summarizing , both schemes coincide mathematically once the super - selected sector in polymer quantum mechanics is fixed . the differences between both quantum models are twofold : the non - separability of the hilbert space which is a distinguishing feature of the polymer quantum mechanics , whereas in lattice quantum mechanics the hilbert space is separable . secondly , the limit of the lattice parameter @xmath41 , which is a mathematical trick in polymer quantum mechanics , is the final procedure in lattice quantization . in this spirit , the present work is directed to establish a link between the known results of lattice quantum mechanics and instanton methods applied to the quantum pendulum and the polymer quantization of the harmonic oscillator . the discrete nature of the parameter @xmath2 in these results offers new insights for instance in the statistical mechanics of the polymer harmonic oscillator and in the polymer quantization of the scalar field as we will discuss in the last section . the paper is organized as follows . in section [ pho ] we express the transition amplitude of the polymer harmonic oscillator in terms of the amplitudes of non - restricted regular hilbert spaces . in section [ instantonsol ] we obtain the instanton solutions and discuss its main properties . the analysis of the quantum fluctuation of the instanton solution together with the penetration barrier amplitude calculation is reported in section [ qfinst ] . finally in section [ discussion ] we discuss our results . let us consider an arbitrary state @xmath47 . this state can be decomposed as @xmath48 , where @xmath28 . as we already mentioned , the hilbert spaces @xmath19 are superselected and therefore , the hamiltonian @xmath49 given in ( [ phamiltonian ] ) acts on this state as @xmath50 and notice that the hamiltonian operator moves inside the summation on the right hand side . as a consequence , the evolution operator of a given state @xmath51 takes the form @xmath52 where the hamiltonian operator is now understood as the one acting on the hilbert space @xmath19 . from now on let us fix the value of @xmath2 and let us make all the calculations on this specific hilbert space @xmath19 . the symbol @xmath2 will be omitted for simplicity unless it is required to avoid confusion . the momentum variable @xmath53 in the quantum configuration space of @xmath19 is confined to the interval @xmath54 and this is a topological constraint @xcite . in this section , we obtain an expression ( [ amplitude ] ) in terms of the amplitude of a system without the topological constraint , i.e. , a system in which the momentum variable @xmath55 . to do so , let us follow the procedure given in @xcite for these systems . first , let us decompose the time interval @xmath56 given in ( [ amplitude ] ) into @xmath57 pieces @xmath58 , where @xmath59 and @xmath60 . the exponential in ( [ amplitude ] ) can now be written as @xmath57 products of infinitesimal time interval exponentials @xmath61 in the polymer construction , within the @xmath62polarization , the completeness relation of the momentum basis @xcite is of the form @xmath63 notice that the integration must end at an infinitesimal piece below @xmath64 to avoid the double - counting of contributions of the identical points @xmath65 and @xmath66 . we now introduce ( [ completep ] ) on each product and obtain @xmath67 \prod^{n+1}_{j=1 } \langle p_j , t_j | p_{j-1 } , t_{j - i } \rangle , \label{tdecomp}\ ] ] where @xmath68 and @xmath69 . this expression allow us to derive separately the infinitesimal amplitudes @xmath70 which can be written as @xmath71 and in order to calculate ( [ finfiam ] ) we consider apart the zeroth hamiltonian infinitesimal amplitude @xmath72 in an unconstrained system , the right hand side of the previous expression is the dirac delta orthonormality condition of the basis elements @xmath73 , that is to say , @xmath74 . in the present case , we can not use this relation because is not well defined . the reason is that it gives rise to a boundary condition different to that in ( [ bcondition ] ) . for our case , the adequate orthonormality condition in @xmath19 reads as @xmath75 to verify that we recover the relation ( [ bcondition ] ) from the relation ( [ driacdin ] ) , let us add @xmath76 to the @xmath53 variable on both sides of ( [ driacdin ] ) . after some algebraic manipulations we obtain @xmath77 and now , removing @xmath78 from both sides and multiplying by @xmath79 we recover ( [ bcondition ] ) . using ( [ driacdin ] ) the zeroth hamiltonian amplitude in ( [ samplitude ] ) can be written as @xmath80 where the dirac deltas inside the summation in the first line are written in terms of its fourier transforms in the second line . we now use the spectral decomposition formula @xcite to obtain the infinitesimal amplitude for the hamiltonian @xmath81 @xmath82 the hamiltonian @xmath83 is given by @xmath84 where @xmath85 . after inserting the amplitude ( [ infinamplitude ] ) in the expression ( [ tdecomp ] ) we arrived at the following expression @xmath86.\end{aligned}\ ] ] on each of the @xmath87 integrals @xmath88 , the momenta variables @xmath89 can be redefined by allowing them to go to a broader range @xmath90 and absorbing one particular summation ( see the appendix [ pathsum ] for details ) . as a result , @xmath87 summations will be absorbed by @xmath87 integrations and therefore , there is only one summation left in this procedure . the total amplitude takes the following form @xmath91 \exp\left\ { \frac{i \epsilon } { \hbar } \sum^{n+1}_{j=1 } \left [ - \frac{2 p^2_c } { m } \sin^2(\frac{\tilde{p}_j}{2 p_c } ) + \right . \nonumber \\ & & \left . + \frac{1}{2k } \left ( \frac { \tilde{p}_j - \tilde{p}_{j-1 } + 2 \pi p_c l \delta_{j , n+1 } } { \epsilon } \right)^2 \right ] \right\ } . \label{amplitudepoly}\end{aligned}\ ] ] the potential @xmath92 is indeed periodic and invariant under this redefinition of the variables @xmath93 . notice that for each value @xmath94 we obtain an amplitude @xmath95 e^ { \frac{i \epsilon } { \hbar } \sum^{n+1}_{j=1}\left [ \frac{1}{2k } \left ( \frac { \tilde{p}_j - \tilde{p}_{j-1 } } { \epsilon } \right)^2 - v(p_j ) \right ] } , \nonumber \\ & = & { \cal n } \int^{p_f + 2 \pi p_c l}_{p_i } \frac{{\cal d } p}{2 \pi p_c}\ , e^ { \frac{i } { \hbar } \int^{t_f}_{t_i } dt \left [ \frac{1}{2k } \dot{p}^2 - v(p ) \right ] } , \label{plpendulo}\end{aligned}\ ] ] in agreement to the general derivation given in @xcite . the symbol @xmath96 in this amplitude indicates that the amplitude corresponds to that of a unconstrained topological system , i.e. , the domain of the momentum variable comprised the entire real line @xmath97 . finally , inserting ( [ plpendulo ] ) in ( [ amplitudepoly ] ) we obtain the amplitude of the polymer harmonic oscillator in the hilbert space @xmath19 @xmath98 from ( [ plpendulo ] ) we can extract out the effective action in momentum variables @xmath99 : = \int^{t_f}_{t_i } dt \left [ \frac{1}{2k } \dot{p}^2 - v(p ) \right ] , \label{paction}\ ] ] with momentum @xmath53 as the dynamical variable instead of the usual @xmath100 and the potential @xmath101 is given by @xmath102 clearly , this potential is periodic with period @xmath33 and its action ( [ paction ] ) corresponds with the action of a simple pendulum in the momentum space . the amplitudes appearing in the right hand side of ( [ polyamplitudesum ] ) can be regularized if we consider the amplitude of the standard harmonic oscillator @xmath103 e^ { \frac{i \epsilon } { \hbar } \sum^{n+1}_{j=1}\left [ \frac{1}{2k } \left ( \frac { \tilde{p}_j - \tilde{p}_{j-1 } } { \epsilon } \right)^2 - v_0(p_j ) \right ] } , \nonumber \\ & = & { \cal n}_0 \int^{p_f}_{p_i } \frac{{\cal d } p } { 2 \pi p_0 } \ , e^ { \frac{i } { \hbar } \int^{t_f}_{t_i } dt \left [ \frac{1}{2k } \dot{p}^2 - v_0(p ) \right ] } , \label{amplitudeho}\end{aligned}\ ] ] where the parameters @xmath104 and @xmath105 were introduced in order to make the measure of the integral dimensionless as well as the factor @xmath106 . in our next analysis we will fix the values of @xmath104 and @xmath105 in terms of @xmath4 and @xmath107 . in the amplitude ( [ amplitudeho ] ) , @xmath108 is the potential of the harmonic oscillator in momentum variables @xmath109 . let us multiply and divide by the standard harmonic oscillator amplitude , given in ( [ amplitudeho ] ) , each summation term on the right hand side of ( [ polyamplitudesum ] ) @xmath110 } } { 2 \pi p_c}}{{\cal n}_0 \int^{p_f } _ { p_i } \frac{{\cal d } p\ , e^ { \frac{i } { \hbar } \int^{t_f}_{t_i } dt \left [ \frac{1}{2k } \dot{p}^2 - v_0(p ) \right ] } } { 2 \pi p_0 } } . \end{aligned}\ ] ] in order to remove the divergence associated with the quotient @xmath111 , we fix the values of the parameters @xmath112 and @xmath113 . this yields @xmath114 and gives the amplitude the form @xmath115 } } { \int^{p_f}_{p_i } { \cal d } p \ , e^ { \frac{i } { \hbar } \int^{t_f}_{t_i } dt \left [ \frac{1}{2k } \dot{p}^2 - v_0(p ) \right ] } } . \label{regamplitude}\end{aligned}\ ] ] now that we have derived the penetration amplitude of the polymer harmonic oscillator in terms of the amplitude of the harmonic oscillator in momentum variables @xmath116 , the next step is to approximate the quotient of integrals in ( [ regamplitude ] ) . to do so , each action on ( [ regamplitude ] ) is expanded around their corresponding instanton solution @xmath117 . the expansion will be carried out up to second order in the quantum fluctuations @xmath118 @xcite . the purpose of the next section is the calculation of the instanton solution @xmath119 of the classical action ( [ paction ] ) . in this section , we study the instanton solution associated with the action ( [ paction ] ) . instantons are solutions of the classical equation of motion in imaginary time , i.e. , on equations written after a wick rotation of the time parameter @xmath120 . the wick rotation of the action ( [ paction ] ) gives the following euclidean action @xmath121 = \int^{\tau_f}_{\tau_i } d\tau \left [ \frac{{p'}^2}{2k } + \frac{2 p^2_c } { m } \sin^2\left ( \frac { p}{2 p_c } \right ) \right ] , \label{mlagrangian}\end{aligned}\ ] ] where @xmath122 . as we already notice , the potential @xmath101 is periodic with period @xmath33 and it is null on the points @xmath123 . see figure [ graphpotential ] . its domain comprises the entire real line as we are dealing with the amplitude without topological constraints given in ( [ regamplitude ] ) . each of these points represents an edge in which the instanton solution can be evaluated for their corresponding value . the potential expanded around @xmath124 is of the form @xmath125 and therefore , behaves as the kinetic energy term of the simple harmonic oscillator . this explains the selection of this system as a regulator in the previous section . to notice the presence of the valleys allowing the existence of instantons , width=377 ] the euler - lagrange equation derived from ( [ mlagrangian ] ) is given by @xmath126 and by defining the dimensionless variable @xmath127 the previous equation takes the familiar classical pendulum equation in imaginary time @xmath128 . the integration of equation ( [ pendulum ] ) gives @xmath129 where @xmath130 is the first integration constant . a requisite for a solution to be an instanton type solution is to yield a finite value for the euclidean action . this imposes the condition @xmath131 on the previous expression . let us denote by a capital @xmath117 the instanton solution with equation of the form @xmath132 selecting the positive root and integrating the equation ( [ pinst ] ) we obtain the instanton solution to be of the form @xmath133 , \label{solop}\ ] ] where @xmath134 is the other integration constant named center of the instanton . it is easy to check that this solution satisfies the equation ( [ ec1 ] ) . we also observe that in the limit when @xmath135 then @xmath136 and in the limit @xmath137 then @xmath138 . these results are adequate in the spirit that on these limits ( of large imaginary time ) the instanton arrives at the edges of the interval ( figure [ instantonpolimerico ] ) , recalling that the potential @xmath101 is such that @xmath139 . the negative root in equation ( [ pinst ] ) gives an anti - instanton solution . @xmath140 and @xmath141 contribute to the amplitude of the full quantum system when the other parts of the periodic potential are considered . the potential @xmath101 evaluated on this solution reads @xmath142 and notice that in the large time intervals @xmath143 the potential @xmath144 , i.e. , remains finite . this has the same effect on the action @xmath145 = \int^{+\infty}_{-\infty } d\tau \ , [ 2v(p^+ ) ] = \frac{4 p^2_c } { m \omega } \mbox{tanh } ( \omega \delta \tau)|^{+\infty}_{-\infty } = \frac{8 p^2_c } { m \omega } , \label{ivaction}\ ] ] and confirms that @xmath119 is in rigor an instanton . observe that the value of the action on this instanton tends to infinity when @xmath41 . this means that in this limit , there is no instanton - like solution . for a fixed but small @xmath4 , this finite value of the action evaluated on the solution @xmath119 is a direct consequence of the ` instantonic ' character of the solution . let us move to the euclidean hamiltonian analysis of this solution . consider the action ( [ mlagrangian ] ) and let us define the coordinate variable in euclidean time given by @xmath146 with this definition , the euclidean hamiltonian is given by @xmath147 , hence , the euclidean polymer hamiltonian takes the form @xmath148 = \frac{k x^2}{2 } - v(p).\ ] ] the hamiltonian evaluated on the instanton solution gives @xmath149 = \frac{k x^2}{2 } - v(p ) = 0,\ ] ] which can be interpreted as an euclidean classical trajectory with null energy . the explicit form of the instanton solution coordinate can be derived using ( [ cinstanton ] ) @xmath150 which peaked in the vicinity of @xmath151 as can be seen in figure ( [ xinstantonpolimerico ] ) . for an infinite time interval , the coordinate tends to zero in accordance to the momentum behavior . , width=302 ] with these expressions we can draw an instanton path in the euclidean phase space with cartesian coordinates @xmath152 \right).\ ] ] in the figure ( [ instantonps ] ) we draw the instanton in the phase space with coordinates @xmath153 . it can be seen that when @xmath154 the instanton starts at @xmath155 and when @xmath156 the instanton arrives at the point @xmath157 . and @xmath158,width=151 ] let us briefly study the instanton solution of the euclidean action of the harmonic oscillator in momentum variables . in this case , the action takes the form @xmath159,\ ] ] and the equation of motion is given by @xmath160 the general solution of this equation is @xmath161 where @xmath162 and @xmath163 are arbitrary coefficients . the action @xmath164 evaluated on @xmath165 in the finite time interval @xmath166 takes the following form @xmath167 = \frac{1}{2m\omega } \left [ c^2_0 e^{2\omega \delta \tau } + c^2_1 e^{-2\omega \delta \tau } \right]^{\tau=\tau_f}_{\tau=\tau_i}.\ ] ] if we consider the limit @xmath168 then only when @xmath169 the euclidean action @xmath170 $ ] is finite with value @xmath171 = 0 $ ] . this probes that in the case of the harmonic oscillator the only instanton solution is the trivial solution , i.e. , @xmath172 . we will see further that although the instanton solution is @xmath173 , the quantum contributions to the action @xmath171 $ ] around the @xmath165 are non - trivial and yields the appropriate regularization of the quotient in ( [ regamplitude ] ) . summarizing this section , we have obtained an instanton solution @xmath119 given in ( [ solop ] ) for the euclidean action ( [ mlagrangian ] ) . the euclidean action evaluated on this solution gives ( [ ivaction ] ) and the instanton fulfills the conditions @xmath174 and @xmath175 . for the harmonic oscillator we found that the instanton solution is @xmath173 and its action takes the value @xmath171 = 0 $ ] . with these results , we are ready to move to the next section and study the quantum fluctuation around these instantons solutions . with the results of the previous section let us return to the amplitude ( [ regamplitude ] ) and let us consider a wick rotation @xmath176 . now let us expand each euclidean action in this amplitude up to second order around their corresponding instanton solution @xmath119 and @xmath165 . the deviations of the trajectories are given as @xmath177 for the euclidean action @xmath178 and @xmath179 for @xmath164 . the amplitude ( [ regamplitude ] ) is now written as @xmath180 } \int { \cal d } \delta p \ , e^ { -\frac{1 } { \hbar } \int^{\tau_f}_{\tau_i } d\tau \left\ { \frac{\delta p}{2 } \left [ - \frac{1}{k } \frac{d^2}{d\tau^2 } + v''(p^+ ) \right ] \delta p+ { \cal o}(\delta p^3 ) \right\ } } } { e^{- \frac{1}{\hbar } s^{h}_e[p^h ] } \int { \cal d } \delta p^h \ , e^ { -\frac{1 } { \hbar } \int^{\tau_f}_{\tau_i } d\tau \,\frac{\delta p^h}{2 } \left [ -\frac{1}{k } \frac{d^2}{d\tau^2 } + \frac{1}{m } \right]\delta p^h } } . \label{ampcla}\end{aligned}\ ] ] where the potential @xmath181 is given by @xmath182 . \label{potentialpho}\ ] ] the zeroth order in this expansion gives the classical values of the euclidean action on the instanton solutions whose values were determined in the previous section . discarding the third order terms @xmath183 , the remaining integrals are gaussian - type integrals in the variables @xmath118 and @xmath184 . to solve them , we first propose that the operators @xmath185 and @xmath186 can be diagonalized @xcite . their corresponding eigenvalues equations are @xmath187 \right ) f_n(\tau ) = \epsilon^{pho}_n f_n(\tau ) , \label{qiequation}\ ] ] for the polymer harmonic oscillator and @xmath188 for the harmonic oscillator . the eigenfunctions @xmath189 and @xmath190 are related with the deviations @xmath118 and @xmath184 as @xmath191 and we assume in this notation that the spectrum is discrete . additionally , we require both systems of eigenfunctions @xmath192 and @xmath193 to be orthonormal in the integration interval @xmath166 . using these definitions , the integrations in the variables @xmath194 and @xmath195 in the amplitude ( [ ampcla ] ) give the following expression for the amplitude @xmath196 } } { e^{- \frac{1}{\hbar } s^{h}_e[p^h ] } } \prod^{+\infty}_{n=0 } \left ( \frac{\epsilon^{pho}_n}{\epsilon^{h}_n } \right)^{- \frac{1}{2}}. \label{pampli}\end{aligned}\ ] ] the eigenvalue problem of the equation ( [ qiequationho ] ) is easily solved . it is a schrdinger - type equation for a particle in a constant potential . in order to obtain a discrete spectrum , we impose the ` temporal box ' boundary conditions , which in this case reads as @xmath197 . the ` length ' of the box is @xmath198 and the eigenvalues are @xmath199 the solution of the eigenvalue problem ( [ qiequation ] ) is a bit more complicated due to the potential ( [ potentialpho ] ) . equation ( [ qiequation ] ) can be written into a more familiar form by giving it a schrdinger type form @xmath200 f_n(\tau ) = 0 , \label{polyinst}\ ] ] where the quantum potential @xmath201 reads as @xmath202 and its graph is given in figure [ gpotencialq ] . remarkably , the form of this potential for the quantum fluctuation is similar to that appearing in the system with the double - well potential @xmath203 @xcite . in this case , the potential of the quantum fluctuation is @xmath204 . as can be seen , it changes by two numerical factors but , as we will see further , this small change is sufficient to modify the discrete spectrum . the spectrum for the potential @xmath201 will be discrete if @xmath205 is such that @xmath206 and will be continuous if @xmath207 . in the appendix [ sqfeq ] we summarize the calculation of the solution to the equation ( [ qiequation ] ) together with the analysis of its eigenvalues . associated to the quantum fluctuation @xmath118,width=377 ] we obtain that the spectrum in the case of the equation ( [ qiequation ] ) is @xmath208 . where the discrete part @xmath209 contains only the zeroth mode contribution @xmath210 and the continuous part @xmath211 is given by the expression ( [ qfmomentum ] ) and is labelled by a continuous variable @xmath212 . the continuous part of the spectrum , @xmath211 , becomes discrete after imposing the ` temporal box ' boundary conditions into its eigenfunctions and the relation ( [ qfmomentum ] ) can be written as @xmath213 where @xmath214 is the relative phase of the eigenfunctions at infinity imaginary time . the zeroth mode @xmath215 must be treated apart because the integration of the variable @xmath216 diverges in the interval @xmath217 . the standard treatment of this mode requires a relation between the variable @xmath162 and the center of the instanton @xmath134 @xcite . in our case , the relation is given by @xmath218}{2\pi\ , \hbar } } \omega\ , d\tau_c , \label{cpint}\ ] ] where the expression ( [ instzeroth ] ) of the appendix ( [ sqfeq ] ) was used . once we have the spectrum of both equations ( [ qiequation ] ) and ( [ qiequationho ] ) we are ready to determine the penetration amplitude given in ( [ pampli ] ) . first of all , recall that within the interval @xmath219 the quantum potential of the polymer harmonic oscillator is null at the point @xmath220 . this implies that the penetration amplitude given in ( [ polyamplitudesum ] ) must be calculated as @xmath221 hence , in order to determine @xmath222 , let us we first consider @xmath223 . in this case ( [ pampli ] ) takes the form @xmath224 } \frac{dc_0}{\sqrt{2\pi}}\prod^{+\infty}_{n=0 } \left ( \frac{\epsilon^{c}_n}{\epsilon^{h}_n } \right)^{- \frac{1}{2}},\ ] ] and notice that for the harmonic oscillator amplitude , the initial and final points are the same because the potential has only one minimum . let us denote by @xmath225 the product of the ratios of the eigenvalues @xmath226 and @xmath227 and notice it can be written in terms of the momenta @xmath228 and @xmath89 as @xmath229 where @xmath230 and @xmath231 . the difference in these momenta for high values of @xmath232 is approximately @xmath233 where in the last relation we inserted @xmath234 . the expression for @xmath225 using this approximation is given by @xmath235 let us now integrate the continuous part ( [ cpint ] ) @xmath236}{2\pi\ , \hbar } } \omega\ , d\tau_c = \sqrt { \frac{s^{pho}_e[p^+]}{2\pi\ , \hbar } } \omega\tau_0,\ ] ] and together with the result for @xmath225 , let us insert them in the amplitude ( [ pieceam ] ) @xmath237 where @xmath238 is the ` density of instantons ' @xcite @xmath239}{\pi \ , \hbar } } \ , \omega \tau_0\,e^{- \frac{s_e[p]}{\hbar}}. \label{rhozero}\ ] ] we now move to the long time regime which allow us the introduction of multiple instanton and anti - instanton solutions . this model is called ` dilute instanton gas approximation ' . in this scenario , the time @xmath232 is large enough to allow widely separated pseudo - particles ( instantons and anti - instantons ) fulfilling the boundary conditions @xcite . this feature leads to the band structure of the spectrum for the case of periodic potentials @xcite . in this context , let us consider the amplitude of a ` dilute instanton gas ' contribution from @xmath240 to the point @xmath241 @xmath242 where @xmath243 is the bessel function of the first kind @xcite . let us insert ( [ micont ] ) into ( [ famplitude ] ) and make explicit the dependence in the parameter @xmath2 @xmath244 \ } , \nonumber \\ & = & \langle 0 , + \frac{\tau_0}{2 } | 0 , - \frac{\tau_0}{2 } \rangle^{(h ) } e^ { 2 \rho(\tau_0 ) \cos(2 \pi \frac{\lambda}{\mu } ) } , \label{famppho}\end{aligned}\ ] ] where in the second line we use the property of the bessel functions @xmath245 finally , recall that in the long time regime ( @xmath246 ) , the euclidean amplitude of the harmonic oscillator can be approximated as @xmath247 where @xmath248 is the vacuum eigenstate of the system at @xmath124 and @xmath249 its eigenvalue . combining this result with ( [ famppho ] ) we obtain that the main contribution to the amplitude in the long time regime takes the form @xmath250 } .\ ] ] the energy inside the square brackets in the previous expression gives the energy of the system @xmath251 , \label{insteigenvalue}\ ] ] where @xmath252 is the characteristic length of the vacuum wavefunction of the standard harmonic oscillator . each eigenvalue @xmath253 , with @xmath254 , is an approximation , in the long time regime , of the zeroth eigenvalue of the hamiltonian ( [ phamiltonian ] ) in the hilbert space @xmath19 . the first point to be notice in the expression ( [ insteigenvalue ] ) is that it gives a simple and compact expression for the the lowest energy eigenvalues of the polymer harmonic oscillator . in order to compare the accuracy of this result , we graph the zeroth eigenvalue for the periodic mathieu equation , i.e. , without using the instanton approximation and @xmath255 given in ( [ insteigenvalue ] ) in terms of @xmath256 it can be seen in figure ( [ comparacion ] ) , that when @xmath257 the eigenvalue @xmath258 of the instanton description coincides with the exact numerical value @xmath249-num of the periodic mathieu solution . therefore , our approximation is valid in the interval @xmath259 . on the other hand , recall that in order to discard polymer effects ( or spatial discreteness ) on the quantum harmonic oscillator , then @xmath260 . if we consider the standard textbooks values for the harmonic oscillator parameters then @xmath261 . combining these values we obtain the condition @xmath262 @xcite . consequently , the instanton analysis can be very well used to described the physics of the polymer harmonic oscillator . a remarkable aspect of the expression ( [ insteigenvalue ] ) is that it shows the band structure mentioned by barbero et al . the minimum eigenvalue of the band is the one corresponding to @xmath40 while the supremum is the one corresponding to @xmath263 ( see figure ( [ gap ] ) ) . as function of the quotient @xmath264 with @xmath265.,width=377 ] any other eigenvalue in the band with different @xmath2 is doubly degenerate ( recall that @xmath266 is not allowed ) and their eigenstates are in the hilbert spaces given by @xmath2 and @xmath267 . these results are in complete agreement with the theory of periodic potentials given in @xcite and with the spectral analysis of almost periodic schrdinger operators @xcite . this double degeneracy of the spectrum ( for @xmath42 and @xmath268 ) implies that any given state @xmath269 within the first energy band can be written as @xmath270 where @xmath271 is the eigenstate in @xmath19 corresponding to the eigenvalue @xmath272 and the arbitrary constants @xmath273 are non - zero only at a countable set of @xmath2 values . the width of this energy band , named @xmath274 , is given by @xmath275 and if we consider @xmath276 , then the width is a very small quantity @xmath277 . a photon with this energy @xmath274 , emitted by a polymer oscillator with frequency @xmath278 has a wavelength which is a million times larger than the diameter of the visible universe . in other words , if we consider @xmath279 then the deviation of the @xmath272 from @xmath249 is not experimentally tested . let us conclude the analysis of the expression ( [ insteigenvalue ] ) by considering the formal limit @xmath280 . in this limit , all the energy eigenvalues take the form @xmath281 i.e. , the first energy band ` gets compressed ' to yield the single vacuum eigenvalue @xmath282 of the standard quantum harmonic oscillator . thus , in this limit , the eigenvalue @xmath283 of the standard quantum harmonic oscillator can be seen as a degenerate eigenvalue and its uncountable degeneracy is labelled by @xmath284 , as was pointed in @xcite . notice , however , that this degeneracy is only apparent : this limit is a mathematical trick and is used to provide a link between the standard quantization of the harmonic oscillator and its polymer version . polymer models allow us to gain understanding of some of the techniques used in the loop quantization program . in the case of known quantum systems , it is crucial to recover their experimental results . this fact can be observed within the formal limit @xmath285 on the polymer quantum harmonic oscillator . the parameter @xmath4 was introduced via the hamiltonian operator @xmath49 . such a hamiltonian allows the splitting of the polymer hilbert space @xmath286 where @xmath287 and @xmath22 is a countable measure on the set @xmath288 . the referred formal limit is usually taken in the super selected hilbert space with @xmath40 . the analysis in the full polymer hilbert space was carried out by barbero et al . @xcite . of particular relevance on barbero s derivation is the pure point spectrum nature of the polymer hamiltonian , @xmath289 $ ] , as a result of the non - regular representation . this is a typical feature of non - regular representations of almost periodic operators @xcite and it is connected to the non - separability of the polymer hilbert space . as was pointed out in @xcite , this feature of the hilbert space renders difficult the analysis of the statistical mechanics of such systems . essentially , the cardinality of the spectrum of the hamiltonian operator @xmath49 in the entire hilbert space @xmath11 is @xmath290 . this implies the partition function @xmath291 to be infinity and therefore , the thermal density matrix @xmath292 is ill - defined . the pure point spectrum of the hamiltonian is present in the band structure which appears as a consequence of the periodicity of the polymer hamiltonian . in standard quantum mechanics for periodic potentials the bands correspond to the continuum spectrum and can be studied as a tunneling effect carried out by pseudo - particles named instantons . motivated by this , and inspired by @xcite , the purpose of this work was to establish a connection between the standard results of quantum mechanics for periodic potentials and lattice quantum mechanics , with those of polymer quantum mechanics . particularly , we payed attention to the instanton methods in order to obtain similar and additional conclusions to those in @xcite although by different ways . to accomplish our task , we first calculated the renormalized propagator of the polymer harmonic oscillator . the ` superselected nature ' of the polymer hilbert spaces @xmath19 allows us to treat the propagator on each of the hilbert spaces separately . the momentum variables within these @xmath19 are topologically constrained . the techniques developed in @xcite were used to achieve the result ( [ polyamplitudesum ] ) together with ( [ regamplitude ] ) . the semiclassical potential ( [ polypot ] ) yields nonlinear equations of motion which brings into consideration , the applicability of the instanton methods . instantons in quantum mechanics are solutions of the hamilton equations in imaginary time . @xmath119 given in ( [ solop ] ) is the instanton of the polymer harmonic oscillator and it renders the finite value of the euclidean action , given in ( [ ivaction ] ) . the quantum fluctuation around @xmath119 is used in order to obtain the quotient of the amplitudes appearing in the renormalized propagator ( [ ampcla ] ) . the dynamic of such fluctuations is ruled by the schrdinger type equation ( [ qiequation ] ) or by its simplified version ( [ polyinst ] ) . the final amplitude is given in ( [ famppho ] ) . the vacuum energy for long ( imaginary ) times can be derived from ( [ famppho ] ) as is given by ( [ insteigenvalue ] ) . notably , vacuum energy depends on @xmath2 due to our calculations were done for a fixed value of @xmath2 . that is to say , we derived the energy eigenvalues within the first allowed energy band of the polymer harmonic oscillator . naturally , with this result we obtain the width of the first allowed band as can be noticed in ( [ bandstructure ] ) . there are some worth mentioning aspects of this band structure that we have obtained . first , it shows directly the point spectrum that was mentioned in barbero s work @xcite . point spectrum here is referred to the fact that the parameter @xmath293 . secondly , in the limit @xmath280 the band width gives rise to the zeroth energy eigenvalue of the standard quantum harmonic oscillator as can be seen in ( [ limit ] ) . this is a particularly interesting outcome which implies that the gap function in figure ( [ gap ] ) ` contracts ' to a point . as a result , emerges an apparent uncountable infinite degeneracy of the eigenvalue @xmath249 . if this analysis is expanded to the other bands , then it will imply that the effective degeneracy of each eigenvalue of the standard harmonic oscillator is again uncountable infinite as was already pointed out in @xcite . when the limit is considered in the polymer amplitude , then the green s function of the standard quantum harmonic oscillator is recovered . this result is independent of the parameter @xmath2 , which is a desired result . in the third place , the band structure is fully consistent with the floquet theory of quantum periodic potentials @xcite . a key point in this aspect is the adequate degeneracy of the zeroth eigenvalues @xcite of the polymer harmonic oscillator . the lowest eigenvalue corresponds to @xmath40 and the highest to @xmath268 . the other eigenvalues are doubly degenerated due to @xmath294 . we compared the zeroth eigenvalue with the exact ( numerical ) solution of the periodic mathieu equation . when @xmath295 , then the eigenvalue obtained with the instanton methods fits the exact ( numerical ) solution . now , recall that , as we mentioned in the introduction , when the parameter @xmath4 is much more smaller than the characteristic length of the standard quantum harmonic oscillator @xmath1 , the mean values of the polymer version of the observables can not be separated of the mean values of the observables within the standard quantization . this will occur only in case that @xmath296 @xcite . this implies that the instanton methods offers a description of the polymer harmonic oscillator which can be used within the interval @xmath297 . of course , when @xmath296 , polymer description is no longer required : the standard harmonic oscillator should be used . when @xmath298 , then instanton methods fail . it would be interesting to understand if these tools can be applied to quantum cosmology . particularly , due to the possibility to derive simple and compact expressions analog to ( [ insteigenvalue ] ) for the energy eigenvalues or other physical quantities . finally , recall that if we are attending a process which is particular to a given super - selected hilbert space , then it is suffice to use lattice quantum mechanics but , if on the other hand , our interest requires the dynamics on the full polymer hilbert space @xmath11 , then lattice quantum mechanics is not enough . an additional physical criterion is required to solve the pathological situation explained above ( for instance in the case of the partition function @xmath299 of the polymer harmonic oscillator ) and such that allows the elimination of most of the eigenstates of the hamiltonian , just leaving a countable number of them . here we present a different scenario in which the non - separability of the hilbert space of the polymer harmonic oscillator plays a non - trivial role . consider for instance the polymer ( fourier ) quantization of the real scalar field given in @xcite . in this model , the quantum harmonic oscillator of each fourier mode @xmath300 of the free scalar field in a flat minkowski spacetime is replaced by its polymer analog , i.e. , by a polymer harmonic oscillator . formally , the hilbert space of this quantum field theory can be written as @xmath301 , where @xmath302 is the polymer hilbert space with frequency @xmath303 and @xmath304 is the mass of the free scalar field . a one particle state of the field is given by @xmath305 where @xmath306 labels the allowed band energy and @xmath307 parametrized the state in the band @xmath306 . if we consider the polymer vacuum state @xcite given by @xmath308 , then transitions of the form @xmath309 will emit a quantum polymer particle with energy given by @xmath310 , \label{tamplitudefield}\ ] ] as can be seen from the expression ( [ insteigenvalue ] ) . here @xmath311 stands for the fundamental scale associated with the polymer quantization of the real scalar field @xmath312 ( see @xcite for details ) and is analog to the lattice parameter @xmath4 for the mechanical system . notice in this example that @xmath2 turns out to be restricted globally @xmath313 , thus , the expression ( [ tamplitudefield ] ) can be used to fix bounds in the parameter @xmath311 . we would like to thank fernando barbero for useful comments . angel garcia - chung acknowledges the total support from dgapa - unam fellowship . the authors acknowledge partial support from conacyt project * 237503 * and dgapa - unam grant in * 103716*. consider the zeroth hamiltonian amplitude for a fixed value of the parameter @xmath2 given by @xmath314 \prod^{n+1}_{j=1 } \langle p_j | p_{j-1 } \rangle^{(\lambda ) } . \label{zamplitude}\ ] ] each infinitesimal amplitude can be written as @xmath315 combining these results , the amplitude ( [ zamplitude ] ) takes the form @xmath316 \prod^{n+1}_{j=1}\left [ \sum_{n_j } e^{2 \pi i \ , n_j \frac{\lambda}{\mu } } \int^{+\infty}_{- \infty } \frac{d x_j}{\mu } \ , e^{i x_j ( p_j - p_{j-1 } - 2 \pi n_j p_c)/ \hbar } \right].\ ] ] let us now consider the first integral , i.e. , the integral in the variable @xmath317 in the previous expression . it is formed with two sums and two integrations in @xmath318 and @xmath319 @xmath320 \left [ \sum_{n_2 } e^{2 \pi i \ , n_2 \frac{\lambda}{\mu } } \int^{+\infty}_{- \infty } \frac{d x_2}{\mu } \ , e^{i x_2 ( p_2 - p_1 - 2 \pi n_2 p_c)/ \hbar } \right].\ ] ] we can move the integration in @xmath53 inside this expression together with one of the summation as @xmath321 } \int^{+\pi p_c}_{- \pi p_c } \frac{dp_1 } { 2 \pi p_c } e^{i \frac { ( x_1 - x_2 ) p_1 } { \hbar } } \sum_{n_1 } e^{2 \pi i \ , n_1 \frac{\lambda}{\mu } } e^{-i \frac{2 \pi n_1 x_1 p_c } { \hbar } } . \label{twos}\ ] ] changing variables @xmath322 yields for the last integral @xmath323 substituting this result in ( [ twos ] ) we obtain @xmath321 } \sum_{n_1 } e^{- 2 \pi i n_1 \frac{\lambda}{\mu } } e^{2 \pi i x_2 n_1 p_c /\hbar } \int^{(2 n_1 + 1 ) \pi p_c}_{(2 n_1 -1 ) \pi p_c } \frac{d \tilde{p}_1 } { 2 \pi p_c } e^{i \frac{(x_1 - x_2 ) \tilde{p}_1}{\hbar}}.\ ] ] let us now rewrite this last expression in the form @xmath324 and redefine the summation label @xmath325 . this gives @xmath326 the summations can now be separated again as @xmath327 the last summation , together with the integral in @xmath328 can be written as @xmath329 inserting this result in ( [ ltwos ] ) we obtain @xmath330 and redefining @xmath331 and @xmath332 gives the more familiar form @xmath333 let us summarize . for each integral in @xmath334 there are two summations @xmath335 and @xmath336 . the one with label @xmath335 is absorbed in the expansion of the interval for the momentum @xmath334 together with the phase in @xmath337 . due to we are dealing with @xmath87 integrals in @xmath53 and @xmath57 summations , the summation with label @xmath338 will still remain . therefore , the amplitude for the zeroth hamiltonian takes the form @xmath339 \prod^{n+1}_{j=1}\left [ \int^{+\infty}_{- \infty } \frac{d x_j}{\mu } \ , e^{i x_j ( p_j - p_{j-1 } - 2 \pi n \delta_{j , n+1 } p_c)/ \hbar } \right].\ ] ] the changes of variable involved in this result does not affect the form of the hamiltonian due to @xmath340 , hence , it can be easily extended to the polymer hamiltonian yielding @xmath339 \prod^{n+1}_{j=1}\left [ \int^{+\infty}_{- \infty } \frac{d x_j}{\mu } \ , e^{i x_j ( p_j - p_{j-1 } - 2 \pi n \delta_{j , n+1 } p_c)/ \hbar - i \epsilon h^{(\mu)}_{poly}(x_j , p_j)/\hbar } \right].\ ] ] in this appendix we are going to summarize the solution to the eigenvalue problem of the equation ( [ polyinst ] ) . we proceed along the notes given in @xcite . to begin with , consider the following potential @xmath341 where @xmath342 and @xmath343 are arbitrary real constants . the equation for the quantum fluctuation in this potential takes the form @xmath344 let us define the dimensionless variable @xmath345 and let us write the equation ( [ feqgen ] ) in terms of @xmath346 @xmath347 + \left\ { \frac{a}{b^2 } - \frac{(1 - m \epsilon_n)}{b^2 \,(1 - \xi^2 ) } \right\ } f_n(\xi ) = 0 . \label{alpoly}\ ] ] this is the equation of the associated legendre polynomials with @xmath348 and @xmath349 , see @xcite for details . this equation can be written in the form of an hypergeometric equation if we define the new function @xmath350 , where @xmath351 is an arbitrary constant . we additionally consider another change of variable and define @xmath352 . as a result , the equation ( [ alpoly ] ) turns into @xmath353 \chi'_n + \left\ { \frac{a}{b^2 } - 2 c - 4 c^2 + \frac{4c^2 + \frac{m\epsilon_n - 1}{b^2}}{4z ( 1 - z ) } \right\ } \chi_n = 0 . \label{heq}\ ] ] the relation between these changes of variable is given as @xmath354 let us impose in the equation ( [ heq ] ) the following condition for @xmath351 @xmath355 which removes the @xmath356dependence of the last coefficient in ( [ heq ] ) . with this condition , the hypergeometric equation takes the following form @xmath357 \frac{d\chi_n}{dz } - \alpha \beta \chi_n = 0 , \label{hgf}\end{aligned}\ ] ] where the parameters @xmath358 , @xmath359 and @xmath360 are defined as @xmath361 the first solution of ( [ hgf ] ) is the hypergeometric function @xmath362 , which is a series in the variable @xmath363 given by @xmath364 and is regular at @xmath365 when @xmath366 . the other independent solution of ( [ hgf ] ) is given by @xmath367 and is singular at @xmath365 . consider now a discrete spectrum within in the interval @xmath368 . in this case , the constant @xmath351 acquires real values @xmath369 . the sign is selected by imposing the condition @xmath370 which implies that @xmath351 is actually @xmath371 . the other limit @xmath372 implies that the series in @xmath373 must be truncated due to the singularity of the hypergeometric function when @xmath374 . this is only possible if @xmath358 or @xmath359 ( ( [ hgeomf ] ) is symmetric in both parameters ) takes a negative natural number value , in other words , @xmath375 . let us consider the potential ( [ fqpot ] ) with the values @xmath376 and @xmath377 . the expression for @xmath358 given in ( [ pdefinition ] ) yields the condition @xmath378 there is only one value @xmath379 for the discrete spectrum and it is also a zeroth mode @xcite . the eigenfunction of this mode is @xmath380 and it is normalized on the infinite time interval . combining the zeroth mode @xmath381 with the derivative of the instanton solution given in ( [ solop ] ) and the value of the action on this solution ( [ ivaction ] ) the zeroth mode can be written as @xmath382 this expression will be used in the calculation of the ratio between the penetration amplitude during the renormalization procedure . it is worth to mention that in the case of the double well system , the parameters @xmath342 , @xmath343 takes the values @xmath383 and @xmath384 . inserting these values in the condition for the discrete spectrum gives @xmath385 which yields the additional discrete eigenvalue @xmath386 . for more general values of @xmath342 and @xmath343 the condition reads as @xmath387 ^ 2\right\}.\ ] ] let us now consider the continuous spectrum . as we already mentioned , in the continuous spectrum the eigenvalues are arbitrary real numbers such that @xmath388 . therefore , each eigenvalue can be labelled by a real ` momentum ' @xmath212 such that @xmath389 the absent of a barrier for this values of @xmath390 allow us to discard possible reflections of the quantum fluctuation as it travels from @xmath391 to @xmath392 . moreover , this values of @xmath390 implies that @xmath351 becomes imaginary @xmath393 . in order to study solutions with asymptotic behavior given by @xmath394 , we consider only the constant @xmath395 . recall now that @xmath396 implies @xmath397 . by taking this consideration , the unique stable solution will be of the form @xmath398^{i \sqrt{|1-m\epsilon_{\tilde{p}}| } } f(-1 + 2c , 2 + 2c , 1 + 2c ; z ) , \label{solutioncs}\ ] ] and notice that for long times it takes the form @xmath399 the term @xmath402 and therefore the first contribution is not considered . notice that the limit @xmath403 gives @xmath404 and it implies that @xmath405 , which is a regular function . when @xmath403 the other contribution can be written as @xmath406 where the phase @xmath407 is given by @xmath408 this is the phase we were looking for and then the next ( and last ) step in this analysis is to impose the ` temporal box ' boundary conditions . these conditions 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it is computed , using instanton methods , the first allowed energy band for the polymer harmonic oscillator . the result is consistent with the band structure of the standard quantum pendulum but with pure point spectrum . an effective infinite degeneracy emerges in the formal limit @xmath0 where @xmath1 is the characteristic length of the vacuum eigenfunction of a quantum harmonic oscillator . as an additional result , it is shown along the article the role played by the lattice reference point @xmath2 in the full quantization of the polymer harmonic oscillator .

superconductivity and long - range magnetic order are two outstanding quantum phenomena ; however these ground states are not generally displayed by the same electrons simultaneously . this is because an internal magnetic field arising from magnetic order usually destroys superconductivity . in the 1970s , a number of materials were found to host both superconductivity and magnetic order , but the two orders were due to different electrons and occurred in spatially - separated regions . this is also true in the recently reported ruthenate - cuprate hybrid compound rusr@xmath13rcu@xmath13o@xmath14 ( r = rare earth ) where the ruo and cuo@xmath13 planes are responsible for the magnetic order and superconductivity , respectively @xcite . an exceptional case is the heavy fermion compound upd@xmath13al@xmath15 in which magnetic order and superconductivity coexist homogeneously @xcite . in this system , however , it is believed that the multiple bands of uranium ( u ) electrons make such coexistence possible . namely , among three u-5f electrons , the two with localized character are responsible for the magnetism and the remaining one is responsible for superconductivity @xcite . such `` duality '' may also be at work in other u - based heavy fermion magneto - superconductors @xcite . it is therefore an outstanding question whether magnetic order and superconductivity due to the same electrons can coexist on a microscopic length scale . although it has been proposed theoretically that magnetism and superconductivity may be viewed as two sub - components of a unified group and that they may coexist under certain conditions @xcite , accumulation of convincing experimental evidence is important . the ce - based heavy fermion compounds and high superconducting transition - temperature ( @xmath7 ) copper oxides are hosts of single - band magnetism or / and superconductivity , and are therefore good candidate materials for exploring this problem . recently , it has been suggested that in the layered heavy fermion compounds ce(rh@xmath1ir@xmath16)in@xmath3 @xcite and ce(rh@xmath1co@xmath16)in@xmath3 @xcite and also cerhin@xmath3 under pressure @xcite , antiferromagnetism and superconductivity coexist . cerh(ir)in@xmath3 crystallizes in a tetragonal structure which consists of cein@xmath15 layers separated by a rh(ir)in@xmath13 block . cerhin@xmath3 is an antiferromagnet with @xmath6=3.7 k , but becomes superconducting under pressures above 1.6 gpa @xcite . ceirin@xmath3 is a superconductor at ambient pressure with @xmath7=0.4 k @xcite and line nodes in the superconducting energy gap @xcite . it is remarkable that the magnetic fluctuations exhibit quasi two - dimensional character as revealed by nqr @xcite and neutron scattering @xcite measurements , probably reflecting the layered crystal structure . upon substituting rh with ir , superconductivity was found in ce(rh@xmath1ir@xmath16)in@xmath3 for @xmath4@xmath170.3 , while magnetic order continued to be observed around 3.8 k in the specific heat for x@xmath180.5 ( ref . @xcite ) and an internal magnetic field was detected by muon spin rotation measurement @xcite . in this paper , we present results obtained from nuclear quadrupole resonance ( nqr ) measurements on ce(rh@xmath1ir@xmath16)in@xmath3 that strongly suggest that antiferromagnetic ( af ) order coexists microscopically with unconventional superconductivity ( sc ) . we find that upon replacing rh with ir in the antiferromagnet cerhin@xmath3 , the neel temperature @xmath6 increases slightly with increasing ir content up to @xmath4=0.45 then decreases rapidly . superconductivity sets in above @xmath190.35 and @xmath7 reaches a maximum of 0.94 k at @xmath4=0.45 . the nuclear spin - lattice relaxation rate @xmath8 shows a broad peak at @xmath6 and follows a @xmath9 variation below @xmath7 , the latter feature indicating that the sc is unconventional as in ceirin@xmath3 . in the coexistence region , @xmath8 becomes proportional to @xmath10 at very low temperatures in the superconducting state and the value @xmath20 increases in the order of x=0.55 , 0.5 and 0.45 , which suggests the existence of low - lying magnetic excitations in addition to the residual density of states ( dos ) due to the presence of disorder . the rest of the paper is organized as follows . the experimental details are described in section ii . in section iii , the nqr spectroscopy that indicates the homogeneous alloying of the samples is presented . the results of the nuclear spin lattice relaxation that evidence the coexistence of antiferromagnetism and superconductivity are also presented in section iii , along with evidence for the unconventional nature of the superconductivity . we conclude in section iv , following a brief discussion of the phase diagram deduced from our nqr measurements . single crystals of ce(rh@xmath1ir@xmath2)in@xmath3 used in this study were grown by the in - flux method @xcite . for nqr measurements , the single crystals were crushed into a powder of moderate particle size to allow maximal penetration of the oscillating magnetic field , @xmath21 , used in the nqr measurements . the measurements below 1.4 k were performed by using a @xmath22he/@xmath23he dilution refrigerator . nqr experiments were performed using a home - built phase - coherent spectrometer . a standard @xmath24/2-@xmath24-echo pulse sequence was used . a small @xmath21 was used to avoid possible heating by the rf pulse below 1 k ; the @xmath24/2 pulse length is about 20 micro - seconds . a cube piston - cylinder device @xcite , filled with si - based organic liquid as a pressure - transmitting medium , was used to generate high pressure . the nqr coil was put inside a teflon cell . to calibrate the pressure at low temperatures , the reduction in @xmath25 of sn metal under pressure was monitored by resistivity measurements @xcite . @xmath7 of the samples was determined from the ac susceptibility measured by using the nqr coil at a frequency of @xmath2632 mhz , and from the @xmath27 data ( see below ) . @xmath8 was measured by the saturation - recovery method . the value of @xmath8 was unambiguously extracted from a good fitting of the nuclear magnetization to the expected theoretical curve @xcite ( discussed in detail below ) . in nqr spectra at @xmath10=4.2 k for ceirin@xmath3 ( upper panel @xcite ) , and for cerh@xmath28ir@xmath28in@xmath3 ( lower panel ) . ] there are two inequivalent crystallographic sites of in in ce(rh@xmath1ir@xmath16)in@xmath3 : the in(1 ) site in the cein@xmath29 plane and the in(2 ) site in the rh(ir)in@xmath13 block . the nqr spectra for the in(1 ) site consist of four equally - spaced transition lines separated by @xmath30 , while the in(2 ) spectra are composed of four un - equally separated lines between 30 and 72 mhz . the spectra of ceirin@xmath3 ( ref . @xcite ) is reproduced in fig . 1(a ) . here @xmath30 is defined as the parameter in the following hamiltonian , @xmath31 where @xmath32 and @xmath33 in nqr line shape ( @xmath345/2 transition ) of the in(1 ) site in cerh@xmath1ir@xmath2in@xmath35 at @xmath10=4.2 k for various ir content . the horizontal line under each spectrum is the position of zero intensity for that spectrum . ] a representative spectra for cerh@xmath28ir@xmath28in@xmath3 is shown in figure 1(b ) . two effects due to alloying are readily seen in this spectra . first , the transition lines for in(1 ) are broadened . second , each transition for in(2 ) is split into three lines . although naively this behavior might suggest phase segregation , we argue below by inspecting the ir - concentration dependence of the spectra , that there is no phase separation in the alloyed sample ; rather the sample is globally homogeneous . figure 2 shows the nqr line shape at t=4.2 k of the 2@xmath30 transition at the in(1 ) site for various ir contents . the @xmath30 decreases monotonically from 6.78 mhz ( @xmath4=0 ) @xcite to 6.065 mhz ( @xmath4=1 ) @xcite , suggesting a smooth evolution of the lattice upon alloying , in agreement with x - ray diffraction measurements @xcite . it should be emphasized that no trace of pure cerhin@xmath3 or ceirin@xmath3 is found in the alloyed samples because no peaks corresponding to @xmath4=0 or @xmath4=1 were observed . in nqr line shape ( @xmath345/2 transition ) of the in(2 ) site of cerh@xmath1ir@xmath2in@xmath35 at @xmath10=4.2 k for various ir content . in this plot , the vertical axis was adjusted so that all samples have the same height for the central peak ( around 32.2 mhz ) . the signal around 35 mhz for low @xmath4 is from the second lowest transition ( @xmath363/2 transition ) ; also see fig . 1(b ) . ] figure 3 shows the spectra corresponding to the lowest transition ( @xmath37 ) line of the in(2 ) site for various ir concentration ranging from @xmath4=0.25 to 0.75 . it is interesting that the positions of the three peaks do not change with ir concentration ( fig . 4(a ) ) , but the relative intensity distribution among these lines does ( fig . 4(b ) ) . also , the left peak is at the same position of the @xmath38 transition for cerhin@xmath35 , while the right peak is at the same position as the corresponding transition for pure ceirin@xmath3 . the central peak is characterized by @xmath39=17.37 mhz and @xmath40=0.473 . 5/2 transition lines of the in(2 ) site of cerh@xmath1ir@xmath2in@xmath35 at @xmath10=4.2 k. ( b ) ir - content dependence of the relative intensity of the three @xmath345/2 transition lines of the in(2 ) site of cerh@xmath1ir@xmath2in@xmath35 . ( c ) ir - content dependence of the peak frequency multiplied by the relative intensity for the three in(2 ) @xmath345/2 transition lines . ] figure 4(c ) depicts a quantity that is the relative intensity shown in fig . 4(b ) multiplied by the corresponding peak position shown in fig . most simply , this corresponds to the `` weighted peak position '' or `` averaged resonance frequency '' for the @xmath41 transition . note that this quantity increases smoothly with increasing ir concentration . the results shown in fig . 4 can be interpreted as follows . in(2 ) has two nearest neighbor @xmath42 ( rh , ir ) sites . there are @xmath4 ir atoms and @xmath43 rh atoms for a given alloy concentration x. if the nqr frequency is sensitive to the local environment , there will be three resonance lines depending on the nearest neighbor configuration of a given in(2 ) , namely , ( rh , rh ) , ( rh , ir ) or ( ir , ir ) . the intensity of each peak will be proportional to the probability that in(2 ) has a corresponding nearest neighbor pair , namely , ( rh , rh ) , ( rh , ir ) or ( ir , ir ) . figure 4 strongly suggests that this is the case , with the central transition corresponding to the case with ( rh , ir ) nearest neighbors . in such a scenario , one might then wonder why in(1 ) only sees an averaged environment . this is probably because the wave function mixing between in(1 ) and the @xmath42 atom is weaker than in the case of in(2 ) , because in(1 ) is farther away from @xmath42 . in addition , in(1 ) has eight nearest neighbor @xmath42 atoms . the effect of having different nearest - neighbor pair is thus further averaged out . as a result , each in(1 ) transition is observed as a broadened line . this is in contrast to the case of in(2 ) whose @xmath44-orbital directly mixes with those of @xmath42 . since @xmath30 is dominated by the on - site electronic configuration @xcite , the stronger coupling between in(2 ) and @xmath42 atoms gives rise to three distinct resonance lines in the alloyed samples rather than a broad single transition as in the case of in(1 ) . although the in(2 ) transition is sensitive to the local atomic configuration , it should be emphasized that globally the electronic states are quite homogeneous , as evidenced by the results of spin - lattice relaxation measurements described in the next subsection . the @xmath8 measurements were performed at the peak of the 2@xmath30 transition ( @xmath37 for the in(1 ) site and at the central peak of the three lowest frequency transition ( @xmath37 ) lines for the in(2 ) site . figure 5 shows the decay curve of the nuclear magnetization for @xmath4=0.45 at three typical temperatures above and below @xmath6 and @xmath7 . at t=0.2k we used a small tipping - angle pulse so that the magnetization is less saturated at small delay time . the decay curve can be fitted by a single component of @xmath27 to the theoretical curve @xcite , @xmath45 the same quality of data were obtained for all alloys and also for the in(2 ) site , whose nuclear magnetization is fitted to the theoretical curve @xcitewith a single component of @xmath27 . @xmath46 ir@xmath47in@xmath3 . the curves are fitting to equation ( 4 ) in the text . ] the successful fitting of the nuclear magnetization to the theoretical curve with a single @xmath27 component is a good indicator of the homogeneous nature of the electronic state . figure 6 shows the temperature dependence of @xmath8 measured at the three peaks of in(2 ) for @xmath4=0.35 . it can be seen that all sites show a quite similar @xmath10 dependence . namely , there is a peak around @xmath10=4 k , although the peak height is reduced as compared to @xmath4=0 @xcite . the absolute value is also very similar . in the figure , the origin for the left and right peaks were shifted for clarity . these results indicate that the three peaks probe the same electronic state despite the fact that they arise from different nearest - neighbor @xmath42 configurations . at the three @xmath345/2 transition lines of the in(2 ) site of cerh@xmath48ir@xmath49in@xmath35 . for clarity , @xmath8 at the left peak was multiplied by 3 , while that for the right peak was divided by 3 . ] figure 7 shows the evolution of the @xmath10 dependence of @xmath8 at the central in(2 ) transition for various ir concentrations . it is evident that the peak temperature and the peak height change with the ir concentration . we associate this peak with the neel ordering temperature , @xmath6 , at which @xmath8 increases due to critical slowing down . @xmath6 determined in this manner correspond well with that inferred from the specific heat @xcite and @xmath50sr measurements @xcite . interestingly , @xmath6 first increases gradually with increasing ir content up to @xmath4=0.45 then decreases rapidly . for @xmath4=0.5 , @xmath6 is reduced to 3 k. for @xmath4=0.55 , no feature is seen in the @xmath10-dependence of @xmath8 ( for clarity of fig . 7 , data are not shown ) , thus it becomes difficult to identify @xmath6 . measured at the central peak of the in(2 ) @xmath345/2 transition in cerh@xmath1ir@xmath2in@xmath35 . ] @xmath6 inferred from the peak in @xmath8 is sensitive to externally - applied hydrostatic pressure , as in pure cerhin@xmath3 . in the right panel of fig . 7 is shown the @xmath27 result under a pressure of 1.02 gpa for the @xmath4=0.5 sample . the broad peak seen at ambient pressure is suppressed , and instead a distinct decrease of @xmath8 is found at 2.5 k , which resembles the case of pure cerhin@xmath3 in which the application of pressure reduces the height of the peak at @xmath6 @xcite and eventually suppresses the peak under @xmath51=1.7 gpa @xcite . thus , as in pure cerhin@xmath3 , @xmath27 can serve as a probe to determine @xmath6 . figure 8 shows typical data sets of @xmath8 measured at the in(1 ) site . the anomaly at @xmath6 is also visible at the in(1 ) site , although it is less clear presumably because the peak at @xmath6 at this site is already rather weak , even in the undoped compound . in @xmath8 measured at the in(1 ) site of cerh@xmath1ir@xmath2in@xmath35 . data for @xmath4=1 and 0 are from ref . @xcite and ref.@xcite , respectively . ] the non - monotonic change of @xmath6 as a function of @xmath4 may be attributed to the increase of exchange coupling between 4f spins which is overcome by the increase of coupling between 4f spins and conduction electrons above @xmath52 , as inferred from doniach s treatment of the kondo necklace @xcite . this result also resembles the behavior of cerhin@xmath3 @xcite as a function of pressure and indicates that the substitution of ir for rh acts as chemical pressure in cerhin@xmath3 . due to the broadening of the spectra upon alloying , it is difficult to estimate precisely the internal magnetic field in the ordered state . the hamiltonion in the presence of magnetic field is given by @xmath53 where @xmath54 is given by eq . ( 1 ) and @xmath55 -axis for in(1 ) site ( a ) and in(2 ) site ( b ) . ] in the present case , @xmath56 is along the crystal c - axis . assuming an internal magnetic field in the ab - plane , which is the case for cerhin@xmath35 , the evolution of the resonance frequency for each transition is calculated for the in(1 ) site ( fig . 9(a ) ) and for the in(2 ) site ( fig . 9(b ) ) . here , the field is assumed to be along x - direction . note that even the @xmath57 transition for the in(2 ) site , which has a fwhm of 0.26 mhz and is the sharpest among all transitions in the alloyed samples , does not show an appreciable change between @xmath10=4.2 k ( above @xmath6 ) and @xmath10=1.4 k ( below @xmath6 ) , see fig . 10 . this suggests that the internal magnetic field at the in(2 ) site is less than 200 oe for @xmath4=0.5 , as inferred from the expected splitting deduced from fig . 9 . such a small internal field , which is samller by a factor of 10 than that in cerhin@xmath3 @xcite , could be due to a moderate reduction of the ordered moment @xcite with a concomitant reduction of the hyperfine coupling @xcite . 3/2 transition for cerh@xmath28ir@xmath28in@xmath3 at @xmath10=4.2 k and 1.4 k. for clarity , the horizon has been shifted . ] next , we discuss the low temperature behavior of ce(rh@xmath1ir@xmath16)in@xmath3 well below @xmath6 . figure 11 shows @xmath8 for both the in(1 ) and in(2 ) sites at low temperatures for the @xmath4=0.5 sample . below @xmath7=0.9 k , @xmath8 decreases sharply with no coherence peak , following a @xmath9 variation down to @xmath10=0.45 k. the observation of the @xmath9 behavior is strong evidence for the existence of line nodes in the superconducting gap function @xcite . for an s - wave gap , @xmath8 would show a coherence peak just below @xmath7 followed by an exponential decrease upon further decreasing @xmath10 . because @xmath8 is measured at the same transition for the entire measured temperature range , our results suggest that antiferromagnetic order and superconductivity are due to the same electronic state derived from the ce-4f@xmath58 electron . if the two ordered states occurred in spatially - separated regions , the nuclear - magnetization decay curve would have been composed of two components ( two @xmath27 s ) below @xmath6 , contradicting the single - component decay curve we observe . it is noteworthy that just above @xmath7 , @xmath8 tends to be proportional to @xmath10 , which suggests that there remains a finite density of states ( dos ) at the fermi level ( @xmath59 ) in the magnetically ordered state , since @xmath60 is dominantly proportional to the square of the low - energy dos at such low-@xmath10 ( see below , eq . this suggests that the gap opening due to the antiferromagnetic order is incomplete , in contrast to the behavior observed in pure cerhin@xmath3 where the gap is more fully developed , leading to a stronger decrease of @xmath8 ( see fig . this remnant of some part of the fermi surface may be important for superconductivity to set in even in the magnetically ordered state . results at low temperatures for cerh@xmath28ir@xmath28in@xmath3 measured at the in(1 ) at in(2 ) sites , respectively . the two solid lines indicate the @xmath9 and @xmath10-linear variations , respectively . ] finally , let us compare the superconducting behavior for @xmath4=0.45 , 0.5 and 0.55 . figure 12 shows the ac - susceptibility ( ac-@xmath61 ) measured using our nqr coil . although it is hard to determine the onset temperature of the superconductivity from ac-@xmath61 , it can be seen that the mid - point of the transition increases in the order of @xmath4=0.55 , 0.5 and 0.45 . @xmath7 determined from the point at which @xmath8 displays a distinct drop is 0.8 k , 0.9 k and 0.94 k for @xmath4=0.55 , 0.5 and 0.45 , respectively . figure 13 shows @xmath8 normalized by its value at @xmath7 plotted against the reduced temperature @xmath62 for @xmath4=0.55 , 0.5 and 0.45 . just below @xmath7 , @xmath8 shows identical behavior for all samples , but at lower temperatures strong variation is observed . in particular , below @xmath11 0.4 k , @xmath8 becomes again proportional to @xmath10 , and the normalized value of @xmath8 increases in the order @xmath4=0.55 , 0.5 and 0.45 . ir@xmath16in@xmath3 ( @xmath4=0.45 , 0.5 and 0.55 ) . ] the most straightforward explanation for @xmath10-linear @xmath8 at low-@xmath10 would be the presence of disorder that produces a finite dos remaining at @xmath59 . by assuming a gap function with line nodes , @xmath63 and with a finite residual dos , @xmath64 ( ref.@xcite ) , we tried to fit the data in the superconducting state to @xmath65 where @xmath66 with @xmath67 being the dos in the normal state and @xmath68 being the fermi function . the resulting fitting parameters are @xmath69=0.32 , 0.45 and 0.63 for @xmath4=0.55 , 0.5 and 0.45 , respectively , with @xmath70=2.5@xmath71 for all samples . in such a case , however , one would expect @xmath64 to be the same for @xmath4=0.55 and 0.45 , because the amount of disorder is expected to be similar . the much larger @xmath64 inferred for @xmath4=0.45 than @xmath4=0.55 suggests an additional mechanism . we propose that this additional @xmath64 comes from low - lying magnetic excitations associated with the coexisting magnetic ordering that is more well developed at lower values of @xmath4 . similar @xmath64 was seen in cerhin@xmath3 under a pressure of 1.6 gpa where magnetism also coexists with superconductivity . in this case the observed behavior was interpreted as due to a gapless @xmath44-wave superconducting state @xcite , or due to additional nodes in the d - wave order parameter @xcite . plotted against the reduced temperature for cerh@xmath1ir@xmath16in@xmath3 at the in(1 ) site . the solid curves are fits to the data as described in the text . @xmath72 is for short of @xmath69 . ] on the other hand , the larger @xmath64 for the in(2 ) site than for in(1 ) site may be due to a larger disorder contribution for this site . this is because the source of disorder in the present case is in the rh(ir)in@xmath13 block . the in(2 ) site is naturally more sensitive to such disorder than the in(1 ) site which is farther removed from this block . a similar case was seen in high-@xmath7 copper oxide superconductors . in tl@xmath13ba@xmath13ca@xmath13cu@xmath15o@xmath73 ( @xmath7=117 k ) @xcite , disorder due to inter - substitution of ca / tl occurs in the ca layer . as a consequence , the cu(1 ) site sandwiched by two ca layers sees a larger @xmath64 than the cu(2 ) site which is adjacent to only one of the ca layers . ir@xmath2in@xmath35 obtained from nqr measurements . af and sc mean antiferromagnetic and superconducting states , respectively . ] the phase diagram shown in fig . 14 summarizes our results . upon doping with ir , the system undergoes a quantum phase transition from an antiferromagnet ( @xmath4=0 ) to a superconductor ( @xmath4=1 ) , with an intervening region where antiferromagnetic and superconducting orders coexist . our results show that this behavior , reported previously based on thermodynamic data @xcite , is confirmed microscopically . @xmath7 reaches a maximum at @xmath4=0.45 ( @xmath7=0.94 k ) , while @xmath6 is found to be the highest ( @xmath6=4.0 k ) . the enhancement of @xmath7 in the antiferromagnetically ordered state is most interesting , suggesting the importance of magnetism in producing the superconductivity . recently , antiferromagnetism and superconductivity was found to coexist also in cerhin@xmath3 under external pressures @xcite , but the coexistent region is rather narrow there . more importantly , in the present case superconductivity develops well inside the ordered state and @xmath7 increases when approaching the maximum of @xmath6 , whereas @xmath7 reaches a maximum after @xmath6 disappears in hydrostatically - pressurized cerhin@xmath3 . the observed phase diagram may be understood in the framework of so(5 ) theory in which the 5-component super - spin can be rotated by a chemical potential from the subspace of antiferromagnetic order to the subspace of d - wave superconductivity and vice versa @xcite . however , a microscopic description of how the same 4f@xmath58 electron can display both magnetic order and superconductivity is still lacking . in conclusion , we have carried out an extensive @xmath0 in nqr study on cerh@xmath1ir@xmath2in@xmath3 . we find that the substitution of ir for rh in the antiferromagnet cerhin@xmath3 acts as chemical pressure . with increasing ir content ( @xmath4 ) , @xmath6 increases slightly up to @xmath4=0.45 , then decreases rapidly . the coexistence of superconductivity with antiferromagnetism for 0.35 @xmath18 @xmath74 0.5 is observed in the temperature dependence of @xmath8 which displays a broad peak at @xmath6 and drops as @xmath9 below @xmath7 . at @xmath4=0.5 , @xmath6 is reduced to 3 k while @xmath7 reaches 0.9 k. our results suggest that the coexisting antiferromagnetic order and superconductivity are due to the same electronic state derived from the ce-4f@xmath58 electron . it is most interesting that the superconducting transition temperature @xmath7 is increased as the system penetrates deeper inside the antiferromagnetically ordered state . @xmath7 for @xmath4=0.45 and 0.5 is more than double that of ceirin@xmath3 . in the coexistence region , @xmath8 shows a @xmath10-linear dependence at low-@xmath10 below @xmath110.4 k. we have argued that this may arise from some magnetic excitations associated with the coexisting magnetism , in addition to the presence of crystal disorder that produces a residual density of states at the fermi level . we thank h. harima for a helpful discussion on the @xmath30 issue , and g. g. lonzarich , n. nagaosa and s .- c . zhang for helpful comments . we also would like to thank w. bao and n.j . curro for useful discussion , and s. kawasaki , k. tanabe and s. yamaoka for assistance in some of the measurements . partial support by japan mext grant no . 14540338 , 16340104 ( g .- q.z ) and no . 10ce2004 ( y.k ) is thanked . work at los alamos was performed under the auspices of the us doe . m. b. maple and o. fisher ( eds ) , _ superconductivity and magnetism _ , ( springer - 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we present a systematic @xmath0 in nqr study on the heavy fermion compounds cerh@xmath1ir@xmath2in@xmath3 ( @xmath4=0.25 , 0.35 , 0.45 , 0.5 , 0.55 and 0.75 ) . the results provide strong evidence for the microscopic coexistence of antiferromagnetic ( af ) order and superconductivity ( sc ) in the range of 0.35 @xmath5 0.55 . specifically , for @xmath4=0.5 , @xmath6 is observed at 3 k with a subsequent onset of superconductivity at @xmath7=0.9 k. @xmath7 reaches a maximum ( 0.94 k ) at @xmath4=0.45 where @xmath6 is found to be the highest ( 4.0 k ) . detailed analysis of the measured spectra indicate that the same electrons participate in both sc and af order . the nuclear spin - lattice relaxation rate @xmath8 shows a broad peak at @xmath6 and follows a @xmath9 variation below @xmath7 , the latter property indicating unconventional sc as in ceirin@xmath3 ( @xmath7=0.4 k ) . we further find that , in the coexistence region , the @xmath9 dependence of @xmath8 is replaced by a @xmath10-linear variation below @xmath110.4 k , with the value @xmath12 increasing with decreasing @xmath4 , likely due to low - lying magnetic excitations associated with the coexisting magnetism .

let @xmath0 be a smooth and quasi - projective variety of dimension @xmath1 , @xmath2 bloch s higher chow group , and @xmath9 the betti cycle class map , where @xmath10 if @xmath11 , then by a standard weight argument ( see @xcite ) , the hodge conjecture implies that @xmath12 is surjective . beilinson once conjectured that @xmath4 is always surjective @xcite . however unless @xmath13 is given by base extension from a smooth quasi - projective variety over a number field , it is known that this conjecture is too optimistic ( * ? ? ? * cor . 9.11 ) . consider these three statements : ( s1 ) @xmath14 is surjective for all smooth complex projective varieties @xmath15 ; ( s2 ) @xmath16 is surjective for all smooth complex quasi - projective varieties @xmath13 ; ( s3 ) @xmath17 is surjective for all smooth complex projective varieties @xmath15 . note that ( s1 ) for @xmath18 is equivalent with the hodge conjecture ( with rational coefficients as opposed to the original version with integral coefficients that was disproven by atiyah - hirzebruch by showing the existence of non - algebraic torsion classes ) , and that it is trivially true for @xmath19 because then @xmath20 . also , @xmath21 for @xmath22 in ( s3 ) because of dimension reasons . when @xmath11 , all three statements ( for all @xmath23 ) are equivalent ( as one sees using a localization sequence argument , and deligne s mixed hodge theory , as on @xcite ) . however as we shall see in this paper , the statements are independent of each other : ( s1 ) is expected to be true , we conjecture ( s3 ) to be true , and show ( s2 ) to be false in general . there is some evidence ( @xcite , @xcite , @xcite , @xcite ) that ( s2 ) holds in the special case @xmath7 , and the results in this paper are consistent with this . in @xcite we provided some evidence that ( s3 ) is always true , and in particular , ( s3 ) can be viewed as an appropriate generalization of the hodge conjecture . in this paper we address a number of issues , namely : @xmath24 necessary and sufficient conditions for @xmath4 , and @xmath25 to be surjective , in terms of kernels of ( reduced ) higher abel - jacobi maps . this is worked out in theorem [ t01 ] and subsequent examples , as well as in corollary [ c116 ] below . naturally this leads to a generalized notion of decomposable classes , which is discussed in section [ sec07 ] . also , in theorem [ t45 ] we exhibit counterexamples to the surjectivity of @xmath4 in ( s2 ) in all cases where it is not trivially true or where one might reasonably expect this surjectivity ( namely @xmath26 or @xmath18 ) . @xmath24 the story can be worked out with integral coefficients in the case @xmath7 , and in particular , we are interested in the nature of the map @xmath27 we prove in section [ sec010 ] that the torsion subgroup @xmath28 of @xmath29 is trivial ( hence this intersection makes sense ! ) . the combination of theorem [ t534 ] and conjecture [ co534 ] below would imply that @xmath30 is surjective . we also relate @xmath30 to the map @xmath31 for @xmath32 a non - zero integer , which is now known to be an isomorphism . ( this is the former bloch - kato conjecture , for the field @xmath33 . ) thus the conjectured surjectivity of @xmath30 can be thought of as a hodge theoretic version of the bloch - kato conjecture . note that @xmath34 equals @xmath35 as mentioned above , the classical hodge conjecture , as originally formulated by hodge with integral coefficients , is false . but we wish to remind the reader that it is false with integral coefficients even _ modulo torsion _ ( see @xcite ) , albeit expected ( by optimists ) to be true with rational coefficients . the following statement , again proven in section [ sec010 ] , therefore seems rather remarkable . [ t534 ] @xmath36 . in particular , @xmath30 is surjective @xmath37 is surjective . what this theorem tells us is that the hodge theoretic analog of the bloch - kato conjecture is the surjectivity of @xmath38 . quite generally we expect that the following is true . [ co534 ] for all @xmath39 , statement ( s3 ) holds . by our earlier remarks this conjecture includes the hodge conjecture and relates it to the ( now proved ) bloch - kato conjecture . the authors wish to thank spencer bloch , hlne esnault , marc levine and chuck weibel for useful conversations and/or correspondence . \(i ) unless otherwise specified , @xmath15 is a smooth complex projective variety of dimension @xmath1 , and @xmath13 is a smooth complex quasi - projective variety . let @xmath40 be a subring . \(ii ) @xmath41 ( tate twist ) . \(iii ) if @xmath42 is an @xmath43-mixed hodge structure ( mhs ) , then we write @xmath44 and @xmath45 . \(iv ) for a quasi - projective variety @xmath46 , @xmath47 is the higher chow group defined in @xcite , and @xmath48 . \(v ) @xmath49 . \(vi ) we write @xmath50 the limit taken over all zariski open subsets of @xmath15 . \(vii ) we let @xmath51 denote the ( algebraic ) deligne - beilinson cohomology @xcite of @xmath52 . we provide a breezy review of some of the ideas in ( * ? ? ? * section 3.1 ) . we first recall the definition of the higher chow groups . let @xmath53 be a quasi - projective variety . put @xmath54 free abelian group generated by subvarieties of codimension @xmath55 in @xmath46 , @xmath56 the standard @xmath57-simplex , and @xmath58 @xmath59 meets all faces properly@xmath60 . we let @xmath61 where @xmath62 is the restriction to the @xmath63-th codimension @xmath64 face . [ defna1 ] ( @xcite ) @xmath65 homology of @xmath66 . we put @xmath67 . we also need to recall the cubical version . let @xmath68 with coordinates @xmath69 and @xmath70 codimension one faces obtained by setting @xmath71 , and boundary maps @xmath72 , where @xmath73 denote the restriction maps to the faces @xmath74 respectively . the rest of the definition is completely analogous for @xmath75 , except that one has to quotient out by a subgroup of degenerate cycles . it is known that both complexes are quasi - isomorphic ( @xcite ) . now write @xmath76 , where @xmath77 is a smooth projective variety of dimension @xmath1 , @xmath78 a ncd with smooth components . for an integer @xmath79 , put @xmath80 } = $ ] disjoint union of @xmath81-fold intersections of the various components of @xmath82 , with corresponding simplicial scheme @xmath83 } \to y \hookrightarrow y^{[0 ] } : = x$ ] . there is a third quadrant double complex @xmath84},-j ) , \ i,\ j \leq 0 ; \begin{matrix } \z_{0}^{i , j+1}\\ \quad\biggl\uparrow\del\\ \z_{0}^{i , j}&{\buildrel { \gy}\over\longrightarrow}&\z_{0}^{i+1,j } \end{matrix}\ ] ] whose differentials are @xmath85 vertically ( @xmath86 as coming from the definition of bloch s higher chow groups ) , and @xmath87 ( @xmath88 gysin ) horizontally . to the corresponding total complex @xmath89 with @xmath90 are associated the two grothendieck spectral sequences @xmath91 and @xmath92 with @xmath93 the second spectral sequence , together with bloch s quasi - isomorphism @xmath94 shows that ( @xcite(sect . 3.1 ) ) @xmath95 the first spectral sequence has @xmath96},-j ) $ ] and @xmath97},-j ) \to \ch^{r+i+1}(y^{[-i-1]},-j)\big)}{\gy\big(\ch^{r+i-1}(y^{[-i+1]},-j)\big ) } \,.\ ] ] the corresponding filtration on @xmath89 also induces a `` weight '' filtration @xmath98 which is characterized by the injection @xmath99},-\ell)\end{matrix}\right\},\ ] ] for @xmath100 , where @xmath101 is called a residue map in @xcite . it is easy to check that @xmath102 our main goal in this section is to prove theorem [ t01 ] , which provides necessary and sufficient conditions for the surjectivity of @xmath103 . the obstruction to surjectivity will be explained in terms of kernels of abel - jacobi maps for the higher chow groups . we fix @xmath52 , @xmath104 and @xmath105 . of particular interest is the top residue @xmath106 for @xmath107 , where @xmath108 } ) \to \ch^{r - m+1}(y^{[m-1]})\big ) } { \gy\big(\ch^{r - m-1}(y^{[m+1]})\big)}.\ ] ] ( in general , @xmath109 . ) we use this to study the cycle class map @xmath4 via the commutative diagram @xmath110 where the injectivity of the map on the bottom row follows from the fact that @xmath111 . ( in general we use @xmath112-coefficients because weight filtration in hodge theory is defined for such coefficients . exceptions to this situation are discussed in sections [ sec012 ] and [ sec010 ] . ) with regard to the morphism @xmath113 } ) \to \ch^{r - m+1}(y^{[m-1]}),\ ] ] for @xmath114 , put @xmath115})^{\circ } : = \gy^{-1}\big(\ch_{\hom}^{r - m+1}(y^{[m-1]})\big),\ ] ] where @xmath116 } ) \subset \ch^{r - m+1}(y^{[m-1]})$ ] is the subgroup of null - homologous cycles on @xmath117}$ ] . notice that for @xmath118 , @xmath119})^{\circ}}{\gy\big ( \ch^{r - m-1}(y^{[m+1]})\big)},\ ] ] and for all @xmath120 , we have @xmath121 for @xmath122 put @xmath123};\qq)^{\circ}}{\gy\big(\ch^{r - m-1}(y^{[m+1]};\qq)\big ) } & \text{if $ k=1$}\\ e_k^{-m,0}\otimes\qq&\text{for $ k=2, ... ,m+1 $ } \end{cases}\ ] ] so that @xmath124 for @xmath125 . ( our main reason for introducing @xmath126 is so that the map @xmath127 in diagram ( [ e004 ] ) below has a chance of being surjective , as implied by the hodge conjecture . ) for @xmath128 we let @xmath129 be @xmath130};\qq ) { \buildrel { \gy}\over\longrightarrow } \ch^{r - m+2}(y^{[m-2]};\qq)\big)&\text{if $ k=1$,}\\ e_k^{-m+k ,- k+1}\otimes\qq & \text{for $ k=2, ... ,m$ , } \end{cases}\ ] ] hence @xmath131 and @xmath132 . then becomes @xmath133 where we abbreviate @xmath134 to @xmath135 and similarly for @xmath136 , and where @xmath137 , which equals @xmath138 because @xmath139 . note that as in @xcite , @xmath140},\qq(r - m))\to\\ h^{r - m+1,r - m+1}(y^{[m-1]},\qq(r - m+1))\end{matrix}\right)}{\gy \big(h^{r - m-1,r - m-1}(y^{[m+1 ] } , \qq(r - m-1))\big)},\ ] ] and , for @xmath141 , @xmath142},\qq(r - m+k ) ) \to\\ h^{2r-2 m + k+2}(y^{[m - k-1]},\qq(r - m+k+1))\end{matrix}\right)}{\gy \big(h^{2r-2m+k-2}(y^{[m - k+1 ] } , \qq(r - m+k-1))\big)}.\ ] ] using the differentials @xmath143 we claim that there is a commutative diagram of exact sequences for each @xmath144 @xmath145 where @xmath146 is the obvious map , @xmath147 is the abel - jacobi map ( defined explicitly in section [ sec05 ] below ) , and where the @xmath148 s are characterized as follows . if we assume @xmath148 is defined , then the definition of @xmath149 is dictated by imposing commutativity in ( [ e003 ] ) . thus we need only define @xmath150 , and show that @xmath151 . the latter will be proven in section [ sec05 ] . note that , as implicit in , @xmath4 is the composition @xmath152 and that there is a map @xmath153 , which is an isomorphism when @xmath7 , and is surjective for all @xmath55 and @xmath57 under the assumption of the hodge conjecture . we let @xmath154 in the diagram ( with @xmath155 the obvious map ) @xmath156\ar@{^{(}->}[r]&\widetilde{e}_1^{-m,0}\ar[d]^-\b\ar[rd]^-{\a_1}\ar[rr]^-{d_1}&&\widetilde{e}_1^{-m+1,0 } \ar[d]^-{\lambda_1 } \\ \gamma\big(h^{2r - m}(u,\qq(r))\big ) \,\ , \ar@{^{(}->}[r]&\gamma(gr^w_0 ) \ar[r]^-\kappa&j(w_{-1})\ar[r]^-{h_1 } & j(gr^w_{-1 } ) \ , . } \ ] ] then and commute ( see theorem [ diagrams - commute ] ) . with regard to the diagram above , it is obvious that @xmath157 from the isomorphisms @xmath158 and @xmath159 , we arrive at the identification @xmath160 we have inclusions @xmath161 where on the left we have equality as @xmath162 and @xmath163 , and the right - most inclusion is an equality if @xmath164 ( e.g. , if @xmath127 is surjective ) . we mention in passing the following result : [ pp1 ] suppose that @xmath165 is injective for @xmath166 . if @xmath127 is surjective , then @xmath167 is surjective . from we get @xmath168 , and we apply the inclusions above a slight tweaking of proposition [ pp1 ] together with diagrams and leads to : [ p001][m . saito @xcite . also see @xcite . ] assume the hodge conjecture , and that the bloch - beilinson conjecture holds , viz . , for all smooth projective @xmath169 , the abel - jacobi map @xmath170 , is injective for all @xmath55 and @xmath171 . if @xmath13 is obtained from a smooth quasi - projective variety over @xmath172 by base change to @xmath173 , then for all @xmath55 and @xmath57 , @xmath16 is surjective . the proof , which is similar to the one given in ( * ? ? ? * prop . 3.7 ) , is omitted . next , @xmath174 by , hence @xmath175 for @xmath176 , with the last equality because of . hence @xmath177 . if @xmath146 is an isomorphism then this gives @xmath178 this is the case when @xmath179 again because @xmath180 . putting all these ideas together , we obtain the following . [ t01 ] \(i ) if @xmath4 is surjective then @xmath181 the converse is true if @xmath182 is surjective ; more precisely : the converse is true if @xmath183 ( e.g. , if @xmath182 is surjective ) ; \(ii ) @xmath4 is surjective implies that @xmath184 \(iii ) if @xmath183 and @xmath185 for all @xmath186 , then there is a short exact sequence @xmath187 note that @xmath188 is automatically injective when @xmath26 by the theory of the picard variety , so that @xmath189 in this case . since @xmath127 is an isomorphism here , we deduce [ c56 ] let us assume that @xmath7 . then @xmath8 is surjective if and only if @xmath190 for all @xmath191 . in particular , @xmath192 is always surjective . ( see also example [ ex1 ] . ) assume @xmath193 . note that @xmath194 . assuming @xmath127 is surjective , then theorem [ t01](iii ) we deduce the short exact sequence @xmath195 recalling @xmath76 of dimension @xmath1 , we have that @xmath196 where the denominator term @xmath197 in the jacobian is identified with its image in @xmath198 , which apparently coincides with @xmath199},\qq(r-1))\big)$ ] by a standard mixed hodge theory argument ( deligne ) . taking limits , we arrive at the short exact sequence @xmath200 @xmath201 where @xmath202 is the @xmath203-th coniveau filtration . however , if for example @xmath204 , then using the fact that a zero - cycle on a projective variety is homologous to zero if and only its degree is @xmath205 , we see @xmath127 is surjective in this case , and that @xmath206 owing to the fact that any finite set of points on @xmath15 lies on a smooth divisor in @xmath15 . therefore we arrive at the following result . [ @xcite ] @xmath207 is surjective . [ ex01 ] the case @xmath208 . we observe that @xmath209 , and that @xmath127 is an isomorphism . thus from theorem [ t01](iii ) and ( [ e003 ] ) , we arrive at the short exact sequence @xmath210 we have @xmath211},1;\qq)\big ) } \xrightarrow{\lambda_{2 } } j\biggl(\frac{h^2(x,\qq(2))}{h^2_y(x,\qq(2))}\biggr).\ ] ] there is an exact sequence @xmath212},1;\qq)\big ) } \to \ch^2(u,1;\qq),\ ] ] hence @xmath213},1;\qq)\big ) } \,.\ ] ] note that @xmath214},1 ) = \big(\cc^{\times}\big)^{\oplus n},\ ] ] and recall we have @xmath215 } = \coprod_{i=1}^n y_i$ ] . _ thus @xmath216 is surjective if and only if @xmath217 is injective on the subgroup of cycles in @xmath218 supported on @xmath82 , modulo the image of the space of decomposables in @xmath219 , supported on @xmath82 . _ now let @xmath220 @xmath221 under the product on the higher chow groups , and @xmath222 then @xmath223 is surjective if and only if @xmath224 is injective . in summary , @xmath225 @xmath226 ( see also corollary [ c100 ] . ) in particular , @xmath227 above is surjective if @xmath228 . we recall bloch s conjecture which says in the case that @xmath15 is a surface , @xmath229 the albanese map @xmath230 alb@xmath231 is an isomorphism . equivalently , this amounts to saying that @xmath229 the motive of @xmath15 degenerates . the degeneration of the motive of @xmath15 implies that @xmath228 ( @xcite ) . so according to bloch s conjecture , if @xmath15 is a surface with @xmath232 , then @xmath227 is surjective . for any smooth quasi - projective variety @xmath0 of dimension @xmath233 we consider the following three regions for the pair @xmath234 ( see figure [ regions ] ) : 1 . @xmath235 and @xmath236 ; 2 . @xmath237 and @xmath238 ; 3 . @xmath239 . the corresponding cycle class map @xmath240 is surjective in regions ii and iii since there the right - hand side is trivial ( see corollary 6.6 on page 85 in @xcite ) . we shall show below that at every point in region i surjectivity fails in general . thus the only open cases are the diagonal ( @xmath26 ) , where surjectivity is the beilinson - hodge conjecture as formulated in @xcite , and the @xmath55-axis , where surjectivity corresponds to the hodge conjecture extended to such @xmath13 . [ t45 ] for a fixed @xmath233 , assume @xmath234 lies in region i. then there exists a smooth quasi - projective variety @xmath0 of dimension @xmath1 such that @xmath4 fails to be surjective . special instances of this are already established in @xcite . let @xmath241 be a hypersurface of sufficiently large degree , @xmath242 general hyperplanes such that if we put @xmath243 then @xmath244 is smooth and @xmath245 where @xmath246 since @xmath237 . fix points @xmath247 in @xmath248 such that the class of @xmath249 is non - trivial in @xmath250 ( possible by mumford@xmath251roitman , see ( * ? ? ? * ch.15 ) ) , and consider the blow - up @xmath252 set @xmath253 , @xmath254 , @xmath255 for @xmath256 , and @xmath257 . observe that @xmath258 finally for @xmath259 , put @xmath260 @xmath261 so that @xmath262 has dimension @xmath263 . note that @xmath264 } = \emptyset,\ ] ] @xmath265 } = \biggl\{\big\{e_1 \cap b_{\{p , q\}}(w)\big\}\times \pp^k\biggr\ } \ \coprod\ \biggl\{\big\{e_2\cap b_{\{p , q\}}(w)\big\ } \times \pp^k\biggr\}\ ] ] @xmath266 and that @xmath267 is an irreducible component of @xmath117}$ ] . then with regard to diagram , @xmath127 is an isomorphism , and yet for @xmath268 there is a class @xmath269};\qq)^{\circ}\ ] ] of the form @xmath270cycle@xmath271 , for which @xmath272};\qq),\ ] ] but @xmath273 . to see why @xmath274 , observe that since @xmath275 has negative weight , it suffices to show that the values of @xmath276 map to zero in @xmath277 for @xmath278 . but the relevant part of @xmath279 involves the cohomology @xmath280})\otimes h^0(\pp^k ) \subset h^{2r-2m+k}(y^{[m - k]})$ ] , which in the end involves the mixed hodge structure of @xmath281 but @xmath282}$ ] is a union of smooth hypersurfaces of dimension @xmath283 . since by lefschetz , the cohomology of hypersurfaces is only `` non - trivial '' in the middle dimension , and in light of the description of @xmath279 in ( [ 111 ] ) , it suffices to show that @xmath284 ( hence @xmath285 , as it reduces to the same thing as the homology of a simplex ) . but @xmath286 which is not the case for region i. thus @xmath274 , hence @xmath287 as well . in particular , @xmath288 , @xmath289 , and @xmath290 . thus by theorem [ t01](i ) , @xmath4 fails to be surjective . this section serves as a necessary forerunner to section [ sec010 ] . along the way we prove some results that are either new , or appear to be known only among experts . let @xmath77 be a smooth projective variety and @xmath291 a proper subvariety . there is a short exact sequence @xmath292 where , for notational simplicity , we write @xmath293 instead of @xmath294 , and let @xmath295 let us assume for the moment that @xmath296 is torsion - free . except for the obvious case @xmath297 this also holds in the following two cases : \(i ) @xmath298 . here @xmath299 is torsion - free , as can be seen from the long exact sequence of cohomology of @xmath15 associated to the short exact sequence @xmath300 \(ii ) @xmath301 . let @xmath82 be a divisor such that the image @xmath302 is precisely the algebraic part @xmath303 . then by the lefschetz @xmath304 theorem , @xmath305 is torsion - free , and @xmath306 is isomorphic to this group . then by purity of negative weight and torsion - freeness , @xmath307 corresponding to this is a commutative diagram ( use the fact that cycle class maps are compatible with localization sequences for the left - hand square , and the commutativity of the right - hand square can be deduced from an extension class interpretation of the abel - jacobi map ( see @xcite ) ) @xmath308 where @xmath309 and @xmath310 @xmath311 [ ex1 ] suppose that @xmath312 . then @xmath313 and @xmath314 are isomorphisms , hence the same holds for @xmath315 by . if we make the identifications @xmath316 , @xmath317 , then we arrive at the short exact sequence @xmath318 where @xmath319 is well - known ( see @xcite ) . this is also a consequence of the identification @xmath320 the surjectivity of @xmath321 in this case is also proven in @xcite . [ ex2 ] suppose that @xmath322 , and @xmath323 is a divisor . we observe that there is a short exact sequence @xmath324 where the first term may be identified with @xmath325 there is a canonical isomorphism @xmath326 by ( * ? ? ? * lemma 3.1 ) . ( in loc . this is formulated in terms of @xmath327-theory , but because of the particular indices , it gives exactly this result . ) we therefore obtain a short exact sequence ( using the fact that @xmath328 factors through deligne cohomology ) @xmath329 so that the snake lemma applied to yields an exact sequence @xmath330 when taking limits over @xmath331 in the above example @xmath332 becomes surjective , and @xmath333 in section [ sec01 ] . so using the description of @xmath334 for each @xmath331 above we obtain the following result . [ c100 ] @xmath335.\ ] ] let us now consider the general situation of @xmath234 with @xmath336 , so that we have the diagram tensored with @xmath337 . then @xmath338 need not be surjective ; moreover a detailed description of this map when @xmath82 is a ncd leads to the same kind of analysis as in section [ sec04 ] . recall that by a weight argument , @xmath339 for @xmath340 . with this in mind let us assume that @xmath341 . then as @xmath291 ranges over all pure codimension one subvarieties , the image of @xmath342 in ( [ e27 ] ) is @xmath343 because of dimensions . referring to @xmath344 , let us put @xmath345 where @xmath291 ranges over all pure codimension one which is finer that the coniveau filtration on @xmath346 . ] algebraic subsets of @xmath15 . note that from example [ ex2 ] , @xmath347 . now fix a @xmath348 of pure codimension one , with desingularization @xmath349 , and composite morphism @xmath350 . by a weight argument ( deligne ) together with the purity of @xmath351 , both gysin images ( @xmath352 ) in the commutative diagram below are the same . @xmath353 ^ -{\sigma_{\ast } } \ar[dd]_-{\lambda_{\ast } } \\ & h^{2r - m}(x,\qq(r ) ) \\ h_y^{2r - m}(x,\qq(r))\ar[ru]_-{j_{\ast } } \ , . } \ ] ] assuming the hodge conjecture , we can find @xmath354 in @xmath355 with @xmath356_{\ast } = { \rm id}_{\im(\sigma_{\ast})}$ ] , where @xmath357_{\ast } : h^{2r - m}(x,\qq(r ) ) \to h^{2r - m-2}(\widetilde{y},\qq(r-1))\ ] ] is induced by @xmath354 ( see ( * ? ? ? note that @xmath358 where @xmath359 . let us similarly write @xmath360 for tensored with @xmath112 , and let @xmath361 be the ( full ) abel - jacobi map . referring to ( [ e27]@xmath362 , we deduce from the hodge conjecture that @xmath363 indeed , if @xmath364 , then @xmath365 by functoriality of the abel - jacobi map . thus @xmath366 by @xmath367 . in particular , under the assumption of the hodge conjecture , for a fixed @xmath331 as above there are short exact sequences @xmath368 where @xmath369 @xmath370 taking the direct limit over @xmath331 we obtain a short exact sequence @xmath371 where @xmath372 @xmath373 [ c116 ] assume the hodge conjecture and let @xmath341 . then @xmath374 \(i ) note that corollary [ c116 ] for @xmath5 is essentially a conjectural type question of jannsen @xcite . \(ii ) let us ( again ) take the direct limit over @xmath82 of diagram ( [ e27]@xmath362 . by applying the snake lemma to the limit diagram , we deduce ( unconditionally ) that for @xmath341 , @xmath375 implies that @xmath376 is injective , which in turn implies by a generalization of beilinson rigidity theorem given in @xcite that @xmath377 is countable for @xmath378 . note that in the case @xmath208 , we have @xmath379 where @xmath380 was defined in example [ ex01 ] , and the statement of countability of @xmath380 is a conjecture of voisin . as always , @xmath381 is a smooth projective variety . this section concerns the following integrally defined map : @xmath382 mentioned in section [ sec01 ] . of course , @xmath383 . we shall prove in this section that @xmath384 is torsion - free , so that by a weight argument @xmath385 clearly , if @xmath386 is surjective then so is @xmath387 , but we shall show that the converse also holds . in fact , we expect the following to be true : [ c0001 ] the map in ( [ e001 ] ) is surjective . for the moment , let us restrict to the case @xmath312 . it is then easy ( and also follows from example [ ex1 ] ) that @xmath388 is surjective , with divisible kernel : @xmath389 . thus for any integer @xmath390 , @xmath391 if @xmath52 is a zariski open part of @xmath262 , then there is an exact sequence @xmath392 where @xmath393 indeed , from mixed hodge theory ( see @xcite(cor . 3.2.13(ii ) ) , the restriction map induces an isomorphism @xmath394 this together with the surjectivity of @xmath395 , implies that the restriction map @xmath396 is surjective ; moreover by a weak purity argument , @xmath397 . via the deligne cycle class maps , we have identifications , @xmath398 , @xmath399 , and hence by a localization sequence argument , the aforementioned identification @xmath400 . now assume given a `` good compactification '' pair @xmath401 , with ncd @xmath402 . there is an exact sequence : @xmath403 further , @xmath404 and @xmath405 next , by shrinking @xmath13 , and using that @xmath406 , we obtain from and a short exact sequence @xmath407 where the last term is uniquely divisible . hence , for @xmath408 , we find an isomorphism @xmath409 next , we observe that @xmath410 is torsion - free , since by the lefschetz @xmath304 theorem the torsion in @xmath411 is algebraic . this implies that @xmath412 finally , we deduce the well - known fact @xmath413 to see this in another context , let us work in the tale topology on a variety @xmath414 , and consider the sheaf @xmath415 on @xmath46 , where for @xmath416 tale , @xmath417 . now let @xmath418 . then by hilbert 90 , @xmath419^l,\ ] ] where @xmath420^l = \{x^l\ |\ x\in \cc(x)^{\times}\}$ ] . there are exact sequences @xmath421 @xmath422^l \xrightarrow{d\log } \omega^1_{\cc(x)/\cc},\ ] ] where @xmath423 are the khler differentials , which induces the short exact sequences @xmath424 and @xmath422^l \xrightarrow{d\log } l\cdot\gamma\big ( h^1(\cc(x),\zz(1))\big)\to 0 \,.\ ] ] thus going from the tale to betti cohomology with finite coefficients can be traced via the isomorphisms : @xmath425^l\ ] ] @xmath426 @xmath427 note that @xmath428 , and @xmath429 from ( [ e36 ] ) are the respective cycle class maps to betti cohomology ( analytic topology ) and tale cohomology . in summary , we have a commutative diagram corresponding to a morphism of sites from the tale to the analytic topologies . @xmath430 where the isomorphism in the bottom row is from . taking cup products , we have a similar diagram @xmath431 where the isomorphism @xmath432 here ( and in the previous diagram for @xmath433 ) , which really arises from the leray spectral sequence associated to a morphism of sites , can be deduced from ( * ? ? ? * thm 3.12 on p.117 ) . the bloch - kato / milnor conjectures ( now theorems @xcite ) tell us that for @xmath32 a non - zero integer , the induced map @xmath434 is an isomorphism .. ] in our situation , this translates to saying [ bk ] for @xmath435 , the map @xmath436 is an isomorphism for any integer @xmath390 . we can now prove the following result . note that part ( i ) for @xmath437 is immediate from the short exact sequence , and for @xmath438 follows from the lefschetz @xmath304 theorem . [ t0001 ] ( i ) @xmath439 is torsion - free for all @xmath63 . in particular , the torsion subgroup of @xmath440 is supported in codimension @xmath64 . @xmath441 in ( [ e001 ] ) is divisible . the groups @xmath442 are uniquely divisible . first observe that the map in theorem [ bk ] is the composition @xmath443 notice that the short exact sequence @xmath444 induces the short exact sequence @xmath445 by theorem [ bk ] , it follows that @xmath446 , thus proving part ( i ) . next observe that @xmath447 and hence @xmath448 using also theorem [ bk ] , we have the commutative diagram @xmath449 \ar[r]^-\simeq & h^m\biggl(\cc(x),\dfrac{\zz(m)}{l\cdot \zz(m)}\biggr ) \\ \dfrac{\im(d\log_m)}{l\cdot\im(d\log_m)}\ar[d ] & \\ \dfrac{\gamma\big(h^m(\cc(x),\zz(m))\big)}{l\cdot \gamma\big(h^m(\cc(x),\zz(m))\big ) } \ar@{^{(}->}[r ] & \dfrac{h^m(\cc(x),\zz(m))}{l\cdot h^m(\cc(x),\zz(m ) ) } \ar[uu]_-\simeq \ , , } \ ] ] where all maps must be isomorphisms . part ( ii ) follows by applying the snake lemma to multiplication by @xmath450 on the short exact sequence @xmath451 as @xmath452 is torsion - free . using the obvious abbreviations , part ( iii ) follows similarly from @xmath453 and @xmath454 . @xmath455 [ c99 ] @xmath456\ ] ] is uniquely divisible . apply theorem [ t0001](iii ) with @xmath457 to corollary [ c100 ] . note that one can define @xmath458 with integral coefficients analogous to . we assume @xmath459 . since @xmath460 is torsion - free by theorem [ t0001](i ) , diagram ( [ e27 ] ) becomes valid after passing to the generic point of @xmath15 . after applying the snake lemma , we arrive at the fact that @xmath461 injects into @xmath462 in the case @xmath7 we deduce [ c66 ] @xmath463 is torsion - free . hence we have an injection of torsion subgroups @xmath464 conjecture [ c0001 ] would imply that the maps @xmath465 in corollaries [ c99 ] and [ c66 ] are injective . fix @xmath466 . if @xmath26 , then according to theorem [ t01 ] , surjectivity of @xmath8 implies that @xmath467 is injective on @xmath468 . thus it makes sense to calculate @xmath469 in general . let @xmath470 be the inclusion of a ncd @xmath82 ( with smooth components ) . then @xmath471 note that @xmath472 can be calculated from the simplicial complex @xmath83}\to y$ ] . so there are residue maps @xmath473};\qq)}{\gy \big(\ch^{r - m-1}(y^{[m+1]};\qq)\big)},\ ] ] @xmath474};\qq)}{\gy \big(\ch^{r - m-1}(y^{[m+1]};\qq)\big)}.\ ] ] [ p004 ] for @xmath475 , @xmath476 if @xmath477 and @xmath8 is surjective , then the abel - jacobi map @xmath478 in induces an injection @xmath479 consider the exact sequence @xmath480 in the weight filtered spectral sequence obtained from involved in computing @xmath481 we can restrict our attention to those @xmath482},-j ) $ ] with @xmath483 , which converges to @xmath484 . it also exists for @xmath485 ( using the column where @xmath486 ) . we use indices to distinguish between the various spectral sequences . note that @xmath487 and that for @xmath488 , @xmath489 since @xmath490 on @xmath491 as its target is trivial . non - canonically , we have @xmath492 thus @xmath493 with the last term isomorphic with @xmath494 . one has a commutative diagram @xmath495 with surjective vertical map by . one sees the bottom row is exact by comparing the spectral sequences @xmath496 and @xmath497 at @xmath498 . hence @xmath499 . _ we now restrict to the case @xmath341 _ because @xmath500 for @xmath239 as @xmath501 } ) = 0 $ ] ( see ( [ e666 ] ) ) . let @xmath502 . then @xmath503 is of codimension @xmath55 , and hence @xmath504 is of codimension @xmath505 in @xmath15 . by hironaka , there is a proper modification diagram @xmath506 where @xmath507 is a ncd . let @xmath508 . we have a corresponding diagram with @xmath509 , @xmath510 where @xmath511 is in @xmath512 . obviously @xmath513 identity implies that @xmath514 is onto . a diagram chase shows that @xmath515 is a surjection as well . as @xmath516 is obtained from @xmath15 by a sequence of blow - ups with non - singular centres , it is clear from the well - known motivic decomposition of @xmath516 with respect @xmath15 , that the ( higher ) chow group of @xmath516 involves that of @xmath15 and of smooth irreducible divisors . hence it follows that @xmath517 . we deduce the isomorphisms @xmath518 @xmath519 let @xmath520 where @xmath521 runs over all proper modifications for all @xmath522 of codimension at least 1 . by construction we have @xmath523 now recall the subspace @xmath524 introduced in equation ( [ e88 ] ) . it follows from the definitions that @xmath525 since , by , @xmath526 in @xmath126 implies @xmath527 in @xmath528 . the above inclusion is an equality if @xmath7 , since in this case @xmath127 is an isomorphism . let @xmath529 be the subspace generated by images of the form @xmath530 under the product where @xmath531 [ p005 ] for @xmath532 we have @xmath533 . it is not clear if equality holds in the above proposition . consider @xmath534 , where @xmath535 . note that @xmath536 is supported in codimension @xmath537 , namely @xmath538 also @xmath539 and @xmath540 @xmath541 , which is not the case as we are assuming @xmath341 . therefore we can assume say @xmath542 . next , if @xmath543 , then @xmath544 , and thus @xmath545 implies @xmath546 is supported on a divisor in @xmath15 . this scenario can be handled in the same way as in the case where we assume that @xmath542 and @xmath547 . namely , since @xmath548 , we can assume that @xmath549 is supported on some @xmath291 of codimension @xmath550 , and by the surjectivity of @xmath515 in diagram ( [ e007 ] ) , we can assume without loss of generality that @xmath291 is a ncd . thus for some @xmath551 , @xmath552 . since @xmath15 is smooth , we have the product ( @xcite ) : @xmath553 which defines @xmath554 . but since @xmath546 can be assumed in general position with respect to @xmath82 , and together with @xmath555 , we have @xmath556 . let @xmath77 be a smooth projective surface , @xmath557 a ncd ( with smooth @xmath558 s ) with open complement @xmath559 . the work of deligne ( @xcite , cor . 3.2.13 and 3.2.14 ) implies that @xmath560 , where @xmath561 is the sheaf of rational @xmath562-forms on @xmath15 , regular on @xmath13 with logarithmic poles along @xmath82 . in particular if we denote by @xmath563 , the space of regular algebraic @xmath562-forms on @xmath13 with @xmath564-periods , then @xmath565 and where we allow for the possibility that the left hand side of ( [ e555 ] ) is non - zero . let @xmath566 be the singular set of @xmath82 . for each @xmath567 we choose distinct points @xmath568 . now let s modify @xmath15 by blowing it up along @xmath569 and call this @xmath570 . so in particular the strict transform @xmath571 of @xmath82 is a copy of @xmath82 itself . on @xmath572 we have for each @xmath573 an interesting real @xmath562-cycle @xmath574 obtained as follows : take the complement of a small disc in the blowup of @xmath575 , the complement of a small disc in the blowup of @xmath576 , and a small tube in @xmath577 along a path in @xmath567 from @xmath575 to @xmath576 that so that the end circles meet the circles of the two previous parts . then by a standard residue argument , integrating an element @xmath578 against @xmath574 gives us essentially @xmath579 times the integral along the part in @xmath567 from @xmath575 to @xmath576 of the residue of @xmath580 along @xmath82 , where we observe that @xmath580 restricts to zero on the other two parts of @xmath574 , viz , there are no non - zero holomorphic @xmath562-forms living on any subset of @xmath581 . picture of @xmath582 @xmath583 note that this integral is determined by the finite dimensional @xmath337-vector space of residues on @xmath82 . so pick @xmath575 and @xmath576 sufficiently general in @xmath567 so that we can never end up in @xmath564 with this integral except with the trivial residue . then doing this for all @xmath584 , we arrive at @xmath585 for which @xmath586 , using the fact that @xmath587 . we deduce : let @xmath588 with corresponding family @xmath589 . then @xmath590 for a very general point @xmath81 in @xmath591 . the purpose of this appendix is to establish the commutativity of diagram above . due to its technical nature , the reader with pressing obligations can easily skip this without losing sight of the main results of this paper . we first digress by describing the abel - jacobi map @xmath592 where @xmath593 is described in . in the case @xmath594 , viz . , @xmath595 this is induced by the classical abel - jacobi map @xmath596};\qq ) \xrightarrow{\xi \mapsto\int_{\del^{-1}\xi}(- ) } j\big(h^{2r-2m+1}(y^{[m-1]},\qq(r - m+1))\big).\ ] ] the abel - jacobi map @xmath597 is defined using the formula in @xcite . by degeneration of the mixed hodge complex spectral sequence at @xmath598 ( deligne ) , and a map of spectral sequences from chow groups to hodge cohomology , together with functoriality of the abel - jacobi map , the map @xmath599 induces @xmath147 for all @xmath600 . so we need only describe the map @xmath599 explicitly . let @xmath601 with coordinates @xmath69 and @xmath70 codimension one faces obtained by setting @xmath71 , and boundary maps @xmath602 , where @xmath73 denote the restriction maps to the faces @xmath74 respectively . here we adopt the notation in @xcite adapted to the cubical description of @xmath603 , with cycles lying in @xmath604 , in general position with respect to the @xmath605 faces of @xmath606 as well as the real part @xmath607^m \subset \square^m$ ] . recall the tate twist @xmath608 , let @xmath77 be smooth projective with @xmath609 , and put @xmath610 sheaf of currents that act on compactly supported @xmath173-valued @xmath611 forms of degree @xmath612 . note that @xmath613 where @xmath614 acts on corresponding @xmath615 forms . let @xmath616 be the sheaf of borel - moore chains of real codimension @xmath617 in @xmath15 with @xmath608 coefficients . one has an inclusion @xmath618 . now put @xmath619.\ ] ] the cohomology of this complex at @xmath620 is precisely the deligne cohomology @xmath621 ( see @xcite ) . we recall the cycle class map @xmath622 in terms of the cubical description of @xmath623 . note that for @xmath118 , @xmath624 and under this identification , @xmath625 is the abel - jacobi map . on @xmath606 we introduce the currents @xmath626 @xmath627^m}(- ) = : \delta_{[-\infty,0]^m},\ ] ] where @xmath628 is integration on @xmath629\big\}$ ] . @xmath630 @xmath631 where @xmath632 has the principle branch . one has @xmath633 we consider a cycle @xmath634 in general position . one considers the projections @xmath635 . we put @xmath636 note that when @xmath11 , @xmath637 ( classical case ! ) . correspondingly @xmath638 up to a normalizing constant , the cycle class map @xmath625 is induced by @xmath639 recall @xmath640 . the abel - jacobi map @xmath641 where we described @xmath642 using the carlson isomorphism , is defined as follows . if @xmath643 , then by a moving lemma ( * ? ? ? * lemma 8.14 ) , we can assume that @xmath59 is in general position with respect to the real cube @xmath607^m\subset \square^m$ ] . furthermore , @xmath644 implies @xmath645 , i.e. , @xmath646^m\}$ ] , @xmath647 , for some @xmath648 , so up to a normalizing constant and for @xmath649 , we have @xmath650 where the latter equality stems from hodge type considerations . we have to unravel the definitions . we use the simplicial complex @xmath83}\to y\hookrightarrow x$ ] as a way of describing @xmath652 . let @xmath653 } ) = \c^{2r-2m+i}(y^{[m - i]},\qq(r - m+i))\ ] ] @xmath654 } ) = \dd^{2r-2m+i}(y^{[m - i]})\ ] ] @xmath655 a class @xmath656 is represented by a @xmath657-closed @xmath658-tuple @xmath659}).\ ] ] with @xmath660 , consider the short exact sequence @xmath661 let us first describe @xmath662 . let @xmath663 and @xmath664 . in this case @xmath665 is in the image of @xmath666 . likewise @xmath667 is in the image of some @xmath668 . the difference @xmath669 maps to a class in @xmath670 which defines the abel - jacobi image of @xmath671 in @xmath670 . we can assume that @xmath672 is represented by the @xmath657-closed @xmath57-tuple : @xmath673}).\ ] ] with @xmath674 . the corresponding value @xmath675 is given by the abel - jacobi membrane integral @xmath676 , as the hodge contribution given by @xmath677 is trivial for hodge type reasons . this is easily seen to be precisely @xmath678 , where we recall that @xmath679 . thus @xmath680 . so now suppose that @xmath681 , i.e. @xmath682 . this means that @xmath683 , where @xmath684},1;\qq)$ ] . then @xmath685 . ( we comment in passing that using the aforementioned moving lemma in @xcite , we can assume that @xmath686}\times [ -\infty,0]$ ] is a proper intersection , and hence that @xmath687 . ) since @xmath688 is a coboundary , it follows that @xmath689 , and hence after removing classes in @xmath690 , [ specifically , @xmath691 , and using @xmath692 , @xmath693 and using @xmath694 , we can assume that @xmath672 is represented by the @xmath657-closed @xmath57-tuple @xmath695}),\ ] ] where @xmath696 note that @xmath697 as @xmath698 ( @xmath699 representing the classical case ! ) . but @xmath700 . hence working modulo the coboundary @xmath701 we can assume that @xmath672 is represented by the @xmath657-closed @xmath702-tuple @xmath703 } ) \,.\ ] ] so modulo @xmath657-coboundary , @xmath704 is represented by the @xmath1-closed current @xmath705 , which is precisely @xmath706 . hence @xmath707 . the general case @xmath708 proceeds in a similar fashion . h. esnault and e. viehweg , _ deligne - beilinson cohomology , _ in beilinson s conjectures on special values of @xmath710-functions , ( rapoport , schappacher , schneider , eds . ) , perspect . math . * 4 * , academic press , 1988 , 43 - 91 . in : the arithmetic and geometry of algebraic cycles , proceedings of the crm summer school , june 7 - 19 , 1998 , banff , alberta , canada ( editors : b. gordon , j. lewis , s. mller - stach , s. saito and n. yui ) , nato science series * 548 * ( 2000 ) , 225 - 260 , kluwer academic publishers .

let @xmath0 be a smooth quasi - projective variety of dimension @xmath1 , @xmath2 bloch s higher chow group , and @xmath3 the cycle class map . beilinson once conjectured @xmath4 to be surjective @xcite ; however jannsen was the first to find a counterexample in the case @xmath5 @xcite . in this paper we study the image of @xmath4 in more detail ( as well as at the `` generic point '' of @xmath6 ) in terms of kernels of abel - jacobi mappings . when @xmath7 , we deduce from the bloch - kato conjecture ( now a theorem ) various results , in particular that the cokernel of @xmath8 at the generic point is the same for integral or rational coefficients .

the understanding of the production of the heavy quarkonium ( bound states of heavy quark ( @xmath8 ) and heavy anti - quark ( @xmath9 ) ) has been a long - term effort both experimentally and theoretically . the different treatments of the non - perturbative evolution of the @xmath10 pair into a quarkonium lead to various theoretical models . there are mainly three models widely used to describe the production of quarkonium : the color singlet model ( csm ) , the color evaporation model ( cem ) and the nrqcd framework . the csm , proposed right after the discovery of the @xmath0 , assumes that the @xmath10 pair evolving in a quarkonium state is in a color - singlet ( cs ) state and the quantum numbers such as spin and angular momentum , are conserved after the formation of the quarkonium . the only inputs required in the model are the absolute value of the colour singlet @xmath10 wave function and its derivatives that can be determined from data of decay processes . once these quantities are provided , the csm has no free parameters @xcite . the csm at leading - order , predicts well the quarkonium production rates at relatively low energy @xcite , but fails to describe the data for charmonium measured by cdf experiment in @xmath11 collisions @xcite probably because it ignores the fragmentation processes from higher states or @xmath5 mesons , dominant at tevatron energies @xcite . recently it has been revived , with the computation at higher orders in the strong coupling constant @xmath12 expansion @xcite , since it was found to better accomodate polarization results from tevatron with respect to nrqcd . the cem @xcite is a phenomenologically successful model and was first proposed in 1977 @xcite . in the cem , the cross - section for a quarkonium state @xmath13 is a fraction @xmath14 of the cross - section of the produced @xmath10 pairs with invariant mass below the @xmath15 threshold , where @xmath16 is the lowest mass meson containing the heavy quark @xmath8 . this cross - section has an upper limit on the @xmath10 pair mass but no constraints on the colour or spin of the final state . the @xmath10 pair is assumed to neutralize its colour by interaction with the collision - induced colour field by `` colour evaporation '' . an important feature is that the fractions @xmath14 are assumed to be universal so that , once they are determined by data , they can be used to predict the cross - sections in other processes and in other kinematical regions . the most basic prediction of the cem is that the ratio of the cross - sections for any two quarkonium states should be constant , independent of the process and the kinematical region . variations in these ratios have been observed : for example the ratio of the cross - sections for @xmath17 and @xmath0 are rather different in photoproduction and hadroproduction and the ratio between different charmonium cross - sections measured at lhc is not constant as a function of @xmath18 . these variations represent a serious challange to the status of the cem as a quantitative phenomenological model for quarkonium production . however , the model is still widely used as simulation benchmark since , once the @xmath14 fractions are determined , it has a full predicting power about cross - sections but it fails to predict the quarkonium polarization . on the other hand , nrqcd can predict both the cross - section and the polarization of quarkonium production . in nrqcd , contributions to the quarkonium cross - section from the heavy - quark pairs produced in a color - octet ( co ) state are also taken into account , in addition to the cs contributions described above . the picture of the nrqcd @xcite formalism is as follows . the orbital splittings in case of quarkonium bound states are smaller than the heavy quark mass @xmath19 , which suggests that all the other dynamical scales of these systems are smaller than @xmath19 . so , the relative velocity @xmath20 between @xmath8 and @xmath9 is believed to be a small quantity , @xmath20 @xmath21 1 . therefore , a hierarchy of scales , @xmath19 @xmath22 @xmath23 @xmath22 @xmath24 , as observed in a non - relativistic ( nr ) system , also holds for quarkonia . here , @xmath19 fixes the distance range for @xmath25 creation and annihilation processes , the momentum @xmath23 is inversely proportional to the spatial size of the bound state and the kinetic energy @xmath24 determines the typical interaction time scale . these different distance scales make the study of quarkonium production interesting and nrqcd calculation incorporates this scale hierarchy . the quarkonia production in nrqcd is calculated in two steps . at first , the creation of the @xmath25 pair in a hard scattering at short distances which is calculated perturbatively as an expansions in the the strong coupling constant @xmath12 . note that @xmath25 states can be in a cs state @xcite as well as in a co state @xcite . then , the @xmath25 pair is evolved into the quarkonium state with the probabilities that are given by the assumed universal nonperturbative long - distance matrix elements ( ldmes ) which are estimated on the basis of the comparison with experimental measurements . for co states , this evolution process also involves the nonperturbative emission of soft gluons to form cs states . the crucial feature of this formalism is that it takes into account the complete structure of the @xmath25 fock space , which is spanned by the states @xmath26}_j$ ] , where @xmath27 , @xmath28 and @xmath4 are the spin , orbital and total angular momenta , respectively and @xmath29 is the color multiplicity . a remarkable progress has been made in quarkonium production studies during last decade based on the nrqcd formalism @xcite . in recent times , the charmonium production in @xmath2 collisions has been measured at @xmath30 and 7 tev by the alice @xcite , atlas @xcite , cms @xcite and lhcb @xcite collaborations at forward , near forward and mid rapidities . it may be noted here that atlas , cms and lhcb collaborations report the prompt production cross - sections while alice measurements include also the @xmath5 feed - down to @xmath0 and @xmath1 . the fonll @xcite formalism has been used to calculate the production cross - sections of @xmath0 and @xmath1 from @xmath5 meson decays which accounts for the feed - down contributions from @xmath5 meson to the @xmath0 and @xmath1 productions . in the present work , the charmonium cross - sections have been calculated at @xmath30 and 7 tev within the framework of lo nrqcd and compared with available experimental data from lhc . the predictions for the production cross - sections of @xmath0 and @xmath1 in @xmath2 collisions at @xmath31 = 13 tev has been made as these collisions are foreseen at lhc in 2015 . in addition , the calculations have also been performed at @xmath31 = 5.1 tev which can be utilized for the normalization of the pb - pb data to be collected at @xmath32 tev . the organization of the paper is as follows . in sec . ii , we give a brief description of the theoretical model of nrqcd . results and comparison with experimental measurements will be presented in sec . iii followed by summary and discussion in sec . the factorization formalism of the nrqcd provides a theoretical framework for studying the heavy quarkonium production and decay . according to the nrqcd factorization formalism , the cross - section for direct production of a resonance @xmath13 in a collision of particle @xmath34 and @xmath5 can be expressed as d_a+bh+x = _ a , b , ndx_a dx_b g_a / a(x_a,^2_f ) g_b / b(x_b,^2_f ) + d(a+bq|q(n ) + x)<^h(n ) > where , @xmath35 is the parton distribution function ( pdf ) of the incoming parton @xmath36 in the incident hadron @xmath37 , which depends on the momentum fraction @xmath38 and the factorization scale @xmath39 as well as on the renormalization scale @xmath40 . however , as we have chosen @xmath39 = @xmath40 , in our case pdfs are function of @xmath41 and @xmath39 only . the tranverse mass of the resonance @xmath13 is @xmath42 , where @xmath43 is the mass of resonance @xmath13 . the short distance contribution @xmath44 can be calculated within the framework of perturbative qcd ( pqcd ) . on the other hand , @xmath45 ( the state @xmath46}_j$ ] ) are nonperturbative ldmes and can be estimated on the basis of the comparison with experimental measurements . the differential cross - section for the short distance contribution i.e. the heavy quark pair production from the reaction of the type @xmath47 , where @xmath48 refer to light incident partons , @xmath49 refers to @xmath25 pair and @xmath50 is the light final state parton , can be written as @xcite = dx_a g_a / a(x_a,^2_f ) g_b / b(x_b,^2_f ) + 2p_t ( abcd ) , where , @xmath31 being the total energy in the centre - of - mass and @xmath51 is the rapidity of the @xmath25 pair . in our numerical computation , we use cteq6 m @xcite for the parton distribution functions . the invariant differential cross - section is given by = , where @xmath52 and @xmath53 are the parton level mandelstam variables . @xmath54 is the feynman amplitude for the process . the value of the momentum fraction @xmath55 can be written as , x_b = . the minimum value of @xmath56 is x_amin = . the ldmes are predicted to scale with a definite power of the relative velocity @xmath20 of the heavy constituents inside @xmath25 bound states . in the limit @xmath57 , the production of quarkonium is based on the @xmath58}$ ] and @xmath59}$ ] ( @xmath4 = 0,1,2 ) cs states and @xmath60}$ ] , @xmath61}$ ] and @xmath62}$ ] co states . in our calculations , we used the expressions for the short distance cs cross - sections given in refs . @xcite and the co cross - sections given in refs . @xcite . in this paper we calculate the @xmath18 distribution of @xmath0 and @xmath1 in @xmath2 collisions at lhc energies . for @xmath0 production in @xmath2 collisions , three sources need to be considered : direct @xmath0 production , feed - down contributions to the @xmath0 from the decay of heavier charmonium states , predominantly from @xmath1 , @xmath63 , @xmath64 and @xmath65 and @xmath0 from @xmath5 hadron decays . the sum of the first two sources is called `` prompt @xmath0 '' and the third source will be called `` @xmath0 from @xmath5 '' . on the other hand , @xmath1 has no significant feed - down contributions from higher mass states . we call this direct contribution as `` prompt @xmath1 '' to be consistent with the experiments . the other source to @xmath1 production is from @xmath5 hadron decays and we call it `` @xmath1 from @xmath5 '' . the sum of the prompt @xmath0(@xmath1 ) and @xmath0(@xmath1 ) from @xmath5 will be called `` inclusive @xmath0(@xmath1 ) '' . the direct production cross - section of @xmath0 can be written as the sum of the contributions @xcite , d(j/ ) = d(q|q ( ^3s_1^[1]))<(q|q ( ^3s_1^[1])j/ ) > + + d(q|q ( ^1s_0^[8]))<(q|q ( ^1s_0^[8])j/ ) > + + d(q|q ( ^3s_1^[8]))<(q|q ( ^3s_1^[8])j/ ) > + + d(q|q ( ^3p_j^[8]))<(q|q ( ^3p_j^[8])j/ ) > + + ... similar expression holds for direct @xmath1 production . the direct production cross - section for @xmath3 can be written as @xcite : d(_cj ) = d(q|q ( ^3p_j^[1]))<(q|q ( ^3p_j^[1])_cj ) > + + d(q|q ( ^3s_1^[8]))<(q|q ( ^3s_1^[8])_cj > + + .... here , we have taken into account the contributions from all three @xmath3 ( @xmath63 , @xmath64 and @xmath65 ) mesons to @xmath0 . .the color - singlet and color - octet matrix elements with numerical values and nrqcd scaling order . [ cols="<,^,^,^,^ " , ] to calculate the direct charmonia and feed - down contributions from heavier states as well as from @xmath5 decays , we use the following branching ratios : @xmath66(@xmath67\rightarrow \mu^{+}\mu^{-}$])=0.0593[0.0078 ] , @xmath66[@xmath68=0.603 . @xmath66(@xmath69)=0.0130 , 0.348 , 0.198 for @xmath4 = 0 , 1 , 2 , respectively and @xmath66(@xmath70$])=0.116[0.283 ] @xcite . to choose the renormalization scale @xmath71 and the factorization scale @xmath39 in this calculations is an important issue and it may cause the uncertainties in the calculations . the choice that @xmath72 = @xmath71 = @xmath73 is the default one in the calculation , with @xmath74 being mass of the charm quark assumed to be 1.4 gev . moreover , it has been shown in @xcite that the scale variation does not improve the result for @xmath0 . thus , in our case we kept @xmath72 = @xmath71 . the ldmes @xcite for cs and co which we have used for our calculations are given in the table i. the central values of ldmes are taken for our calculations . for fonll @xcite calculations , pdf used is cteq6.6 and the central values of the factorization and renormalization scales are chosen to be @xmath75 = @xmath76 , where @xmath18 and @xmath77 are the transverse momentum and mass of the b - quark and central value of @xmath77 = 4.75 gev is used . in order to estimate the uncertainty on the calculated values , four possible sources have been considered namely , the factorization scale , the mass of the charm quark , the branching ratios for the feed - down to @xmath0 and the pdfs . the largest uncertainty in the branching ratio is 5% which corresponds to @xmath78 channel . the uncertainty due to the assumed pdf was estimated by performing the calculations with different pdfs namely , cteq6 m , cteq6l and cteq6l1 . the results of these calculations agree within 10% . the uncertainty due to the charm quark mass was estimated by carrying out the numerical calculations for @xmath74 = 1.2 and 1.6 gev . the variation was found to be about 12% . on the other hand , the uncertainty due to the variation in the values of factorization and renormalization scales was found to be as large as 45% and 30% when the value is reduced and enhanced by a factor of two , repectively . thus , this is the most dominant source of uncertainy and in all the subsequent plots for the numerical values , the uncertainty bands correspond only to this source . this assumption is valid in case the four sources of uncertainty are assumed to be uncorrelated and can be added in quadrature . the nrqcd calculations have been carried out for the differential cross - sections of @xmath0 and @xmath1 as a function of @xmath18 at @xmath31 = 2.76 and 7 tev . the numerical results have been compared with experimental data available from cms ( @xmath79 , @xmath80 , @xmath81 and @xmath82 ) , atlas ( @xmath83 and @xmath84 ) , lhcb ( @xmath85 ) and alice ( @xmath86 ) . thus , this comprehensive study explores the validity of nrqcd calculation at mid , near forward and forward rapidities at lhc energies . as a function of @xmath87 compared with the atlas @xcite , cms @xcite , lhcb @xcite and alice @xcite data . for data , the vertical error bars represent the statistical errors while the boxes correspond to the systematic uncertainties . we have shown the sum of all contributions with a green band . the direct and feed - down contributions to @xmath0 are shown only by lines which are for the central values.,title="fig:",width=232,height=188 ] as a function of @xmath87 compared with the atlas @xcite , cms @xcite , lhcb @xcite and alice @xcite data . for data , the vertical error bars represent the statistical errors while the boxes correspond to the systematic uncertainties . we have shown the sum of all contributions with a green band . the direct and feed - down contributions to @xmath0 are shown only by lines which are for the central values.,title="fig:",width=232,height=188 ] + as a function of @xmath87 compared with the atlas @xcite , cms @xcite , lhcb @xcite and alice @xcite data . for data , the vertical error bars represent the statistical errors while the boxes correspond to the systematic uncertainties . we have shown the sum of all contributions with a green band . the direct and feed - down contributions to @xmath0 are shown only by lines which are for the central values.,title="fig:",width=232,height=188 ] as a function of @xmath87 compared with the atlas @xcite , cms @xcite , lhcb @xcite and alice @xcite data . for data , the vertical error bars represent the statistical errors while the boxes correspond to the systematic uncertainties . we have shown the sum of all contributions with a green band . the direct and feed - down contributions to @xmath0 are shown only by lines which are for the central values.,title="fig:",width=232,height=188 ] + as a function of @xmath87 compared with the atlas @xcite , cms @xcite , lhcb @xcite and alice @xcite data . for data , the vertical error bars represent the statistical errors while the boxes correspond to the systematic uncertainties . we have shown the sum of all contributions with a green band . the direct and feed - down contributions to @xmath0 are shown only by lines which are for the central values.,title="fig:",width=232,height=188 ] as a function of @xmath87 compared with the atlas @xcite , cms @xcite , lhcb @xcite and alice @xcite data . for data , the vertical error bars represent the statistical errors while the boxes correspond to the systematic uncertainties . we have shown the sum of all contributions with a green band . the direct and feed - down contributions to @xmath0 are shown only by lines which are for the central values.,title="fig:",width=232,height=188 ] in fig . [ fig1 ] , the numerical values from the nrqcd calculations for differential cross - section of @xmath0 as a function of @xmath18 have been compared with the experimental values obtained by the four experiments at lhc namely , atlas @xcite , cms @xcite , lhcb @xcite and alice @xcite at @xmath31 = 7 tev . it may be noted that the @xmath5 feed - down contribution in case of alice has been accounted using fonll . it is observed from fig . [ fig1 ] , that the calculated values show good agreement with all the experimental data for @xmath88 4 gev . in a recent publication by yan - qing ma and raju venugopalan @xcite , it has been demonstrated that the low @xmath18 cross - section can be reproduced by inclusion of color glass condensate ( cgc ) effects within the nrqcd framework . the calculated values of the differential cross - section of @xmath1 as a function of @xmath18 using the nrqcd framework have been compared with cms @xcite , lhcb @xcite and alice @xcite and are shown in fig . it is important to note that for @xmath1 there is no contribution from the higher excited charmonium states . thus , the prompt and direct production is the same . again for alice , @xmath5 feed - down to @xmath1 has been calculated from the fonll . the calculated and measured values for @xmath1 are also in good agreement . as a function of @xmath87 compared with the cms @xcite , lhcb @xcite and alice @xcite.,title="fig:",width=232,height=188 ] as a function of @xmath87 compared with the cms @xcite , lhcb @xcite and alice @xcite.,title="fig:",width=232,height=188 ] + as a function of @xmath87 compared with the cms @xcite , lhcb @xcite and alice @xcite.,title="fig:",width=232,height=188 ] as a function of @xmath87 compared with the cms @xcite , lhcb @xcite and alice @xcite.,title="fig:",width=232,height=188 ] the numerical calculations for inclusive @xmath0 production were also carried out for @xmath31 = 2.76 tev and compared with the reported inclusive measurements from lhcb @xcite and alice @xcite in fig . [ fig3 ] . in this case the calculated and measured values for @xmath0 are in good agreement for @xmath6 3 gev . as a function of @xmath87 compared with the lhcb @xcite and alice @xcite.,title="fig:",width=232,height=188 ] as a function of @xmath87 compared with the lhcb @xcite and alice @xcite.,title="fig:",width=232,height=188 ] the alice collaboration has also reported the ratio of the differential cross - sections of @xmath1 to @xmath0 at @xmath31 = 7 tev @xcite . the measured and calculated values are shown in fig . the agreement is reasonable for @xmath89 4 gev and the increasing trend of the value for the ratio has been well reproduced . it is worth noting that the prediction for this ratio from cem is independent of @xmath18 . to @xmath0 production cross - section ratio as a function of @xmath87 compared to the alice @xcite.,width=245,height=188 ] at @xmath31 = 2.76 , 5.1 and 13 tev at mid and forward rapidity.,title="fig:",width=232,height=188 ] at @xmath31 = 2.76 , 5.1 and 13 tev at mid and forward rapidity.,title="fig:",width=232,height=188 ] + at @xmath31 = 2.76 , 5.1 and 13 tev at mid and forward rapidity.,title="fig:",width=232,height=188 ] at @xmath31 = 2.76 , 5.1 and 13 tev at mid and forward rapidity.,title="fig:",width=232,height=188 ] + at @xmath31 = 2.76 , 5.1 and 13 tev at mid and forward rapidity.,title="fig:",width=232,height=188 ] at @xmath31 = 2.76 , 5.1 and 13 tev at mid and forward rapidity.,title="fig:",width=232,height=188 ] at @xmath31 = 2.76 , 5.1 and 13 tev at mid and forward rapidity.,title="fig:",width=232,height=188 ] + at @xmath31 = 2.76 , 5.1 and 13 tev at mid and forward rapidity.,title="fig:",width=232,height=188 ] at @xmath31 = 2.76 , 5.1 and 13 tev at mid and forward rapidity.,title="fig:",width=232,height=188 ] + at @xmath31 = 2.76 , 5.1 and 13 tev at mid and forward rapidity.,title="fig:",width=232,height=188 ] this success of nrqcd calculations in describing the @xmath2 collisions data at various rapidities and energies , have prompted the predictions at @xmath31 = 2.76 , 5.1 and 13 tev . these predictions have been shown in fig . [ fig5 ] and [ fig6 ] . it will be interesting to test applicability of these calculations at much higher centre - of - mass energy of 13 tev in 2015 . on the other hand , the predictions at @xmath31 = 2.76 and 5.1 tev may be used for the normalization of the pb - pb collisions data . in summary , the prompt and inclusive production cross - sections of @xmath0 and @xmath1 at lhc energies have been calculated within the framework of lo nrqcd and fonll . these calculations include the contributions from direct production and from the decays of heavier charmonium states such as @xmath1 , @xmath63 , @xmath64 and @xmath65 . the feed - down to @xmath0 and @xmath1 from @xmath5 meson decays has been implemented using the fonll calculation . the comparisons with experimental data from lhc at different energies and rapidity windows show the lo nrqcd calculations give a good description of the production cross - sections of @xmath0 and @xmath1 for @xmath6 4 gev . the calculations for the prediction of production cross - sections of @xmath0 and @xmath1 at @xmath31 = 2.76 , 5.1 and 13 tev has been carried out . it may be noted that the fragmentation process contributes to the charmonium production at high @xmath18 @xcite and inclusion of this process may further improve the calculations . the production cross - sections at low @xmath18 has been well reproduced with the cgc+nrqcd formalism @xcite which is important for alice and lhcb data . in future , we intend to adopt the cgc formalisms @xcite for quarkonium production in the low @xmath87 region to cover the entire @xmath87 range with the inclusion of all the feed - down contributions . it is a pleasure to thank matteo cacciari for helpful discussion about fonll . the work of b. p. was supported by csir , india ( file no . 09/489(0085)/2010-emr-i ) . 50 c. chao - hsi , nucl . phys . * b172 * , 425 ( 1980 ) . g. a. schuler , arxiv : hep - ph/9403387 . et al . _ , lett . * 69 * , 3704 ( 1992 ) . e. braaten _ et al . _ , rev . lett . * 71 * , 1673 ( 1993 ) . p. artoisenet , j.p . lansberg , and f. maltoni , phys . b * 653 * , 60 ( 2007 ) . j. campbell , f. maltoni , and f. tramontano , phys . * 98 * , 252002 ( 2007 ) . p. artoisenet , j. campbell , j. p. lansberg , f. maltoni , and f. tramontano , phys . * 101 * , 152001 ( 2008 ) . bin gong and jian - xiong wang , phys . d * 78 * , 074011 ( 2008 ) bin gong and jian - xiong wang , phys . lett * 100 * , 232001 ( 2008 ) . bin gong , xue qian li and jian - xiong wang , phys . b * 673 * , 197 ( 2009 ) . y. q. ma , k. wang and k. t. chao , phys . d * 84 * , 114001 ( 2011 ) . yan - qing ma , kai wang and kuang - ta chao , phys . rev . lett * 106 * , 042002 ( 2011 ) . mathias butenschon and bernd a. kniehl , phys . lett * 106 * , 022003 ( 2011 ) . bin gong , lu - ping wan , jian - xiong wang and hong - fei zhang , phys . lett * 110 * , 042002 ( 2013 ) . g. t. bodwin , h. s. chung , u - rae kim and j. lee , phys . lett * 113 * , 022001 ( 2014 ) . r. aaij _ et al . _ , ( lhcb collaboration ) eur.phys.j . c * 71 * , 1645 ( 2011 ) . r. aaij _ et al . _ , ( lhcb collaboration ) eur.phys.j . c * 72 * , 2100 ( 2012 ) . r. aaij _ et al . _ , ( lhcb collaboration ) jhep * 02 * , 041 ( 2013 ) .

we have performed a systematic study of @xmath0 and @xmath1 production in @xmath2 collisions at different lhc energies and at different rapidities using the leading order ( lo ) non - relativistic qcd ( nrqcd ) model of heavy quarkonium production . we have included the contributions from @xmath3 ( @xmath4 = 0 , 1 , 2 ) and @xmath1 decays to @xmath0 . the calculated values have been compared with the available data from the four experiments at lhc namely , alice , atlas , cms and lhcb . in case of alice , inclusive @xmath0 and @xmath1 cross - sections have been calculated by including the feed - down from @xmath5 meson using fixed - order next - to - leading logarithm ( fonll ) formalism . it is found that all the experimental cross - sections are well reproduced for @xmath6 4 gev within the theoretical uncertainties arising due to the choice of the factorization scale . we also predict the transverse momentum distributions of @xmath0 and @xmath1 both for the direct and feed - down processes at the upcoming lhc energies of @xmath7 5.1 tev and 13 tev for the year 2015 .

spin exchange interaction between atoms in a spin-1 bose - einstein condensate ( bec ) causes complex spin mixing dynamics and spin diffusion , which is a major obstacle to realize experimentally a high precision magnetometer based on spinor bec @xcite . in order to improve the sensitivity of the magnetometer , a smaller spin exchange interaction is required , which may be implemented effectively by dynamical decoupling method using optical feshbach resonance techniques @xcite . in addition , the small spin exchange interaction can be utilized to resolve the ambiguity of the spin texture in ferromagnetically interacting @xmath0rb spin-1 bec , where the spatial texture structure may be induced by the spin exchange interaction , the magnetic dipolar interaction , or both of them @xcite . however , a more experimentalist - friendly proposal to suppress the spin exchange interaction is employing the magnetic pulses and the microwave pulses , which are much easier to implement and tune experimentally @xcite . by applying a magnetic field to an atomic spin-1 bec , only considered is the quadratic zeeman effect @xmath1 which is proportional to the square of the field , because the linear zeeman effect can be eliminated mathematically by adopting the rotating reference frame , due to the conservation of the total magnetization of the spin-1 condensate @xcite . under current experimental conditions , the effective quadratic zeeman energy of either the magnetic field or the microwave driving field can be adjusted from -240 hz to + 240 hz , which is about 10 times larger than the spin exchange interaction for typical densities of a @xmath0rb spin-1 condensate , @xmath2 @xmath3 @xcite . in this paper , we propose to localize the spin dynamics of a spin-1 bec by periodically applying magnetic and/or microwave field pulses , which effectively suppress the spin exchange interaction . by applying two - step pulse cycles with positive @xmath1 only , the condensate dynamics is localized if the relative phase of the initial state is close to zero ; by applying four - step pulse cycles with both positive and negative @xmath1 , the condensate dynamics is localized for an _ arbitrary _ initial state . the exploration of the robustness of the protocols shows that a wide parameter regime exists for a spin-1 condensate under current experimental conditions . this proposal may find its potential application to improve the sensitivity of a practical high - resolution magnetometer based on spin-1 bec . the paper is organized as follows . in sec . [ sec : td ] , we review the theoretical description of the free spin mixing dynamics under the single spatial mode approximation ( sma ) in a spin-1 bec in a magnetic field , whose quadratic zeeman splitting @xmath1 ranges from large negative values to large positive values . in sec . [ sec : ld ] , we analytically design and numerically confirm the control protocols of magnetic / microwave pulses to localize the condensate spin dynamics , where either a two - step or a four - step pulse cycle is employed . furthermore , the robustness of the control protocols is explored in sec . [ sec : r ] by assuming 5% random error of the pulse amplitude @xmath4 . finally , a brief summary is presented in sec . [ sec : con ] . within the mean field theory , the free spin mixing dynamics in a spin-1 bec with either ferromagnetic or antiferromagnetic spin exchange interaction under the sma in a magnetic field is described by the following equation of motion @xcite @xmath5 where @xmath6 with @xmath7 being the total number of atoms in the condensate and @xmath8 a normalized spatial mode function under the sma , which is determined by a scalar gross - pitaevskii equation with a spin - independent interaction , @xmath9\phi(\vec{r})= \mu \phi(\vec{r})$ ] where @xmath10 is the atomic mass and @xmath11 is the external harmonic trapping potential . the spin independent coefficient @xmath12 and spin exchange coefficient @xmath13 are given , respectively , by @xmath14 and @xmath15 with the @xmath16-wave scattering length @xmath17 for two spin-1 atoms in the compound symmetric channel of total spin 0(2 ) . for two popular ultracold spin-1 atomic gases in experiments , @xmath0rb and @xmath18na , @xmath19 is always satisfied and thus guarantees the validity of the sma in most experimental situations @xcite . the fractional population of spin component @xmath20 satisfies @xmath21 . the magnetization @xmath22 is a constant during the evolution , due to the isotropic nature of the spin exchange interaction . the relative phase among the three components is @xmath23 with @xmath24 being the phase of the spin wave function . the quadratic zeeman energy is @xmath25 with @xmath26 the zeeman energy shift of the component . in general , @xmath27 hz / g@xmath28 for @xmath0rb becs and @xmath29 hz / g@xmath28 for @xmath18na becs , where the magnetic field @xmath30 is in unit of gauss . due to the conservation of the magnetization @xmath31 , the linear zeeman energy @xmath32 can be eliminated mathematically by adopting a rotating reference frame . the total spin energy is a constant during the free evolution of the spin-1 condensate in a magnetic field @xmath33+\delta(1-\rho_0).\nonumber\ ] ] starting from a given initial state , which is usually a ground state in a magnetic field in experiments , the condensate evolves according to an iso - energy trajectory in the plane of @xmath34-@xmath35 , by changing abruptly the magnetic field to a different value . by taking into account of the energy conservation , the eq . ( [ ea ] ) is further simplified as @xmath36 [ ( 2c\rho_0+\delta)(1-\rho_0)-\varepsilon]-(c\rho_0m)^2\}$ ] thus the time evolution of @xmath34 can be analytically expressed in terms of the jacobian elliptic function cn ( . , . ) if @xmath37 and sinusoidal function if @xmath38 @xcite @xmath39\nonumber\\\end{aligned}\ ] ] for @xmath38 ; @xmath40 for @xmath41 ; @xmath42 for @xmath43 . we have set @xmath44 . here @xmath45 ( @xmath46 for @xmath47 ) are the roots of @xmath48 , @xmath49 if @xmath41 , and @xmath50 if @xmath43 . @xmath51 is determined by the initial state . hereafter we assume @xmath52 thus the energy unit is @xmath53 and the time unit is @xmath54 and @xmath55 . -@xmath35 plane for @xmath56 ( dashed line ) , @xmath57 ( solid lines ) , @xmath58 ( dotted line ) , from bottom to top . all the trajectories with solid lines evolve in a clockwise direction . the initial state ( asterisk ) is @xmath59 and @xmath60 . ( b ) dependence of the oscillation amplitude @xmath61 ( blue solid line ) and the period @xmath62 ( green dashed line ) of @xmath34 on @xmath1 . the running phase modes corresponds to the region i and iv and the oscillatory modes corresponds to region ii and iii . the red dotted lines marked by @xmath63 and @xmath64 denotes , respectively , the ground state quadratic zeeman energy ( the initial state coincides with the ground state and @xmath61 is zero ) and the resonant quadratic zeeman energy ( @xmath62 is infinite and @xmath65 or @xmath66 ) . , width=312 ] typical trajectories are illustrated in fig . [ x_delta](a ) for different @xmath1 . although starting from the same initial state , the trajectories could cover the whole @xmath34-@xmath35 plane by continuously varying the quadratic zeeman energy @xmath1 from negative infinity to positive infinity . all the trajectories are classified into two modes : the oscillatory mode where @xmath35 is between @xmath67 $ ] and the running phase mode where @xmath35 goes beyond @xmath67 $ ] . as shown obviously in fig . [ x_delta](a ) , the oscillatory mode trajectories are evolving in a clockwise ( counterclockwise ) direction if @xmath68 ( @xmath69 ) , while the running phase mode trajectories for large @xmath70 may take one of two opposite directions , depending on the sign of @xmath1 . this is a key point in order to localize the condensate spin dynamics . the boundaries between the oscillatory modes and the running phase modes satisfy one of the two requirements , @xmath71 ( @xmath72 ) or @xmath66 ( @xmath73 ) if time is long . the corresponding period @xmath62 becomes infinite [ see also fig . [ x_delta](b ) ] . another special point @xmath74 denotes the coincidence of the initial state with the ground state thus the oscillation amplitude @xmath61 is zero but the period @xmath62 is finite . the oscillation amplitudes and the periods are shown in fig . [ x_delta](b ) . there are clearly four regions : ( i ) running phase mode with increasing @xmath75 ; ( ii ) oscillatory mode with @xmath76 ; ( iii ) oscillatory mode with @xmath77 ; ( iv ) running phase mode with decreasing @xmath75 . the amplitude of the oscillations @xmath61 monotonically increases in regions ( i ) and ( iii ) but decreases in regions ( ii ) and ( iv ) with @xmath1 increasing . the period of the oscillations @xmath62 shows two resonant peaks at @xmath78 where @xmath79 or @xmath66 at long enough time . the period is almost a constant between these two peaks but decreases rapidly outside the peaks . similar oscillation behaviors were observed also in antiferromagnetically interacting @xmath18na spin-1 condensates ( @xmath69 ) @xcite . we observe from fig . [ x_delta](a ) that in the oscillatory mode @xmath35 increase or decrease with time if @xmath80 and @xmath34 is around its extremes . we may utilize this property to localize the condensate dynamics around @xmath81 by canceling @xmath35 in a period with @xmath35 increasing ( decreasing ) during the first ( second ) part . for an arbitrary state , however , we may utilize both the oscillatory and the running phase modes to localize the dynamics since @xmath35 may increase or decrease for a given @xmath34 , depending on the value of @xmath1 . each modulation cycle includes a free evolution with @xmath38 ( dashed line ) and a controlled evolution with @xmath82 ( solid lines , from bottom to top ) . the initial state ( red asterisk ) is @xmath59 and @xmath83 . ( b ) dependence of the amplitude ( blue solid line ) and period ( green dashed line ) of @xmath34 on @xmath84 , the nonzero quadratic zeeman splitting . circles and diamonds are the period and amplitude calculated analytically with eqs . ( [ approt ] ) and ( [ approa ] ) for large @xmath84s , respectively . , width=312 ] we consider first that the control period consists of two steps , a free evolution ( @xmath38 ) for a time slot @xmath85 followed by an evolution in a magnetic field @xmath86 for a time slot @xmath87 . we refer hereafter this protocol as two - step control . for a given initial state with @xmath88 , it is easy to prove analytically that @xmath85 depends on the initial state and @xmath87 depends uniquely on @xmath84 , which indicates that there is only one free parameter in the two - step control protocol . the time dependence of the magnetic field for the two - step control is @xmath89 where @xmath90 is an integer denoting the number of control cycles . typical controlled trajectories are illustrated in fig . [ 2step](a ) for three values of @xmath84 , where the initial state is @xmath91 and @xmath92 . the calculated times are @xmath93 and @xmath94 for @xmath95 . we see clearly that the oscillations of both @xmath34 and @xmath35 under the two - step control are smaller than that during free evolution , indicating that the condensate dynamics is indeed localized by the two - step control protocol . starting from the same initial state , the lager the @xmath84 is , the smaller the oscillation of @xmath34 is . the condensate spin average is @xmath96 for a state with @xcite @xmath97 once we localize @xmath98 and @xmath75 , the condensate spin @xmath99 is obviously localized . for a nonzero @xmath100 , the localization occurs similarly . the cycle period @xmath62 depends on the free evolution time @xmath85 and the controlled evolution time @xmath87 . the free evolution time is determined by the evolution time of the system from its initial state @xmath101 and @xmath102 to the symmetric state @xmath103 and @xmath104 . in this way , the time @xmath85 is calculated analytically by using eq . ( [ eq:1 ] ) @xmath105 in the limit of small @xmath106 , @xmath107|$ ] where we have used @xmath108 . similarly , the controlled evolution time @xmath87 is the evolution time of the system in the magnetic field @xmath86 and can be calculated , by using the conditions @xmath109 and @xmath110 , @xmath111 in the limit of small @xmath106 and large @xmath84 , @xmath112|$ ] . in total , the cycle period is approximated as @xmath113}\right|\ ] ] for small @xmath102 and large @xmath84 . we notice that @xmath114 for large @xmath84 , as shown also in fig . [ 2step](b ) . we define the control oscillation amplitude as @xmath115 , which depends obviously on the initial state and the magnetic field @xmath84 . the amplitude can be calculated analytically but is too lengthy to present here . in the limit of large @xmath84 and small @xmath102 , the amplitude is approximately @xmath116 where @xmath117 we see that @xmath61 approaches to @xmath118 as @xmath84 goes to infinity . in fig [ 2step](b ) , we present the dependence of @xmath61 and @xmath62 on the control magnetic field @xmath84 . we see clearly that @xmath61 and @xmath62 decrease monotonically as @xmath84 increases , manifesting the fact that better localization of the condensate dynamics is achieved in a higher magnetic field . we note that @xmath61 and @xmath62 approach their _ nonzero _ asymptotic values at large values of @xmath84 . actually , to reduce the oscillation amplitude @xmath61 further down to zero , we have to employ the following four - step control protocol . and @xmath75 under a four - step pulse sequence of @xmath4 for a four - step protocol . the red asterisk marks the initial state . the parameters are @xmath59 , @xmath119 , @xmath120 , and @xmath121 . the values of @xmath87 and @xmath122 are calculated , @xmath123 and @xmath124 . ( c ) amplitude of @xmath34 under four - step pulse sequences . better localization of @xmath34 ( smaller @xmath61 ) is achieved for larger @xmath84 and smaller @xmath125.,width=312 ] we consider next that the control period consists of four steps , ( i ) a free evolution for a time @xmath85 , ( ii ) an evolution in a magnetic field with @xmath126 for a time @xmath87 , ( iii ) a second free evolution for the time @xmath127 , and ( iv ) a second controlled evolution in another magnetic field with @xmath128 for a time @xmath122 , as shown in fig . [ 4step](a ) . we refer this protocol as four - step control . for simplicity but without loss of generality , we limit to the symmetric situations where @xmath129 and @xmath130 . it will be analytically proved that @xmath87 and @xmath122 are uniquely determined by @xmath84 and @xmath125 . thus there are only two free parameters , @xmath84 and @xmath125 , in the four - step control we considered here . it is straightforward to find the analytical solution to @xmath87 and @xmath122 , by using the initial state and the eqs . ( [ eq:1]-[eq:1b ] ) , @xmath131 where @xmath132 is the elliptic integral of the first kind . @xmath133 and @xmath134 are determined by @xmath135 and @xmath136 . we note here that the initial state for @xmath87 is @xmath135 and @xmath137 , and that for @xmath122 is @xmath101 and @xmath138 . the total period for a complete cycle is @xmath139 typical controlled evolution of the condensate is illustrated in fig . [ 4step](b ) , where the parameters are given in the caption . compared to the two - step control protocol , there are two advantages . the first is that the initial state is arbitrary , particularly , @xmath102 goes beyond the smallness requirement . the second is that the oscillation amplitude and period approach to zero if @xmath84 is large enough and @xmath125 is short enough , as shown in fig . [ 4step](c ) and eq . ( [ eq:4 t ] ) . from bottom to top . the insert shows the zoom - in view near the dip in the main panel . ( b ) fidelity after 1 cycle ( red lines ) and 10 cycles ( blue lines ) under the four - step protocol for @xmath140 ( dash - dotted lines ) , @xmath141 ( dashed lines ) , and @xmath142 ( solid lines ) . the results show that both the two - step protocol and the four - step protocol are robust , i.e. , the fidelity @xmath143 is close to 1 , in the presence of relatively 5% magnetic pulse errors . , title="fig:",width=312 ] from bottom to top . the insert shows the zoom - in view near the dip in the main panel . ( b ) fidelity after 1 cycle ( red lines ) and 10 cycles ( blue lines ) under the four - step protocol for @xmath140 ( dash - dotted lines ) , @xmath141 ( dashed lines ) , and @xmath142 ( solid lines ) . the results show that both the two - step protocol and the four - step protocol are robust , i.e. , the fidelity @xmath143 is close to 1 , in the presence of relatively 5% magnetic pulse errors . , title="fig:",width=312 ] we have assumed the magnetic control pulses are perfect in previous sections , but there are always uncontrollable errors in practical experiments , e.g. , the microwave field @xmath1 may have 5% relative uncertainty @xcite . since the timing is pretty accurate in current experiments , we next evaluate the robustness of the two - step and four - step protocols only under the 5% uncertainty of @xmath1 for many control cycles . we define the fidelity of a protocol after many control cycles as @xmath144 where @xmath145 is the initial and the final state of the spin-1 condensate and satisfies @xmath146 . the state has three components , @xmath147 with @xmath148 and @xmath149 and @xmath24 being the fraction and the phase of the component @xmath150 , respectively . the fidelity measures how close the initial and the final states are . the fidelity is 1 for ideal pulses but lower than 1 in the presence of pulse errors . the larger the errors are , the lower the fidelity is . higher fidelity indicates more robustness of the protocol to pulse errors . we assume the magnetic field error is distributed with equal probability in the range @xmath151 d$ ] with an average of @xmath84 . for the four - step protocols , the errors for @xmath152 and @xmath153 are independent . we numerically calculate the dependence of the fidelity @xmath143 on @xmath84 and show the results in fig . [ fidelity](a ) for two - step protocols and [ fidelity](b ) for four - step protocols . as shown in fig . [ fidelity](a ) , the fidelity is above 99% for most @xmath84 , except a special dip near @xmath154 where the period @xmath62 is most sensitive to the change of @xmath84 [ i.e. , the largest derivative of @xmath62 with respect to @xmath84 in fig . [ 2step](b ) ] . this result manifests that the two - step control protocols are pretty robust under the uncertainty of the magnetic field , if we choose a field away from the dip . for the four - step protocols , as shown in fig . [ fidelity](b ) , we find that the fidelity is very close to 1 , though it decreases as @xmath84 increasing or @xmath125 decreasing . by taking into account of the requirement of small @xmath125 and large @xmath84 to better localize the condensate dynamics , a delicate balance between the localization and the robustness is required in practical experiments . we propose to to localize the spin mixing dynamics in a spin-1 bose condensate by periodically applying magnetic pulse sequences , according to the two - step protocol for an initial state with small initial relative phase or the four - step protocol for an arbitrary initial state . numerical calculations confirm the validity of the proposal for a ferromagnetically interacting spin-1 condensate under the single spatial mode approximation . we further illustrate the robustness of the localization protocol with numerical calculations by assuming 5% uncertainty of the magnetic pulse amplitude , which might occur in practical experiments @xcite . our proposal may be utilized to realize higher precision magnetometers based on spinor bec @xcite or to explore the weak dipolar interaction effects in @xmath0rb spin-1 condensates by suppressing the spin dynamics induced by the spin exchange interaction @xcite . this work is supported by the national basic research program of china grant no . 2013cb922003 , the national natural science foundation of china grant no . 11275139 , the nsaf grant no . u1330201 , and the fundamental research funds for the central universities . 28ifxundefined [ 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 * * , ( ) link:\doibase 10.1103/physrevlett.98.200801 [ * * , ( ) ] link:\doibase 10.1103/physreva.67.013605 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.102.125301 [ * * , ( ) ] link:\doibase 10.1103/physreva.88.063809 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.85.1191 [ * * , ( ) ] link:\doibase 10.1103/physreva.81.033602 [ * * , ( ) ] link:\doibase 10.1103/physreva.84.013606 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.100.170403 [ * * , ( ) ] link:\doibase 10.1103/physreva.81.053612 [ * * , ( ) ] link:\doibase 10.1103/physreva.82.043627 [ * * , ( ) ] link:\doibase 10.1103/physreva.85.053646 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.112.185301 [ * * , ( ) ] link:\doibase 10.1103/physreva.73.041602 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.111.090403 [ * * , ( ) ] link:\doibase 10.1103/physreva.89.023608 [ * * , ( ) ] link:\doibase 10.1103/physreva.90.023610 [ * * , ( ) ] link:\doibase 10.1103/physreva.90.013626 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.81.5257 [ * * , ( ) ] link:\doibase 10.1103/physreva.66.011601 [ * * , ( ) ] link:\doibase 10.1103/physreva.60.1463 [ * * , ( ) ] link:\doibase 10.1103/physreva.88.031602 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.110406 [ * * , ( ) ] @noop ( )

spin exchange interaction between atoms in a spin-1 bose - einstein condensate causes atomic spin evolving periodically under the single spatial mode approximation in the mean field theory . by applying fast magnetic pulses according to a two - step or a four - step control protocol , we find analytically that the spin dynamics is significantly suppressed for an arbitrary initial state . numerical calculations under single mode approximation are carried out to confirm the validity and robustness of these protocols . this localization method can be readily utilized to improve the sensitivity of a magnetometer based on spin-1 bose - einstein condensates .

pareto optimality is a natural extension of the concept of maximum to multi - objective optimization problems . a solution is part of the pareto optimal set , or pareto front , if it is impossible to improve one objective without worsening another . instead of imposing an arbitrary aggregation of the different objectives into a single scalar function , pareto optimality keeps track of all potentially interesting solutions in the presence of trade - offs . the pareto approach , originally introduced in economics @xcite , has shown to be useful in many engineering applications @xcite , decision - making analysis @xcite , more recently in biology @xcite . here we apply the pareto approach to the optimization of the response of multi - input monotone systems , which are widely used to describe input - output systems in control theory @xcite . consider a system which processes a multi - input vector @xmath4 into a single output @xmath5 , each coordinate of @xmath6 being monotone . consider now a list of such input vectors @xmath7 . the natural order on @xmath8 induces a partial order between the elements @xmath9 , which in turn , due to the monotonicity of @xmath6 , induces partial order constraints between the values @xmath10 . these constraints limit the space accessible to the vector @xmath11 . we then introduce an optimization problem : certain values of @xmath12 have to be maximized while other have to be minimized . the question is to know how far the function @xmath6 can be optimized to fullfill this multi - objective problem , while keeping its monotonicity properties unaffected . this problem can be formulated beyond the framework of monotone systems , as @xmath6 and the @xmath13 can be essentially seen as a source of partial order constraints . the most general formulation of our problem is then to start directly from partial order constraints between bounded values @xmath10 , with either a minimization or a maximization objective associated to each of the @xmath12 , and search for the pareto optimal solutions of this problem . geometrically , setting the bounded @xmath12 between @xmath14 and @xmath15 , the space compatible with the constraints is a convex polytope in the hypercube @xmath16^n$ ] , and we look for the pareto front of this polytope given an ideal point located on a corner of the hypercube . the monotone systems formalism applies in particular to the evolution of gene regulatory networks in biology , which is the initial motivation of the present authors @xcite . in this context : each @xmath13 represents a combination of environmental signals ; @xmath6 is the signal processing performed by the gene regulatory network , where mutations impact the strength of the interactions but not their sign @xcite ; the @xmath17 are the output of the network ( ex : the expression of a specific gene ) , which can be either beneficial or detrimental , setting selective pressure toward respectively either maximization or minimization . the broader formulation of the problem can find applications in fields more specific to operations research , such as task scheduling : some tasks have to be realized in a certain temporal order relative to each other due to design constraints ( e.g. the assembly of the different parts of a car ) @xcite , with some having to be realized as soon as possible and others as late as possible due to externalities ( e.g. stock management , supply constraints , or processing unit occupancy ) . in section 2 , we demonstrate general properties of pareto sets in relation with set operations , introduce the concept of pareto self - dominance . we demonstrate a number of properties to break multi - objective optimization problems into simpler sub - problems , useful for our later demonstration , notably a maximality property valid for compact set ( proposition [ maximality ] ) . in section 3 , we provide a graph formulation of our multi - objective optimization under partial order constraints problem , using a hasse diagram @xcite , which we colour according to optimization objectives . we then describe and prove a dynamic programming algorithm based on successive edge contractions with appropriate vertex colouring rules . in section 4 we discuss the complexity of our algorithm as a function of the number of variables @xmath0 and to the complexity of the pareto front @xmath2 . we propose an improved version of the algorithm in @xmath1 . we finally discuss how a parameterized complexity approach @xcite can provide a combinatorial description of the pareto front , with a complexity of @xmath3 in the case of series - parallel partial orders with monochromatic connected components of bounded size . we consider a partially ordered set ( @xmath18 ) and define the corresponding strict order @xmath19 and @xmath20 . for @xmath21 , we denote the _ upper contour set _ and the _ strictly upper contour set _ respectively by @xmath22 and @xmath23 the ( strong ) pareto optimal set of @xmath24 , denoted by @xmath25 , contains the elements of @xmath26 which are not stricly dominated by another element of @xmath26 : @xmath27 equivalently we have @xmath28 the following propositions on pareto optimal sets will be of use in the proof of our algorithm in section [ sec : proof ] . [ intersection ] let @xmath29 and @xmath30 be subseteqs of @xmath31 . then @xmath32 . _ proof : _ consider @xmath33 . @xmath34 because @xmath35 . as @xmath36 , by definition of pareto optimality @xmath37 . [ union ] consider @xmath38 . @xmath39 _ proof : _ consider @xmath40 . we have @xmath41 this implies @xmath42 . given that @xmath43 , @xmath44 such that @xmath45 . considering that in particular @xmath46 , we have @xmath47 . [ dominator ] we say that a set @xmath48 is _ self - dominated _ if @xmath49 such that @xmath50 . we will now prove a generalization of the following property of the maximum in @xmath8 to pareto optimality in @xmath31 : take two closed segments @xmath29 and @xmath30 in @xmath8 , such that @xmath51 and @xmath52 . this imposes @xmath53 . [ maximality ] consider @xmath54 , @xmath30 self - dominated . if @xmath51 and @xmath55 then @xmath56 . _ proof : _ we only need to show @xmath57 . for that consider @xmath58 . @xmath59 by hypothesis . by definition [ dominator ] , @xmath60 as @xmath61 , in particular @xmath62 by relation ( [ eq : maxequal ] ) , @xmath63 . [ necessary ] consider @xmath48 self - dominated . suppose @xmath64 is a necessary condition for @xmath35 , then @xmath65 . _ proof : _ the hypothesis @xmath66 implies @xmath67 by proposition [ intersection ] . the result comes from proposition [ maximality ] replacing @xmath29 and @xmath30 by respectively @xmath68 and @xmath29 . [ parpar ] consider @xmath24 self - dominated . suppose there exist @xmath69 such that @xmath70 . then @xmath71 _ proof : _ we just need to apply proposition [ maximality ] to @xmath72 and @xmath73 , the initial inclusion hypothesis being given by proposition [ union ] . * remark* proposition [ parpar ] is in particular true when @xmath74 and indicates that @xmath75 [ parparpar ] suppose that in addition to the hypotheses of proposition [ parpar ] , @xmath76 is self - dominated and verifies @xmath77 such that @xmath78 . we then have @xmath79\ ] ] . _ proof : _ applying successively proposition [ parpar ] to @xmath76 instead of @xmath26 , the remark of proposition [ parpar ] in the backward direction , then in the forward direction and finally using the fact that @xmath80 for any set @xmath81 , @xmath82 \\ & = par\left [ par\left ( \bigcup_{j=1}^{j}par(b_{j})\right ) \cup \bigcup_{i=2}^{i}par(a_{i})\right ] \\ & = par\left [ \bigcup_{j=1}^{j}par(b_{j})\cup\bigcup_{i=2}^{i}a_{i}\right ] \\ & = par\left [ \bigcup_{j=1}^{j}par(par(b_{j}))\cup\bigcup_{i=2}^{i}% par(a_{i})\right ] \\ & = par\left [ \bigcup_{j=1}^{j}par(b_{j})\cup\bigcup_{i=2}^{i}par(a_{i}% ) \right ] \end{split}\ ] ] finally , we note that if @xmath29 is a closed , non - empty subseteq of @xmath31 which has an upper bound in the sense of the pareto order ( _ i.e. _ @xmath83 ) , then @xmath84 @xcite . as a consequence @xmath29 is self - dominated , as we show below . [ rn ] when @xmath85 closed , non - empty and pareto bounded from above then @xmath29 is self - dominated . _ proof : _ consider @xmath86 . @xmath87 is closed , non - empty and pareto bounded from above . thus @xmath88 and we can consider @xmath89 so that @xmath90 and @xmath91 now , as @xmath92 @xmath93 which proves @xmath94 proposition [ rn ] ensures that we can use propositions [ intersection ] , [ union ] , [ maximality ] and [ necessary ] for the proof of the algorithm in the context of @xmath16^{n}$ ] presented in the next part . propositions [ parpar ] and [ parparpar ] will not be used for the demonstration of our specific algorithm , but to discuss the complexity of more naive algorithms in section [ discusscomplexity ] . indeed , they can be used generally in multi - objective optimization problems for reducing the pareto set search into sub - problems . proposition [ parpar ] indicates that the pareto front can be found by taking the pareto front of the assembled pareto fronts of any decomposition of the search space . proposition [ parparpar ] extends this result to any finite number of recursive splittings of the search space , and shows that it is necessary to search only one time the pareto set of the assembled pareto sets of the terminal sub - problems . _ * pareto order * _ we consider @xmath0 bounded variables @xmath95^{n}$ ] and introduce for convenience the notation @xmath96 and @xmath97 . our multi - objective optimization consist of minimizing some of the variables and maximizing all the others . formally , we have a partition @xmath98 of the index set @xmath99 . we define a pareto ordering @xmath100 on @xmath31 with the _ signature _ @xmath101 as follows : if @xmath102 @xmath31and @xmath103 @xmath31 then@xmath104 i.e. for the variables in the _ ascending _ set @xmath105 smaller is better . in other terms , the ideal point has coordinates @xmath14 for indexes in @xmath106 and @xmath15 for indexes in @xmath107 . we define the corresponding ( weak ) strict pareto order through @xmath108 . partial order constraints correspond to a set of ( weak ) inequalities of the type @xmath109 between coordinates of @xmath21 . _ * vertex colouring * _ we introduce a graph stucture which vertices @xmath110 are associated with a group of variables @xmath111 , called _ aggregate _ , and noted @xmath112 . an aggregate @xmath113 represents a state such that @xmath114 , a value noted @xmath112 . a vertex can be of one of the four following natures ( or colour indicated in parenthesis as used in the figures ) : * if @xmath115 , @xmath110 is a _ descending vertex _ ( red ) ; * if @xmath116 , @xmath110 is an _ ascending vertex _ ( blue ) ; * if @xmath117 comprises indexes from both @xmath106 and @xmath107 , @xmath110 is a _ trade - off vertex _ or _ tov _ ( grey ) ; * if @xmath118 or @xmath119 , @xmath110 is a _ boundary vertex _ ( black ) . _ * partial order constraints * _ edges express order constraints between variables , according to the convention of so - called hasse diagrams : edge @xmath120 points from @xmath121 to @xmath122 if and only if @xmath109 , where @xmath123 ( resp . @xmath124 ) . the inequality constraints can be consistently carried over to aggregates provided their index - sets @xmath117 and @xmath125 are disjoint , and we will use the obvious notation @xmath126 . variables engaged in cyclical inequalities are trivially equal to each other and thus are assumed to be aggregated , resulting in acyclic diagrams . _ * relations between vertices * _ two vertices @xmath121 and @xmath122 connected by @xmath120 are said to be _ conflicting _ if @xmath121 is descending and @xmath122 is ascending . we say that a vertex @xmath121 _ aims for _ another vertex @xmath122 if : either @xmath121 is descending and points to @xmath122 via an edge @xmath120 ; or the other way round , @xmath121 is ascending and is pointed from @xmath122 via an edge @xmath127 , independently of the colour of @xmath122 . for example , conflicting vertices necessarily aim for each other . a maximal connected component of the hasse diagram exclusively comprising ascending or descending vertices is called a _ monotone connected components_. a vertex qualified as _ extremal _ ( in the sense of the monotone connected components ) if it only aims for vertices of different colour . note that conflicting vertices are not necessarily extremal as they may point to other vertices of the same nature , and extremal vertices are not necessarily conflicting as they may point to tovs or to boundary vertices . _ * edge contractions rules * _ an edge contraction between two vertices @xmath110 and @xmath128 consist of removing @xmath129 , and replacing @xmath110 and @xmath128 by a unique vertex @xmath130 , where @xmath131 . while all vertices that are not affected by the contraction are of constant colour , the colour of @xmath130 is determined consistently with the colours definitions : * if @xmath110 and @xmath128 are ascending , @xmath130 is ascending ; * if @xmath110 and @xmath128 are descending , @xmath130 is descending ; * if @xmath110 or @xmath128 is a boundary vertex , @xmath130 is a boundary vertex ; * if @xmath110 and @xmath128 are different and none of them is a boundary vertex , @xmath130 is a tov ; _ * resolution tree * _ we have just introduced all the notions for representing our initial problem and any state resulting from edge contractions in the form of a 4 colours directed acyclic graphs , for which we reserve the term _ hasse diagrams _ , or _ diagrams _ , noted @xmath132 with an index . the stages of the dynamic algorithm described below generate another sort of graph , which we call the _ resolution tree_. to avoid any confusion between the two sorts of graphs , the vertices of the resolution tree are called _ nodes_. each node of the resolution tree corresponds to a hasse diagram , and each edge of the resolution tree corresponds to an operation of edge contraction applied to a hasse diagram . a _ branch _ will only refer to a branch of the resolution tree . a vertex aiming for several other vertices in @xmath132 will be called a _ junction _ , whereas a node connected to several downstream nodes in the resolution tree will be called a _ branching_. _ * re - formulation of the problem * _ we finally call @xmath133 the subspace of @xmath134 which verifies the partial order relations represent by the hasse diagram @xmath132 . with this representation , our main goal can be formulated in the following way : _ let _ @xmath135^{n}$ ] _ _ be the space of all vectors respecting the partial order constraints represented by _ _ @xmath132 . _ determine _ @xmath136 _ _ the set of vectors optimal under the pareto order _ _ @xmath100 _ _ on _ _ @xmath31 . the algorithm starts by setting @xmath137 steps 1 to 4 described below ( and illustrated in figure [ fig : illustration ] ) are then recursively applied to all diagrams @xmath138 , @xmath139 , generated at depth @xmath140 until step 2 can no longer be performed , _ i.e. _ the diagram in question no longer contains any ascending or descending vertex ( eqivalently , only contains tovs and boundary vertices ) : 1 . perform a _ transitive reduction _ of @xmath141 , _ i.e. _ remove any direct edge @xmath142 if there exists a longer _ path _ from @xmath143 to @xmath144 on @xmath141 . 2 . select an extremal vertex @xmath121 . 3 . consider the vertices @xmath145 which @xmath121 aims for . there is always at least one such vertex , in limiting cases provided by boundary vertices . define the diagrams @xmath146 by respectively contracting the edge connecting @xmath121 and @xmath147 according to the colouring rules defined in section [ definitions ] . at the end of this branching process , we are left with a collection of terminal graphs @xmath148 , and we posit that the solution of the initial problem is : @xmath149 and identification of an extremal vertex @xmath150 . left : aggregation of @xmath150 with two conflicting vertices @xmath151 and @xmath152 , yielding a branching of the resolution tree into two diagrams , the union of which solutions results in the pareto front of the parent diagram . color code : filled blue circle : ascending vertex , filled red circle : descending vertex , filled grey circle : tov.[fig : illustration ] ] our proof of the algorithm described above proceeds in three steps : step 1 : : we first show that for each iteration of the algorithm , @xmath153 . step 2 : : next we show that @xmath154 . step 3 : : finally we show that the terminal graphs satisfy @xmath155 . for brevity s sake we treat only the case of an ascending connected component , the demonstration being easily adapted for a descending connected component . consider an ascending subgraph @xmath156 in @xmath141 and a maximal vertex @xmath157 . there are two possibilities : 1 . the vertex @xmath158 representing the upper bound is directly connected to @xmath150 . in this case @xmath158 is the unique vertex pointing to @xmath150 : if there would be another vertex @xmath159 pointing to @xmath150 , there would exist a chain of vertices pointing from @xmath158 to @xmath159 , as @xmath158 is the global upper bound , which would contradict the fact that we have taken the transitive reduction of the diagram . one or more vertices @xmath160 point to @xmath161 , none of which has constant value . we then define the sets @xmath162 a necessary condition for @xmath163 is @xmath164 . suppose otherwise that @xmath165 ( or in the case 1 above @xmath166 ) . then there exists @xmath167 such that for all @xmath168 . if we denote by @xmath169 the vector with coordinates @xmath170 for @xmath171 and otherwise @xmath14 , we have for all @xmath172 and @xmath173 as by assumption @xmath174 . this contradicts @xmath175 . as by definition , @xmath176 , we can use proposition [ necessary ] and have that @xmath177 . we only have to show that @xmath178 , the inclusion in the other direction directly following from proposition [ union ] . we only discuss the case @xmath179 as the result is trivial otherwise ( in particular in case 1 of the first step ) . consider two distinct indices from the set @xmath180 which we can take to be @xmath15 and @xmath181 without loss of generality . consider @xmath182 and suppose there exists @xmath183 such that @xmath90 . then by virtue of @xmath150 being an ascending vertex and by definition of the pareto order : @xmath184 next , as @xmath150 was maximal within its ascending subgraph , any vertex pointing to it must contain at least one descending variable , from which follows that @xmath185 indeed , the latter inequality is trivially true if all @xmath186 label descending variables , i.e. @xmath187 . if not , we can choose an @xmath188 and @xmath189 for which the statement @xmath90 implies both @xmath190 and @xmath191 and hence , together with @xmath192 and @xmath193 by definition of the aggregates , we obtain the equality @xmath194 . we also have @xmath195 implying : @xmath196 finally , @xmath197 and in @xmath141 , vertex @xmath198 points to @xmath150 by hypothesis , which implies : @xmath199 examining the above relations in the order ( [ c2]-[c4]-[c1]-[c3 ] ) , we see that all the variables at play must be equal , in particular @xmath200 . then @xmath201 and consequently @xmath202 . to summarize , we have just demonstrated : @xmath203 now , if we take in particular @xmath204 in relation ( [ inter ] ) , relation ( [ eq : maxequal ] ) implies that @xmath205 by maximality of @xmath206 in @xmath207 . this gives : @xmath208 applying this time relation ( [ eq : maxequal ] ) in the backward direction demonstrates the announced result : @xmath209 by construction a terminal graph @xmath210 contains only boundary vertices and tovs . now consider @xmath211 , and consider a vertex @xmath212 in @xmath210 . if @xmath212 is a boundary vertex , then the variables in @xmath117 are already at their optimum bounds , and @xmath213 otherwise , @xmath110 is a tov and as in step 2 above we can choose an @xmath214 and @xmath215 for which the statement @xmath90 implies both @xmath190 and @xmath191 and hence @xmath216 . as the aggregates form a partition of the initial index set @xmath217 , the above immediately implies @xmath218 , and hence , by relation ( [ eq : maxequal ] ) , @xmath219 . as this is true for any @xmath220 , we have demonstrated our result : @xmath221 . in order to illustrate the working of the algorithm presented here we have implemented it in mathematica 9 , using the built - in graph primitives @xcite . the program generates random dags with a specified number of internal nodes , randomly chosen to be ascending or descending , using the layer method and then applies the algorithm . in figure [ fig : example ] we show an example of a typical input graph , and the output of the algorithm , in this case the union of three terminal graphs . due to spatial limitations we can not display the intermediate steps the algorithm makes to produce the final result . internal vertices , randomly chosen to be either ascending or descending . bottom : the resulting pareto optimal set obtained from our algorithm , consisting of the union of the set parameterized by three distinct terminal graphs . colour code as in figure [ fig : illustration ] , in addition , black circles represent boundary values . ] we discuss here the complexity of different approaches and propose two further improvements of our algorithm . complexity will be discussed relative to the number of variables @xmath0 , and to the complexity of the pareto front , as quantified by its number of faces @xmath2 . a _ face _ is defined as a maximal convex subset of the pareto front , which itself is a subset of the convex polytope @xmath133 . note that the number of faces of the pareto front can be exponential in the number of variables , and that faces do not necessarily all have the same dimension . the first improvement consist of introducing an additional contraction rule with forbidden steps , leading to a complexity linear in the number of dimensions of the inital problem , relatively to the size to pareto front . this lead to resolution tree size @xmath1 , where the depth and the number of leaves of the tree respectively equal @xmath0 and @xmath2 . the second improvement consist of a combinatorial description of the pareto front from the set of solutions of specific components of the diagram . for this we will introduce the interface @xmath222 , which comprises all the conflicting vertices of the initial diagram . for each resolution of @xmath222 , the solutions of the monotone connected components can be computed independently , then assembled combinatorially . parameterization by the different resolutions @xmath222 can exponentially reduce the computing time and the size of the description of the full solution . in particular for series - parallel partial orders , the resolution of @xmath222 is unique . under the additional assumption that the size of monotone ( or monochromatic ) connected components is bounded , one obtains a resolution and a description of the pareto front in @xmath3 , eventhough the pareto front may comprise an exponential number of faces . an exhaustive seach of the pareto front would be to consider the @xmath223 corners of the @xmath224 hypercube , check for each whether the coordinates of the coresponding n - dimensional vector respect the partial order conditions ( test up to @xmath225 conditions ) , assess the pareto optimality of each admissible corner relative the other admissible corners ( @xmath226 , where @xmath227 is the potentially exponential number of admissible corners ) , and finally examine the pareto optimality of all possible interpolations(@xmath228 ) between the @xmath229 pareto optimal admissible corners . given that @xmath227 and @xmath229 can both scale like @xmath223 , the complexity of this process can reach @xmath230 . the edge contraction algorithm as it is described so far provides several benefits compared to the exhaustive search . first , contraction operations on the hasse diagram ensure to only explore solutions which are consistent with the partial order constraints.the complexity of the program can then be taken as examining the validity of subsets of contracted edges ( @xmath231 , as the number of edges is @xmath232 ) , then checking the resulting admissible subspacesfor their relative pareto optimality ( @xmath233 ) . second , the restriction of contractions to vertices which are maximum within monotone connected components ensures that the final result of each contraction process is indeed pareto optimal . finally , each terminal diagram provides a full parametrization of a face of the pareto front , the union of all which covers the full pareto front , without requiring to assess pareto optimality a second time within the set of potential solutions as propositions [ parparpar]and [ parpar ] would suggest in the general case . the worst case complexity of the resolution tree is bounded by the a case where all possible subsets of edges have to be contracted and the number of edges is maximum , the first chosen vertex is in conflict with the @xmath234 other vertices , the second with @xmath235 vertices remaining after contraction , and so on , thus resulting in a complexity of @xmath236 . however , the version of the algorithm described so far has potential sources of increased complexity : ( i ) duplicated diagrams representing the exact same parameterization of a face of the pareto front ; ( ii ) diagrams @xmath237 which aggregates the initial variables into a sub - partition of another diagram @xmath238 , so that the space of the pareto front parameterized by @xmath237 is only a subspace of the space parameterized by @xmath238 . case ( i ) happens in particular when an extremal vertex conflicts with two or more other vertices , because the two corresponding contractions occur in a certain order along a resolution branch , and in another order along another branch . case ( ii ) happens in particular when a vertex @xmath239 aims to a tov @xmath240 and a conflicting vertex @xmath241 : along a first resolution branch , @xmath239 aggregates with @xmath240 , then @xmath241 aggregates with the resulting tov , whereas along a second branch , @xmath239 aggregates with @xmath241 , resulting in two distinct tovs . consequently , the parameterization given by two distinct tovs in the second branch includes the solution obtained in the first . a way to fix the redundancies due to case ( i ) would be to store known nodes of the resolution tree in a hash table , using a so - called _ memoization _ strategy @xcite . we propose instead an improved version of the contraction rules , which ensures that every terminal diagram represents a distinct face of the pareto front and which also removes sub - representations of the pareto front due to case ( ii ) . the resulting algorithm has a complexity of @xmath242 where @xmath2 is the complexity of the pareto front and @xmath0 the number of variables . we define _ frozen edges _ as edges which can not be contracted . furthermore , we impose this property to be inherited downstream of the resolution tree , i.e. a frozen edge remains frozen after contraction of other edges . otherwise , an edge is qualified as _ free_. the improved version of the algorithm consists of modifying step 3 of the original algorithm as follows : _ * modified contraction rule * _ : if possible , contract extremal vertices @xmath121 which aim for other vertices via a single free edge . otherwise : ( i ) contract in priority @xmath121 with conflicting vertices , then with tovs , and , ( ii ) for each @xmath243 , the edges contracted to obtain @xmath244 , @xmath245 , are frozen in @xmath246 . + as shown below , these modifications ensure that the terminal diagrams @xmath247 describe all _ distinct _ faces of the pareto front . note that the total size of the resolution tree can be further minimized by treating in priority junctions comprising as few as possible alternatives , which generalizes the priority treament of single free edge contractions . however this changes neither the depth of the tree @xmath0 , nor its number of leaves @xmath2 . an elementary point is that the algorithm can be consistently run until the leaf of each branch is reached . a potential issue would indeed be that the creation of a frozen edge leads to the existence of an extremal vertex not connected to any free edge . however , this never happens due to the priority contraction of single free edges , which implies that frozen edges are generated only at stages when every extremal vertex is connected to two or more free edges . as the treatment of single free edge contractions does not differ from the rules of the initial algorithm , we set ourselves at a branching of the resolution tree corresponding to @xmath248 alternative contractions of an extremal vertex @xmath239 aiming to @xmath249 , where the @xmath250 first @xmath121 conflict with @xmath239 , and the remaining @xmath121 are tovs , where @xmath251 . we show first that the edge freezing rules leads to all admissible ( in the sense of other rules ) and distinct partitions of the initial variables . consider the first contraction of @xmath252 . the resulting graph @xmath253 induces all admissible partitions such that @xmath254 and @xmath255 are in the same set . consider then the contraction of @xmath256 , where @xmath257 is frozen according to the modified algorithm . the resulting branch @xmath258 induces all partitions such that @xmath254 and @xmath259 are in the same set but @xmath255 is not . for each @xmath260-th iteration of this process , the sub - tree stemming from @xmath244 induces partitions such that @xmath254 and @xmath123 are in the same aggregate but @xmath261 are not . therefore , throughout the different branches , the contractions with @xmath239 enumerate without redundance all the accessible subsets of @xmath262 containing @xmath254 . we now want to show that the face parameterized by every terminal diagram is embedded in a stricly different direction of the euclidian space , thus parameterizes a distinct face of the pareto front . for this , we have to show that none of the partitions is included in another . this is obtained thanks to the prioritization of conflicting vertices contractions : when @xmath239 aggregates with a tov @xmath263 , @xmath264 , any edge @xmath265 , @xmath266 , joining @xmath239 to a conflicting @xmath121 is frozen in @xmath267 . consequently , at this stage , none of the conflicting @xmath121 can be aggregated to the tov resulting from the contraction of @xmath268 . this implies that two conflicting vertices susceptible to form a tov can not both aggregate to another tov , at any stage of the process . therefore , a tov can only contain a single pair of conflicting variables , whereas at least two such pairs would be necessary to form a sub - partition of the aggregate . we define the interface @xmath222 of the initial problem @xmath269 as the set of all conflicting vertices . @xmath222 contains all extremal vertices of @xmath269 which do not directly aim for the maximum or minumum bound . while @xmath222 can be composed of several connected components , a monotone connected component of @xmath269 may intersect several connected components of @xmath222 . we call @xmath270 the diagrams obtained by aggregating first all the extremal vertices of @xmath271 in @xmath222 . as all conflicting vertices have been aggregated into tovs at this point , the algorithm only results in aggregation of extremal vertices with existing tovs . in this sense , the remaining monotone connected components of each @xmath270 are isolated from each other by tovs . now call @xmath272 the montone connected components of @xmath270 taken together with the tovs they aim for . each @xmath273 can be solved separately , leading to its own set of leaves @xmath274 . the parametrizations of the different parts of the pareto front of @xmath270 can be obtained by concatenating all possible combinations of the @xmath275 indexes of the @xmath276 . here the concatenation @xmath277 between diagrams is defined as the merging of vertices which represent aggregated variable sets with a non - empty intersection . with these notations , we have : @xmath278 with for every @xmath279 : @xmath280 where @xmath281 . such a combinatorial representation of the pareto front can be exponentially smaller than the number of faces of the pareto front itself . in particular when the size of the @xmath273 is bounded , the number of terms of the concatenation representing @xmath282 increases linearly , while they represent an exponentially increasing number of faces of the pareto front . in the case of series - parallel @xmath132 , we have @xmath283 . this is due to series - parallel graphs being characterized by the absence of fence subgraph ( n " shaped motif ) . this property implies that there can not be conflicting vertices which each participate to a junction . in other words , at least one of the two has no other alternative than contracting with its conflicting vertex , leading to the absence of branching process during the resolution of the interface of the hasse diagram . under the additional condition that monochromatic connected components have bounded size , all @xmath284 have a bounded size , and the complexity of the resolution of these sub - diagrams is bounded . thus the pareto front of the full problem admits a representation which computation time grows linearly with the number of @xmath284 , which itself increases at much linearly with the number of initial variables . we have described and demonstrated an algorithm which allows to find an exact parameterization of the pareto front of any polytope defined by partial order relations within the hypercube @xmath16^n$ ] , given an ideal point located at a corner of this hypercube . the solution is obtained in a linear number of steps with the number of parameters @xmath0 , relative to the number @xmath2 of faces of the pareto front . this result is obtained by establishing a mapping between hasse diagrams and polytopes , where a colouring of the graph encodes the location of the ideal point . more explicitly , vertices represent sets of aggregated variables , edges correspond to ordering relations , and colours correspond to the optimization objectives associated with the variables . following a dynamic programming approach in the space of coloured graphs , the initial polytope is successively projected onto smaller dimension spaces , corresponding to edge contractions in its diagram representation . the pareto front ultimately consists of the union of spaces corresponding to each terminal hasse diagram obtained after @xmath0 contractions . a major advantage of this approach is that assembling the solution from the decomposed sub - problems is a direct operation , which does not require any further pareto computation than taking a union of sets . we have furthermore introduced a parmeterized complexity approach , by introducing a specific subgraph , which we call the interface and which corresponds to the smallest set containing all the potential trade - offs . the edge contraction algorithm can be applied to this subgraph before all other vertices , until all conflicting parameters have been aggregated . this partial computation is sufficient to determine the dimension of the pareto front . in particular , given that it is necessary to combine at least two nodes with conflicting objectives to generate a trade - off , the dimension of the pareto front will be lower than @xmath285 , where @xmath286 is the number of vertices in the interface . as soon as the interface is resolved , the diagram has a specific structure , where some coloured connected components are isolated from others . due to this property , given a resolution of the interface , the pareto front can be represented combinatorially from the solutions of the connected components which share a single objective . for series - parallel partial orders , the interface has a unique resolution . when , additionally , coloured connected components are of bounded size , the pareto front , though of exponential complexity , can be computed and represented in @xmath3 . the propositions of the section 2 of the article are generic and can be applied to reduce multi - objective optimization problems and re - assemble the pareto front , similarly to other exact approaches to the pareto set @xcite . specifically , the maximality property expressed in proposition [ maximality ] and proposition [ necessary ] shows how to decrease the search space while conserving the pareto set , and proposition [ parparpar ] can then be used to break up the problem recursively into smaller subproblems . in the context of the optimization of multi - input monotone systems , it would be important to investigate generalization of the algorithm developed here to linear inequality constraints or to ideal points other than corners of the search space . pn was supported by fom programme 103 `` dna in action : physics of the genome '' . this work is part of the research programme of the foundation for fundamental research on matter ( fom ) , which is part of the netherlands organisation for scientific research ( nwo ) . we would like to thank olivier spanjaard from the laboratoire dinformatique de paris 6 ( lip6 ) and the members of the laboratoire dinformatique fondamentale de lille ( lifl ) for their unvaluable advices . shoval o , sheftel h , shinar g , hart y , ramote o , mayo a , dekel e , kavanagh k , alon u ( 2012 ) phenotype space evolutionary trade - offs , pareto optimality , and the geometry of phenotype space . _ science _ 336:1157 - 1160 .

we developed a graph - based method to solve the multi - objective optimization problem of minimizing or maximizing subsets of @xmath0 bounded variables under partial order constraints . this problem , motivated by the optimization of the response of multi - input monotone systems applied to biological gene networks , can find applications in other contexts such as task scheduling . we introduce a mapping between coloured graphs ( hasse diagrams ) and polytopes associated with an ideal point , and find an exact closed - form description of the pareto optimal set using a dynamic program based on edge contractions . the proof of the algorithm is based on decomposition properties of pareto optimal sets that follow from elementary set operations , notably a maximality property valid for compact ensembles . in the general case , the pareto front is found in @xmath1 steps , where @xmath0 is the number of variables and @xmath2 the number of faces of the pareto front . using a parameterized complexity approach , the computation and the representation of the solution reaches @xmath3 for series - parallel graphs when the size of monochromatic connected components is bounded .

in this paper , we study traveling waves for monotone ( i.e. , order - preserving ) semiflows @xmath0 with the bistability structure on some subsets of the space @xmath1 consisting of all continuous functions from the habitat @xmath2(@xmath3 or @xmath4 ) to the banach lattice @xmath5 , where @xmath6 or @xmath7 is the set of evolution times . here the bistability structure is generalized from a number of studies for various evolution equations . it means that the restricted semiflow on @xmath5 admits two ordered stable equilibria , between which all others are unstable . we focus on the existence of traveling waves connecting these two stable equilibria , which are called bistable traveling waves . this setting allows us to study not only autonomous and time - periodic evolution systems in a homogeneous habitat ( media ) , but also those in a periodic habitat . besides , the obtained results can be extended to the semiflows with weak compactness on some subsets of the space @xmath8 consisting of all monotone functions from @xmath9 to @xmath5 . to explain the concept of the bistability structure , we recall some related works on typical evolution equations . fife and mcleod @xcite proved the existence and global asymptotic stability of monotone traveling waves for the following reaction - diffusion equation : @xmath10 where @xmath11 . clearly , the restriction of system on @xmath12 is the ordinary differential equation @xmath13 , which admits a unique unstable equilibrium between two ordered and stable ones . the same property is shared by the nonlocal dispersal equation in @xcite and the lattice equations in @xcite . chen @xcite studied a general nonlocal evolution equation @xmath14 , which also possess the above bistability structure . some related investigations on discrete - time equations can be found in @xcite . for the time - periodic reaction - diffusion equation @xmath15 , the spatially homogeneous equation is a time - periodic ordinary differential equation . in this case , the equilibrium in the bistability structure should be understood as the time - periodic solution . under such bistability assumption , alikakos , bates and chen @xcite obtained the existence of bistable time - periodic traveling waves . recently , yagisita @xcite studied bistable traveling waves for discrete and continuous - time semiflows on the space consisting of all left - continuous and non - decreasing functions from @xmath9 to @xmath12 under the assumption that there is exactly one intermediate unstable equilibrium . it should be mentioned that the result in @xcite for continuous - time semiflows requires an additional assumption on the existence of a pair of upper and lower solutions . note that the restrictions on @xmath12 of the afore - mentioned systems are all scalar equations , and hence , there is only one unstable equilibrium in between two stable ones . but in the case where @xmath16 , there may be multiple unstable equilibria . this is one of the main reasons why some ideas and techniques developed for scalar equations can not be easily extended to higher dimensional systems . volpert @xcite established the existence and stability of traveling waves for the bistable reaction - diffusion system @xmath17 by using topological methods , where @xmath18 is a positive definite diagonal matrix . fang and zhao @xcite further extended these results to the case where @xmath18 is semi - positive definite via the vanishing viscosity approach . consider the following parabolic equation in a cylindrical domain @xmath19 : @xmath20 where @xmath21 is of the same type as the nonlinearity in and @xmath22 is a bounded domain with smooth boundary in @xmath23 . obviously , the restriction of the solution semiflow of on @xmath24 gives rise to the following @xmath25-independent system : @xmath26 one can see from matano @xcite ( or casten and holland @xcite ) that any nonconstant steady state of is linearly unstable when the domain @xmath22 is convex . it follows that if @xmath22 is convex , then admits the bistability structure : its @xmath25-independent system has two ( constant ) linearly stable steady states , between which all others are linearly unstable . in such a case , berestycki and nirenberg @xcite obtained the existence and uniqueness of bistable traveling waves . in the case where @xmath22 is an appropriate dumbbell - shaped domain , matano @xcite constructed a counterexample to show that has stable non - constant steady states , and berestycki and hamel @xcite also proved the nonexistence of traveling waves connecting two stable constant steady states . for bistable traveling waves in time - delayed reaction - diffusion equations , we refer to @xcite . for such an equation with time delay @xmath27 , one can choose @xmath28,{\mathbb { r}})$ ] so that its solution semiflow has the bistability structure . recently , there is an increasing interest in reaction diffusion equations in periodic habitats . a typical example is @xmath29 where @xmath30 is a positive periodic function with period @xmath31 . define @xmath32,{\mathbb { r}})$ ] and @xmath33 . it is easy to see that @xmath34 and that any element in @xmath35 is a constant function in @xmath36 . thus , the solution semiflow of on @xmath37 can be regarded as a conjugate semiflow on @xmath36 , and hence , the bistability structure should be understood as : the restriction of the solution semiflow of on @xmath35 has two ordered @xmath38-periodic steady states , between which all others are unstable . assuming that the function @xmath21 is of bistable type , xin @xcite obtained the existence of spatially periodic ( pulsating ) traveling wave as long as @xmath39 is sufficiently close to a positive constant in a certain sense ( see also @xcite ) . however , whether the solution semiflow of admits the bistability structure remains an open problem . we will give an affirmative answer in section 6.3 and further improve xin s existence result . meanwhile , a counterexample will be constructed to show that the solution semiflow of has no bistability structure in the general case of varying @xmath40 . more recently , chen , guo and wu @xcite proved the existence , uniqueness and stability of spatially periodic traveling waves for one - dimensional lattice equations in a periodic habitat under the bistability assumption . there are also other types of bistable waves ( see , e.g.,@xcite ) . for monostable systems in periodic habitats , we refer to @xcite and references therein . in general , there are multiple intermediate unstable equilibria in between two stable ones in the case where the space @xmath5 is high dimensional . meanwhile , it is possible for the given system to have intermediate unstable time - periodic orbits in @xmath5 . these give more difficulties to the study of bistable semiflows than monostable ones , whose restricted systems on @xmath5 have only one unstable and one stable equilibria . to overcome these difficulties , we will show that all these unstable equilibria and all points in these periodic orbits are unordered in @xmath5 under some appropriate assumptions . with this in mind , a bistable system can be regarded as the union of two monostable systems although such a union is not unique . from this point of view , we establish a link between monostable subsystems and the bistable system itself , which plays a vital role in the propagation of bistable traveling waves . this link is stated in terms of spreading speeds of monostable subsystems ( see assumption ( a6 ) ) . for spreading speeds of various monostable evolution systems , we refer to @xcite and references therein . in our investigation , we consider seven cases : ( i ) @xmath6 and @xmath41 ; ( ii ) @xmath6 and @xmath42 ; ( iii ) @xmath43 and @xmath41 ; ( iv ) @xmath43 and @xmath44 ; ( v ) periodic habitat ; ( vi ) weak compactness ; ( vii ) time periodic . for the case ( i ) , we combine the above observations for general bistable semiflows and yagisita s perturbation idea in @xcite to prove the existence of traveling waves . for the case ( iii ) , we use the bistable traveling waves @xmath45 of discrete - time semiflows @xmath46 to approximate the bistable wave of the continuous - time semiflow @xmath47 . this new approach heavily relies on an estimation of the boundedness of @xmath48 as @xmath49 , which is proved by the bistability structure of the semiflow ( see inequalities and ( [ bistabilityinequ2 ] ) ) . it turns out that our result does not require the additional assumption on the existence of a pair of upper and lower solutions as in @xcite . in the case ( ii ) , both the evolution time @xmath50 and the habitat @xmath2 are discrete , a traveling wave @xmath51 of @xmath52 can not be well - defined in the usual way because the wave speed @xmath53 and hence , the domain of @xmath54 is unknown . so we define it to be a traveling wave of an associated map @xmath55 . however , @xmath55 has much weaker compactness than @xmath56 . to overcome this difficulty , we establish a variant of helly s theorem for monotone functions from @xmath9 to @xmath5 in the appendix , which is also of its own interest . this discovery also enables us to study monotone semiflows in a periodic habitat and with weak compactness , respectively . further , we can deal with the case ( iv ) by the similar idea as in the case ( iii ) because now traveling waves in the case ( ii ) are defined on @xmath9 . traveling waves for a time - periodic system can be obtained with the help of the discrete - time semiflow generated by the associated poincar map . motivated by the discussions in ( * ? ? ? * section 5 ) , we can regard a semiflow in a periodic habitat as a conjugate semiflow in a homogeneous discrete habitat , and hence , we can employ the arguments for the cases ( ii ) and ( iv ) to establish the existence of spatially periodic bistable traveling waves . the rest of this paper is organized as follows . in section 2 , we present our main assumptions . section 3 is focused on discrete - time , continuous - time , and time - periodic compact semiflows on some subsets of @xmath57 . in section 4 , we extend our results to compact semiflows in a periodic habitat . in section 5 , we further investigate semiflows with weak compactness . in section 6 , we apply the abstract results to four classes of evolution systems : a time - periodic reaction - diffusion system , a parabolic system in a cylinder , a parabolic equation with periodic diffusion , and a time - delayed reaction - diffusion equation . a short appendix section completes the paper . throughout this paper , we assume that @xmath5 is an ordered banach space with the norm @xmath58 and the cone @xmath59 . further , we assume that @xmath5 is also a vector lattice with the following monotonicity condition : @xmath60 where @xmath61 . such a banach space is called a banach lattice . we use @xmath62 to denote the set of all continuous functions from the compact metric space @xmath63 to the @xmath39-dimensional euclidean space @xmath64 . we equip @xmath62 with the maximum norm and the standard cone consisting of all nonnegative functions . then @xmath62 is a special banach lattice , which will be used in this paper . for more general information about banach lattices , we refer to the book @xcite . let the spatial habitat @xmath2 be the real line @xmath65 or the lattice @xmath66 for some positive number @xmath38 . for simplicity , we let @xmath67 . we say a function @xmath68 is bounded if the set @xmath69 is bounded . throughout this paper , we always use @xmath70 to denote the set of all bounded functions from @xmath9 to @xmath5 , and @xmath57 to denote the set of all bounded and continuous functions from @xmath2 to @xmath5 . moreover , any element in @xmath5 can be regarded as a constant function in @xmath70 and @xmath57 . in this paper , we equip @xmath57 with the compact open topology , that is , a sequence @xmath71 converges to @xmath72 in @xmath57 if and only if @xmath73 converges to @xmath74 in @xmath5 uniformly for @xmath25 in any bounded subset of @xmath2 . the following norm on @xmath57 can induce such topology : @xmath75 clearly , if @xmath76 , then @xmath77 with respect to the compact open topology if and only if @xmath78 for every @xmath79 . we assume that @xmath80 is not empty . for any @xmath81 , we write @xmath82 provided @xmath83 , @xmath84 provided @xmath82 but @xmath85 , and @xmath86 provided @xmath87 . a set @xmath88 is said to be totally unordered if any two elements ( if exist ) are unordered . for any @xmath89 , we write @xmath90 provided @xmath91 for all @xmath92 , @xmath93 provided @xmath90 but @xmath94 , and @xmath95 provided @xmath96 for all @xmath92 . for any @xmath97 with @xmath98 , we define @xmath99 , @xmath100 and @xmath101 . for any @xmath89 , we write the interval @xmath102_{{\mathcal}{c}}$ ] to denote the set @xmath103 , @xmath104_{{\mathcal}{c}}$ ] to denote the set @xmath105 , and similarly , we can write the intervals @xmath102]_{{\mathcal}{c}}$ ] and @xmath107_{{\mathcal}{c}}$ ] . and for any @xmath108 in @xmath5 , we can write the intervals @xmath109_{{\mathcal}{x}},[[u , v]]_{{\mathcal}{x}},[[u , v]_{{\mathcal}{x}}$ ] and @xmath109]_{{\mathcal}{x}}$ ] in a similar way . let @xmath111 and @xmath56 be a map from @xmath112 to @xmath112 . let @xmath113 be the set of all fixed points of @xmath56 restricted on @xmath114 . [ defstabfix ] for the map @xmath115 , a fixed point @xmath116 is said to be strongly stable from below if there exist a number @xmath117 and a unit vector @xmath118 such that @xmath119\gg \alpha-\eta e\quad \text{for any}\quad \eta\in ( 0,\delta].\ ] ] strong instability from below is defined by reversing the inequality . similarly , we can define strong stability ( instability ) from above . given @xmath120 , define the translation operator @xmath121 on @xmath70 by @xmath122(x)=\phi(x - y)$ ] . assume that @xmath123 and @xmath124 are in @xmath113 . we impose the following hypothesis on @xmath56 : 1 . ( _ translation invariance _ ) @xmath125=q\circ t_y [ \phi ] , \forall \phi\in{\mathcal}{c}_\beta , y\in\mathcal{h}$ ] . ( _ continuity _ ) @xmath126 is continuous with respect to the compact open topology . ( _ monotonicity _ ) @xmath56 is order preserving in the sense that @xmath127\ge q[\psi]$ ] whenever @xmath90 in @xmath112 . ( _ compactness _ ) @xmath126 is compact with respect to the compact open topology . 5 . ( _ bistability _ ) two fixed points @xmath123 and @xmath124 are strongly stable from above and below , respectively , for the map @xmath128 , and the set @xmath129 is totally unordered . note that the above bistability assumption is imposed on the spatially homogeneous map @xmath115 . we allow the existence of other fixed points on the boundary of @xmath114 so that the theory is applicable to two species competitive evolution systems . the non - ordering property of @xmath129 can be obtained by the strong instability of all fixed points in this set if the semiflow is eventually strongly monotone . more precisely , a sufficient condition for hypothesis ( a5 ) to hold is : 1 . ( _ bistability _ ) @xmath115 is eventually strongly monotone in the sense that there exists @xmath130 such that @xmath131\gg q^{m}[v]$ ] for all @xmath132 whenever @xmath133 in @xmath114 . further , for the map @xmath115 , two fixed points @xmath123 and @xmath124 are strongly stable from above and below , respectively , and each @xmath134 ( if exists ) is strongly unstable from both below and above . the following figures illustrate the bistability structures in ( a5 ) and ( a5@xmath135 ) . next we show that the assumption ( a5@xmath135 ) implies ( a5 ) . in applications , however , one may find other weaker sufficient conditions than ( a5@xmath135 ) for ( a5 ) to hold . [ bistabilitymap ] if ( a5 @xmath135 ) holds , then for any @xmath136 , we have @xmath137 and @xmath138 . without loss of generality , we only show @xmath139 . assume , for the sake of contradiction , that @xmath140 . then @xmath141\ll q^{m_1}[\alpha_2]=\alpha_2 $ ] . since @xmath142 is strongly unstable from above , there exists @xmath143 and @xmath144 such that @xmath145_{{\mathcal}{x}}$ ] and @xmath146\gg u_0 $ ] . define the recursion @xmath147,n\ge 0 $ ] . then @xmath148 is convergent to some @xmath149 with @xmath150 due to hypothesis ( a4 ) . by the eventual strong monotonicity of @xmath56 , we see that @xmath151\ll q^{m_1}[u_{n+1-m_1}]=u_{n+1}\ll q^{m_1}[\alpha]=\alpha , \forall n\ge m_1.\ ] ] since @xmath152 is strongly unstable from below , we can find @xmath153 and @xmath154 such that @xmath155\ll \alpha-\delta e_\alpha,\forall \delta\in(0,\delta_\alpha]$ ] . choose @xmath156 such that @xmath157 . define @xmath158 : u_{n_1}\le \alpha-\delta e_\alpha\}$ ] . thus , @xmath159 . on the other hand , we have @xmath160\le q[\alpha-\eta e_\alpha]\ll \alpha-\eta e_\alpha,\ ] ] a contradiction . due to assumption ( a5 ) , a bistable system @xmath52 can be regarded as the union of two monostable systems . more precisely , assuming that @xmath161 , we have two monostable sub - systems : @xmath52 restricted on @xmath162_{{\mathcal}{c}}$ ] and @xmath163_{{\mathcal}{c}}$ ] , respectively . with this in mind , next we construct an initial function @xmath164 so that we can define the leftward asymptotic speed of propagation of @xmath164 , and hence , present our last assumption . note that in ( a5 ) we do not require @xmath165 or @xmath166 . but ( a5 ) is sufficient to guarantee that @xmath152 and @xmath124 can be separated by two neighborhoods in @xmath163_{{\mathcal}{x}}$ ] , and a similar claim is valid for @xmath123 and @xmath152 ( see lemma 3.1 ) . in view of assumption ( a5 ) , we can find a positive number @xmath167 and a unit vector @xmath168 such that @xmath169\gg \beta-\eta e_\beta,\quad \forall \eta\in(0,\delta_\beta].\ ] ] define @xmath170:\theta \alpha+(1-\theta)\beta\in [ \beta-\delta_\beta e_\beta,\beta]_{\mathcal}{x}\right\}.\ ] ] let @xmath171 choose a nondecreasing initial function @xmath172 with the property that @xmath173 it then follows from assumptions ( a1)-(a2 ) and ( a5 ) that @xmath174(x)=q[\phi_\alpha^-(+\infty)](0)=q[v_\alpha^-]\ge q[\beta-\delta_\beta e_\beta]\gg \beta-\delta_\beta e_\beta,\ ] ] and hence , there exits @xmath175 such that @xmath176(x)\gg \beta-\delta_\beta e_\beta , \forall x\ge \sigma-1.\ ] ] define a sequence @xmath177 of points in @xmath5 as follows : @xmath178(\sigma n),\quad n\ge 1.\ ] ] then we have @xmath179(2\sigma)=q[q[\phi_\alpha^- ] ( \cdot+\sigma)](\sigma)\ge q[\phi_\alpha^-](\sigma)=a_{1,\sigma}.\ ] ] by induction , we see that @xmath177 is nondecreasing in @xmath180 . thus , assumption ( a4 ) implies that @xmath177 tends to a fixed point @xmath181 with @xmath182 . therefore , @xmath183 . by the above observation , we have @xmath184(x)\ge \lim_{n\to \infty } q^n[\phi_\alpha^-](\sigma n)= \lim_{n\to \infty } a_{n,\sigma}=\beta,\ ] ] and hence , @xmath185\subset \lambda(\phi_\alpha^-):=\left\{c\in{\mathbb { r}}:\lim_{n\to \infty , x\ge -cn}q^n[\phi_\alpha^-](x)=\beta\right\}.\ ] ] define @xmath186 clearly , @xmath187 $ ] and @xmath188 . we further claim that @xmath189 is independent of the choice of @xmath164 as long as @xmath164 has the property . indeed , for any given @xmath72 with the property , we have @xmath190 it then follows that for any @xmath191 and @xmath192 , @xmath193(x)=\lim_{n\to \infty , x\ge -(c-\epsilon ) n}q^n[\phi_\alpha^-](x-1 ) \nonumber\\ & & \le \lim_{n\to \infty , x\ge -(c-\epsilon ) n}q^n[\phi](x ) \le \lim_{n\to \infty , x\ge -(c-\epsilon ) n}q^n[\phi_\alpha^-](x+1)\nonumber\\ & & = \lim_{n\to \infty , x\ge -c n}q^n[\phi_\alpha^-](x)=\beta,\end{aligned}\ ] ] which implies that @xmath194 and hence , @xmath195 . for convenience , we may call @xmath189 as the leftward asymptotic speed of propagation of @xmath164 . following the above procedure , we can find @xmath196 such that @xmath197\ll \eta e_0 , \forall \eta\in ( 0,\delta_0].\ ] ] here we emphasis that @xmath198 above and @xmath199 will play a vital role in the whole paper because they describe the local stability of fixed points @xmath123 and @xmath124 . similarly , we can define @xmath200 : \theta \alpha\in [ 0,\delta_0e_0]_{\mathcal}{x}\}$ ] and @xmath201 . let @xmath202 be a nondecreasing initial function with the property that @xmath203 due to the same reason , we can define the number @xmath204(x)=0\right\},\ ] ] which is called the rightward asymptotic speed of propagation of @xmath205 . as showed above , these two speeds are bounded below , but may be plus infinity . to better understand these two spreading speeds , we use figure [ fig2](left ) to explain them . now we are ready to state our last assumption on @xmath56 : 1 . ( _ counter - propagation _ ) for each @xmath134 , @xmath206 . assumption ( a6 ) assures that two initial functions in the left hand side of figure [ fig2 ] will eventually propagate oppositely although one of these two speeds may be negative . it is interesting to note that assumption ( a6 ) is nearly necessary for the propagation of a bistable traveling wave . indeed , if a monotone evolution system admits a bistable traveling wave , then it is usually unique ( up to translation ) and globally attractive ( see , e.g. , remark [ stabilityremark ] ) . this implies that the solution starting from the initial data @xmath207 converges to a phase shift of the bistable wave . if @xmath208 , then the comparison principle would force the solutions starting from @xmath209 to split the bistable wave comparing with the definition of spreading speeds ( short for asymptotic speeds of spread / propagation ) for monostable semiflows ( see , e.g. , @xcite ) , one can find that the leftward spreading speed of the monostable subsystem @xmath210 restricted on @xmath163_{\mathcal}{c}$ ] is shared by a large class of initial functions , and in many applications , it equals @xmath189 . a similar observation holds for @xmath211 . thus , for a specific bistable system , the assumption ( a6 ) can be verified by using the properties of spreading speeds for monostable subsystems . if we consider the non - increasing traveling waves , then we can similarly define the numbers @xmath212 and @xmath213 ( see figure [ fig2](right ) ) . as such , ( a6 ) should be stated as @xmath214 . we say a habitat is homogeneous for the semiflow @xmath0 on a metric space @xmath215 if @xmath216(x - y)=q_t[\phi(\cdot - y)](x),\quad \forall \phi\in{\mathcal}{e},x , y\in{\mathcal}{h } , t\in { \mathcal}{t}.\ ] ] in this section , we will establish the existence of bistable traveling waves for the semiflow @xmath0 on @xmath217 in the following order : discrete - time semiflows in a continuous habitat , discrete - time semiflows in a discrete habitat , time - periodic semiflows , continuous - time semiflows in a continuous habitat , and continuous - time semiflows in a discrete habitat . in this case , time @xmath50 is discrete and habitat @xmath2 is continuous : @xmath6 and @xmath41 . for convenience , we use @xmath56 to denote @xmath218 , and consider the semiflow @xmath52 , where @xmath219 is the @xmath180-th iteration of @xmath56 . [ defdiscont ] @xmath220 with @xmath221 is said to be a traveling wave with speed @xmath222 of the discrete semiflow @xmath210 if @xmath223(x)=\psi(x+cn),\forall x\in{\mathbb { r } } , n\ge 0 $ ] . we say that @xmath54 connects @xmath123 to @xmath124 if @xmath224 and @xmath225 . we first show that @xmath123 and @xmath124 are two isolated fixed points of @xmath56 in @xmath114 if ( a5 ) holds . [ islatedequilibria ] let @xmath198 and @xmath226 be chosen such that and hold , respectively . then @xmath227 and @xmath228_{\mathcal}{x}=\{\beta\}$ ] . assume , for the sake of contradiction , that @xmath229 . define the number @xmath230 $ ] by @xmath231 : \alpha\in [ 0,\delta e_0]_{\mathcal}{x}\}.\ ] ] then it follows that @xmath232 but @xmath233_{\mathcal}{x}$ ] . however , by the monotonicity of @xmath56 and the fact that @xmath123 is strongly stable , we have @xmath234\le q[\bar{\delta}e_0]\ll \bar{\delta}e_0.\ ] ] this contradicts @xmath233_{\mathcal}{x}$ ] . and hence , @xmath235 . similarly , we have @xmath236_{\mathcal}{x}=\{\beta\}$ ] . choose @xmath117 such that @xmath237 assume that @xmath238 and @xmath239 are two nondecreasing functions in @xmath240 with the properties that @xmath241 clearly @xmath242 . and we have the following observation . [ upperlowersolu1 ] assume that @xmath56 satisfies ( a1)-(a3 ) and ( a5 ) . then there exists a positive rational number @xmath243 such that for any @xmath244 , we have @xmath245(x)\ge { \underline}{\psi}(x - c)\quad \text{and}\quad q[\bar{\psi}](x)\le \bar{\psi}(x+c)\quad \text{for any $ x\in{\mathbb { r}}$}.\ ] ] assume that @xmath246 be an increasing sequence in @xmath65 . then the sequence @xmath247 converges to @xmath248 in @xmath112 since @xmath249 . it then follows from ( a1)-(a2 ) and ( a5 ) that @xmath245(+\infty)=\lim_{n\to \infty } q[{\underline}{\psi}](x_n)=\lim_{n\to \infty}q[{\underline}{\psi}(\cdot+x_n)](0)=q[\beta-\delta e_\beta]\gg \beta-\delta e_\beta.\ ] ] therefore , there exists a positive @xmath250 such that @xmath251(y_0)\ge \beta-\delta e_\beta$ ] . note that @xmath251(x)$ ] is nondecreasing in @xmath25 . then for any @xmath252 we have @xmath245(x)\ge q[{\underline}{\psi}](y_0)\ge \beta-\delta e_\beta \ge { \underline}{\psi}(x - c ) , \forall x\ge y_0\ ] ] and @xmath245(x)\ge 0={\underline}{\psi}(0 ) \ge { \underline}{\psi}(x - y_0)\ge { \underline}{\psi}(x - c),\forall x < y_0,\ ] ] which means @xmath251(x ) \ge { \underline}{\psi}(x - c),\forall c\ge y_0 $ ] . similarly , we have @xmath253(-\infty)=\lim_{n\to \infty } q[{\underline}{\psi}](-x_n)=\lim_{n\to \infty}q[\bar{\psi}(\cdot - x_n)](0)=q[\delta e_0]\ll \delta e_0=\bar{\psi}(-\infty),\ ] ] and hence , there exists @xmath254 such that @xmath255(x)\le \bar{\psi}(x+c),\forall c\ge z_0 $ ] . choosing @xmath256 , we complete the proof . let @xmath257 . clearly , @xmath258 , is a rational number . for any @xmath259 , define the map @xmath260 by @xmath261(x)=\phi(\xi x),\forall x\in\mathbb{r}$ ] . define @xmath262 by @xmath263 [ fixedpoint1 ] assume that @xmath56 satisfies ( a1)-(a5 ) . then for each @xmath264 , @xmath265 has a fixed point @xmath71 in @xmath112 such that @xmath71 is nondecreasing and @xmath266 . we first show that @xmath267 $ ] . indeed , when @xmath268 we have @xmath269(x);\ ] ] when @xmath270 we have @xmath271(x)={\underline}{\psi}_n(\kappa_n x)={\underline}{\psi}_n(x+{\frac}{\bar{c}}{n}x)\ge { \underline}{\psi}_n(x+\bar{c}),\ ] ] and hence , @xmath272(x)$ ] for all @xmath273 . consequently , by the monotonicity of @xmath56 and @xmath274(x+\bar{c})$](see lemma [ upperlowersolu1 ] ) we obtain @xmath275(x+\bar{c})\le q\circ a_{\kappa_n}[{\underline}{\psi}_n ] ( x)=g_n[{\underline}{\psi}_n ] ( x).\ ] ] similarly , we have @xmath276 $ ] . it then follows that @xmath277\le g_n^k[\bar{\psi}_n]\le \bar{\psi}_n,\quad \forall k\in \mathbb{n}.\ ] ] for any @xmath278 , we have @xmath279=g_n\circ g_n^{k-1}[{\underline}{\psi}_n]\in g_n[{\mathcal}{c}_\beta].\ ] ] since @xmath280 is order preserving and @xmath281 is nondecreasing in @xmath25 , we know that @xmath282(x)$ ] is nondecreasing both in @xmath283 and @xmath25 . recall that @xmath280 is compact due to assumption ( a4 ) . it then follows that @xmath282 $ ] converges in @xmath112 . denote the limit by @xmath71 . by inequality , we also get @xmath266 . moreover , @xmath73 is also nondecreasing due to proposition [ basicprop](2 ) . and obviously , @xmath284=g_n [ \lim_{k\to \infty}g_n^k[{\underline}{\psi}_n]]=g_n [ \phi_n].\ ] ] this completes the proof . the following lemma reveals a relation between the wave speeds of monostable traveling waves in the sub - monostable systems and the numbers defined in and . [ miniwavespeeddsf ] let @xmath189 and @xmath211 be defined as in and . assume that @xmath56 satisfies ( a3 ) . then the following statements are valid : 1 . if @xmath285 is a monotone traveling wave connecting @xmath152 to @xmath124 of the discrete semiflow @xmath286 , then the speed @xmath287 . 2 . if @xmath285 is a monotone traveling wave connecting @xmath123 to @xmath152 of the discrete semiflow @xmath286 , then the speed @xmath288 . we only prove the statement ( 1 ) since the proof for ( 2 ) is similar . in view of lemma [ islatedequilibria ] , we see that @xmath152 and @xmath124 can be separated by balls @xmath289 and @xmath290 with radius @xmath291 in the metric space @xmath163_{\mathcal}{x}$ ] . then @xmath292 . we write @xmath293 as the form @xmath294 . recall the definition of @xmath295 in , it then follows that @xmath296 which implies that @xmath297 since @xmath298 and @xmath299 , there must exist a nondecreasing initial function @xmath164 with property such that @xmath300 . assume , for the sake of contradiction , that @xmath301 . choose a rational number @xmath302 with @xmath303 . it then follows from that @xmath304(-{\frac}{q}{p}\times pn)\le \lim_{n\to \infty}q^{pn}[\psi](-qn)\nonumber\\ & = & \lim_{n\to \infty } \psi(-q n+cpn)=\psi(-\infty)=0,\end{aligned}\ ] ] a contradiction . thus , we have @xmath305 . now we are ready to prove the main result of this subsection . [ thdiscont ] assume that @xmath56 satisfies ( a1)-(a6 ) . then there exists @xmath222 such that the discrete semiflow @xmath286 admits a non - decreasing traveling wave with speed @xmath53 and connecting @xmath123 to @xmath124 . we spend three steps to complete the proof . firstly , we construct @xmath306 , @xmath307 such that @xmath308(x)=\phi_+(x+c_+)\quad \text{and}\quad q[\phi_-](x)=\phi_-(x+c_-)\ ] ] with @xmath309_{\mathcal}{x}\quad \text{and}\quad \phi_+(0)\in [ \beta-\delta e_\beta,\beta)_{\mathcal}{x}.\ ] ] indeed , let @xmath71 be obtained in lemma [ fixedpoint1 ] . since @xmath310 and @xmath311 , we have @xmath312 and @xmath313 now we define @xmath314 as follows : @xmath315_{{\mathcal}{x}}\},\quad b_n:=\inf_{x\in\mathbb{r}}\{\phi_n(x)\in [ \beta-\delta e_\beta,\beta]_{{\mathcal}{x}}\}.\ ] ] it then follows that @xmath316 and @xmath317 define @xmath318 and @xmath319 . then @xmath320(\cdot+a_n)=q[\phi_n(\kappa_n\cdot)](\cdot+a_n)=q[\phi_n(\kappa_n(\cdot+a_n))]\in q[{\mathcal}{c}_\beta].\ ] ] similarly , @xmath321\in q[{\mathcal}{c}_\beta]$ ] . thus , there exists a subindex ( still denoted by @xmath180 ) , two nondecreasing functions @xmath322 and @xmath323 $ ] with @xmath324 such that @xmath325 obviously , @xmath326 and @xmath327 . by the definitions of @xmath328 and @xmath329 , we immediately have @xmath330 and @xmath331 , and hence @xmath332 and @xmath333 . define @xmath334 and @xmath335 . obviously , @xmath336 because @xmath337 . now we want to only prove @xmath338(x)=\phi_-(x+c_-)$ ] because the proof of the other one is similar . note that the following limit is uniform for @xmath25 in any bounded subset @xmath339 @xmath340 it then follows that for any @xmath341 , we have @xmath342(x+c_-+a_n)\nonumber\\ & & = \lim_{n\to \infty } q[\phi_n(\kappa_n\cdot)](x+c_-+a_n)=\lim_{n\to \infty}q [ \phi_n(\kappa_n(\cdot+a_n))](x+c_-)\nonumber\\ & & = \lim_{n\to \infty}q [ \phi_{-,n}(\kappa_n(\cdot+a_n)-a_n)](x+c_-)=q[\phi_-](x),\end{aligned}\ ] ] where the last equality is obtained from proposition [ sequenceinc](2 ) and the continuity of @xmath56 . secondly , we prove that @xmath343 obtained in the first step have the following properties : 1 . @xmath344 and @xmath345 ; 2 . @xmath346 and @xmath347 are ordered . indeed , let @xmath246 be an increasing sequence in @xmath65 . note that @xmath348(0)\in q[{\mathcal}{c}_\beta](0)$ ] , which is precompact in @xmath114 . it then follows that there exists a subindex @xmath349 and @xmath350 such that @xmath351 , which , together with the fact that @xmath352 is nondecreasing and proposition [ sequenceinc](1 ) , implies that @xmath353 . besides , from we see that @xmath354 is a fixed point of @xmath56 . similar results hold for @xmath355 and @xmath356 . recall that @xmath357 and @xmath358 , which , together with the choice of @xmath359 , implies that @xmath344 and @xmath345 . further , since any two real numbers are ordered , we see that there exist sequences @xmath360 such that for each @xmath278 , @xmath361 define @xmath362 and @xmath363 . then either @xmath364 or @xmath365 has infinitely many elements . if @xmath364 does , then there holds @xmath366 this implies that @xmath367 , and hence , @xmath368 . if @xmath365 has infinitely many elements , then we have @xmath369 by a similar argument . thus , @xmath346 and @xmath347 must be ordered in @xmath114 . finally , we prove that either @xmath352 or @xmath370 connects @xmath123 to @xmath124 . indeed , we have shown in the second step that @xmath346 and @xmath347 are ordered . it then follows from the bistability assumption ( a5 ) that there are only three possibilities : 1 . @xmath371 ; 2 . @xmath372 ; 3 . @xmath373 for some @xmath134 . we further claim that the possibility ( iii ) can not happen . otherwise , lemma [ miniwavespeeddsf ] implies that @xmath374 and @xmath375 . since @xmath376 , it then follows that @xmath377 which contradicts assumption ( a6 ) . thus , either ( i ) or ( ii ) holds , and hence , we complete the proof . in this case , both time @xmath50 and habitat @xmath2 are discrete : @xmath6 and @xmath42 . without confusion , we consider the semiflow @xmath52 in a metric space @xmath378 . since the habitat is discrete , we can not use the definition of traveling waves with a unknown speed as in definition [ defdiscont ] . this is because the wave profile @xmath379 may not be well - defined for all @xmath273 . so we start with the modification of the definition of traveling waves in a discrete habitat . [ defdisdis ] @xmath220 with @xmath380 is said to be a traveling wave with speed @xmath222 of the discrete semiflow @xmath210 if there exists a countable set @xmath381 such that @xmath382(i)=\psi(i+x+c),\forall i\in{\mathbb { z } } , x\in{\mathbb { r}}\setminus \gamma$ ] . by definition [ defdisdis ] and proposition [ setprop1 ] , it follows that there exists @xmath383 such that @xmath384(i)=\psi(i+x_0+cn),\forall i\in{\mathbb { z } } , n\ge0 $ ] . define @xmath385 . then , with a little abuse of notation , we have @xmath386(i)=\phi(i+cn),\forall i\in{\mathbb { z } } , n\ge0 $ ] . such a definition of traveling waves is motivated by the idea employed in the proof of theorem [ thdisdis ] . let @xmath387 be a fixed point of @xmath56 . define @xmath388 by @xmath389(x)=q[\phi(\cdot+x)](0),\quad \forall x\in { \mathbb { r}}.\ ] ] then we see from ( * ? ? ? * lemma 2.1 ) that @xmath55 satisfies ( a1)-(a3 ) and ( a5 ) with @xmath390 and @xmath391 if @xmath56 itself satisfies ( a1)-(a3 ) and ( a5 ) . further , if @xmath56 satisfies ( a4 ) , then the set @xmath392(x)\subset { \mathcal}{x}_\beta$ ] is precompact for any @xmath273 . for @xmath388 , we have similar results as in lemma [ upperlowersolu1 ] and [ fixedpoint1 ] . [ upperlowersolu2 ] assume that @xmath56 satisfies ( a1)-(a3 ) and ( a5 ) . then there exists a positive rational number @xmath243 such that for any @xmath244 , we have @xmath393(x)\ge { \underline}{\psi}(x - c)\quad \text{and}\quad \tilde{q}[\bar{\psi}](x)\le \bar{\psi}(x+c)\quad \text{for any $ x\in{\mathbb { r}}$}.\ ] ] [ fixedpoint2 ] assume that @xmath56 satisfies ( a1)-(a5 ) . then for each @xmath264 , @xmath394 has a fixed point @xmath395 in @xmath396 such that @xmath395 is nondecreasing and @xmath397 . by the same arguments as in the proof of lemma [ fixedpoint1 ] , we can obtain a similar inequality as : @xmath398\le \tilde{g}_n^k[\bar{\psi}_n]\le \bar{\psi}_n,\quad \forall k\in\mathbb{n}.\ ] ] define @xmath399 and @xmath400,k\ge 1 $ ] . then @xmath401(x)=\tilde{q}[w_{n , k}(\kappa_n\cdot)](x)=q[w_{n , k}(\kappa_n(\cdot+x))](0).\ ] ] note that @xmath402 $ ] is compact and @xmath403 is nondecreasing in @xmath283 . it then follows that for any fixed @xmath404 , @xmath405 converges in @xmath114 . denote the limit by @xmath406 . then @xmath406 is nondecreasing in @xmath404 and @xmath407 . taking @xmath408 in , we arrive at @xmath409(0)$ ] . consequently , @xmath410=\tilde{q}\circ a_{\kappa_n}[\tilde{\phi}_n]=\tilde{g}_n[\tilde{\phi}_n].\ ] ] this completes the proof . to overcome the difficulty due to the lack of compactness for @xmath55 , we will use the properties of monotone functions established in the appendix to show the convergence of a sequence in @xmath392 $ ] . [ thdisdis ] assume that @xmath411 and @xmath56 satisfies ( a1)-(a6 ) . then there exists @xmath222 such that the semiflow @xmath412 on @xmath112 admits a nondecreasing traveling wave @xmath220 with speed @xmath53 and connecting @xmath123 to @xmath124 . further , @xmath54 is either left or right continuous . as in the proof of theorem [ thdiscont ] , we define @xmath413_{{\mathcal}{x}}\},\quad \tilde{b}_n:=\inf_{x\in{\mathbb { r}}}\{\tilde{\phi}_n(x)\in[\beta-\delta e_\beta,\beta]_{{\mathcal}{x}}\}.\ ] ] then @xmath414 . note that for any @xmath273 , we have @xmath415(x ) = \tilde{q}[\tilde{\phi}_n(\kappa_n\cdot)](x)=q[\tilde{\phi}_n ( \kappa_n(\cdot+x))](0)\in q[{\mathcal}{c}_\beta](0).\ ] ] since @xmath402(0)$ ] is precompact in @xmath114 , it then follows that for any @xmath273 , @xmath416 and @xmath417 both exist . and hence , by the definitions of @xmath418 and @xmath419 , we have @xmath420 but @xmath421_{{\mathcal}{x}}\quad \text{and}\quad \tilde{\phi}_n(\tilde{b}_n^-)\not\in [ [ \beta-\delta e_\beta,\beta]_{{\mathcal}{x}}.\ ] ] define @xmath422 and @xmath423 . then @xmath424 but @xmath425_{{\mathcal}{x}}\quad \text{and}\quad \tilde{\phi}_{+,n}(0 ^ -)\not\in [ [ \beta-\delta e_\beta,\beta]_{{\mathcal}{x}}.\ ] ] since @xmath426 $ ] , we have @xmath427 ( x+\tilde{a}_n ) = q[\tilde{\phi}_n(\kappa_n(\cdot+\tilde{a}_n+x))](0)\in q[{\mathcal}{c}_\beta](0).\ ] ] similarly , @xmath428(0)\in q[{\mathcal}{c}_\beta](0)$ ] . let @xmath429 be the set of all rational numbers , and @xmath430 be an increasing sequence converging to @xmath25 . using @xmath426 $ ] again , we see that for any @xmath431 and @xmath432 , @xmath433(0)\in q[{\mathcal}{c}_\beta](0).\ ] ] similarly , @xmath434(0)$ ] . note that @xmath402(0)$ ] is precompact in @xmath114 and that @xmath429 is countable . it then follows that there exists a subindex ( still denoted by @xmath435 ) and @xmath436 such that @xmath437 and that for any @xmath438 , @xmath431 and @xmath432 , sequences @xmath439 and @xmath440 converge in @xmath114 . and hence , the following limits @xmath441(0)\ ] ] and @xmath442(0)\ ] ] both exist . this means the limits @xmath443 exist for all @xmath273 . define @xmath444 and @xmath445 clearly , @xmath446 are nondecreasing functions in @xmath396 and for any @xmath447 , @xmath448 all exist . hence , we see from theorem [ hellyth ] , that there exists a countable subset @xmath364 of @xmath9 such that @xmath449 converges to @xmath450 for all @xmath451 . define @xmath452 and @xmath453 thus , @xmath454 is left continuous and @xmath455 is right continuous . note that @xmath456 for all @xmath457 . it then follows that @xmath449 converges to @xmath458 for @xmath459 , where @xmath460 is also countable . let @xmath461 be an increasing sequence converging to @xmath123 and @xmath462 be an increasing sequence converging to @xmath463 , respectively . note that @xmath464 and @xmath465_{{\mathcal}{x}}.\ ] ] similarly , we have @xmath466 but @xmath467_{{\mathcal}{x}}$ ] . define @xmath334 and @xmath335 . obviously , @xmath336 since @xmath324 . now we want to prove @xmath468(0)=\tilde{\phi}_-(x+c_- ) , \ , \forall x\in{\mathbb { r}}\setminus \gamma_2 $ ] . note that @xmath469 it then follows that @xmath470(x+c_-+\tilde{a}_n)\nonumber\\ & & = \lim_{n\to \infty } \tilde{q}[\tilde{\phi}_n(\kappa_n\cdot)](x+c_-+\tilde{a}_n)=\lim_{n\to \infty}\tilde{q } [ \tilde{\phi}_n(\kappa_n(\cdot+a_n))](x+c_-)\nonumber\\ & & = \lim_{n\to \infty}\tilde{q } [ \tilde{\phi}_{-,n}(\kappa_n(\cdot+\tilde{a}_n)-\tilde{a}_n)](x+c_-)\nonumber\\ & & = \lim_{n\to \infty } q [ \tilde{\phi}_{-,n}(\kappa_n(\cdot+x+c_-+\tilde{a}_n)-\tilde{a}_n)](0).\end{aligned}\ ] ] in view of proposition [ convergence ] , we obtain that @xmath471(0)$ ] @xmath472 . a similar result also holds for @xmath473 . now , the same argument as in the proof of theorem [ thdiscont ] completes the proof . let @xmath474 be a positive number , where @xmath43 or @xmath475 . recall that a family of mappings @xmath476 is said to be an @xmath477-time periodic semiflow on a metric space @xmath478 provided that it has the following properties : 1 . @xmath479=\phi,\ , \forall \phi\in { \mathcal}{e}$ ] . 2 . @xmath480=q_{t+\omega}[\phi],\ , \forall t\ge 0,\ , \phi\in { \mathcal}{e}$ ] . 3 . @xmath481 $ ] is continuous jointly in @xmath482 on @xmath483 . the mapping @xmath484 is called the poincar map associated with this periodic semiflow . [ defpertime ] 1 . in the case where @xmath41 , @xmath485 is said to be an @xmath477-time periodic traveling wave with speed @xmath53 of the semiflow @xmath476 if @xmath486(x)=u(t , x+ct)$ ] and @xmath487 for all @xmath488 . 2 . in the case where @xmath42 , @xmath485 is said to be an @xmath477-time periodic traveling wave with speed @xmath53 of the semiflow @xmath476 if there exists a countable subset @xmath381 such that @xmath489(0)=u(t , x+ct)$ ] for all @xmath488 and @xmath487 for all @xmath490 . [ thtimeperi ] let @xmath491 be a strongly positive @xmath477-time periodic orbit of @xmath0 restricted on @xmath5 . assume that @xmath492 satisfies hypotheses ( a1)-(a6 ) with @xmath493 . then @xmath0 admits a traveling wave @xmath485 with @xmath494 and @xmath495 uniformly for @xmath496 . furthermore , @xmath497 is nondecreasing in @xmath273 . case 1 . @xmath41 . since the map @xmath484 satisfies ( a1)-(a6 ) , there exits @xmath498 and a nondecreasing function @xmath499 connecting @xmath123 to @xmath500 such that @xmath501(x)=\phi(x+c\omega)$ ] . clearly , @xmath502=\phi$ ] . define @xmath503(x)$ ] . then we have @xmath504(x)=q_t[u(0,\cdot)](x)$ ] , and @xmath505(x)=t_{ct } q_t t_{c\omega } q_\omega[\phi](x)=t_{ct } q_t [ \phi](x)=u(t , x).\ ] ] note that @xmath506=\beta(t)$ ] and that @xmath72 is nondecreasing and connecting @xmath123 to @xmath500 . it then follows that @xmath494 and @xmath495 . case 2 . @xmath42 . since the map @xmath484 satisfies ( a1)-(a6 ) , there exits @xmath498 , a countable subset @xmath381 and a nondecreasing function @xmath507 connecting @xmath123 to @xmath500 such that @xmath508(x)=\phi(x+c\omega),\forall x\in { \mathbb { r}}\setminus \gamma$ ] . clearly , @xmath509(x)=\phi(x),\forall x\in { \mathbb { r}}\setminus \gamma$ ] . define @xmath510(x)$ ] . thus , we have @xmath511(x)=\tilde{q}_t[u(0,\cdot)](x)=q_t[u(0,\cdot+x)](0 ) , \ , \ , \forall x\in { \mathbb { r}},\ ] ] and @xmath512(x)=t_{ct } \tilde{q}_t t_{c\omega } \tilde{q}_\omega[\phi](x)=t_{ct } \tilde{q}_t [ \phi](x)=u(t , x),\ ] ] for all @xmath513 . note that @xmath506=\beta(t)$ ] and that @xmath72 is nondecreasing and connecting @xmath123 to @xmath500 . it then follows that @xmath494 and @xmath495 . in this subsection , we consider continuous - time semiflows in the continuous habitat @xmath41 . recall that a family of mappings @xmath47 is said to be a semiflow on a metric space @xmath478 provided that @xmath514 satisfies the following properties : 1 . @xmath479=\phi,\ , \forall \phi\in { \mathcal}{e}$ ] . @xmath515=q_{t+s}[\phi],\ , \forall t , s\ge 0 , \ , \phi\in { \mathcal}{e}$ ] . 3 . @xmath481 $ ] is continuous jointly in @xmath482 on @xmath483 . before moving to the study of traveling waves of the semiflow @xmath47 , we first investigate the spatially homogeneous system , that is , the system restricted on @xmath5 . let @xmath387 be an equilibrium in @xmath5 . for each @xmath516 , we use @xmath517 to denote the set of all fixed points of the map @xmath518 restricted on @xmath114 . clearly , the equilibrium set of the semiflow is @xmath519 , which is a subset of @xmath517 for any @xmath516 . the subsequent results indicates that the instability of intermediate equilibria of the semiflow implies the nonordering property of all intermediate fixed points of each time-@xmath520 map . [ bistabilitysemiflow ] for any given @xmath516 , if the map @xmath518 satisfies the bistability assumption ( a5@xmath135 ) with @xmath521 , then @xmath518 satisfies ( a5 ) with @xmath522 . let @xmath523 be given . we first show that any two points @xmath524 and @xmath525 are unordered . assume , for the sake of contradiction , that @xmath526 and @xmath527 are ordered . without loss of generality , we also assume that @xmath528 . then the eventual strong monotonicity implies that @xmath529 . since @xmath526 is strongly unstable from above , there exist a unit vector @xmath530 and a number @xmath117 such that @xmath531\gg u+\delta e$ ] with @xmath532_{{\mathcal}{x}}$ ] . from ( * theorem 1.2.1 ) , we see that @xmath533 $ ] is eventually strongly increasing and converges to some @xmath534 . note that @xmath535_{{\mathcal}{x}}$ ] is strongly unstable from below . hence , by the same arguments as in the proof of proposition [ bistabilitymap ] , we obtain a contradiction . next we show the set @xmath536 is unordered . for this purpose , we see from the first step that it suffices to prove that for any two ordered elements @xmath528 in @xmath536 , @xmath109_{{\mathcal}{x}}\cap \sigma \neq\emptyset$ ] . indeed , by the eventual strong monotonicity , we have @xmath529 . then , we can choose a sequence @xmath537 on the segment connecting @xmath526 and @xmath527 such that @xmath538 . by ( * ? ? ? * theorem 1.3.7 ) , it follows that @xmath539 . clearly , we have @xmath540\}$ ] and @xmath541\}$ ] , and hence @xmath542 . note that @xmath543 is contained in the compact set @xmath544}$ ] . in the compact metric space consisting of all nonempty compact subsets of @xmath544}$ ] with hausdorff distance @xmath545 , the sequence @xmath546 has a convergent subsequence . without loss of generality , we assume that for some nonempty compact set @xmath547}$ ] , @xmath548 . since each @xmath549 is invariant for the semiflow @xmath550 , so is the compact set @xmath551 , that is , @xmath552 . for any given @xmath553 , there exist two sequences of points @xmath554 such that @xmath555 and @xmath556 as @xmath557 . since @xmath558 , we have @xmath559 and @xmath560 . letting @xmath557 , we then have @xmath561 and @xmath562 , and hence @xmath563 . this implies that @xmath551 is a singleton , that is , @xmath564 . by the invariance of @xmath551 for the semiflow , we see that @xmath152 is an equilibrium . since @xmath542 , it follows that @xmath565_x$ ] . for a continuous - time semiflow @xmath47 , we need the following definition of traveling waves . [ defcontcont ] @xmath285 with @xmath221 is said to be a traveling wave with speed @xmath222 of the continuous - time semiflow @xmath566 if @xmath567(x)=\psi(x+ct),\forall x\in{\mathbb { r } } , t\ge 0 $ ] . we say that @xmath54 connects @xmath123 to @xmath124 if @xmath568 and @xmath299 . [ thcontcont ] assume that for each @xmath516 , the map @xmath518 satisfies assumptions ( a1),(a3)-(a5 ) with @xmath522 , and the time - one map @xmath218 satisfies ( a6 ) with @xmath521 . then there exists @xmath222 such that @xmath47 admits a non - decreasing traveling wave with speed @xmath53 and connecting @xmath123 to @xmath124 . let @xmath569 and @xmath359 be chosen as in [ delta0 ] , [ deltabeta ] and [ delta ] , respectively . we proceeds with three steps . firstly , we show that there exists @xmath570 such that each discrete semiflow @xmath571 admits two nondecreasing traveling waves @xmath572 with @xmath573 and @xmath574 has the following properties : @xmath575 but @xmath576_{{\mathcal}{x}}\quad \text{and}\quad \psi_{+,s_k}(0)\not\in[[\beta-\delta e_\beta,\beta]_{{\mathcal}{x}}.\ ] ] indeed , since for each @xmath577 the map @xmath578 satisfies ( a1)-(a5 ) , from the first two steps of the proof for theorem [ thdiscont ] , we see that for the discrete semiflow @xmath46 , there exists two nondecreasing traveling waves @xmath579 with the following properties : 1 . @xmath580 connects @xmath123 to some @xmath581 and @xmath582 connects some @xmath583 to @xmath124 ; 2 . @xmath584 and @xmath585 are ordered and @xmath586 . by a similar argument as in ( * theorem 1.3.7 ) , it then follows that both @xmath587 have a subsequence @xmath588 which tends to an equilibrium of the semiflow as @xmath589 , say the limit @xmath590 and @xmath591 , respectively . since @xmath584 and @xmath585 are ordered , it follows from proposition [ bistabilitysemiflow ] that there are only three possibilities for the relation of @xmath590 and @xmath591 : @xmath592 if @xmath593 , then for sufficiently large @xmath283 we can define @xmath594_{{\mathcal}{x}}\},\quad b_{s_k}:=\inf\{x\in\mathbb{r}:\phi_{-,s_k}(x)\in[\beta-\delta e_\beta,\beta]_{{\mathcal}{x}}\}.\ ] ] and hence , @xmath595 and @xmath596 are the required traveling waves . if @xmath597 , then for sufficient large @xmath283 we can define @xmath598_{{\mathcal}{x}}\},\ , b_{s_k}:=\inf\{x\in{\mathbb { r}}:\phi_{+,s_k}(x)\in[\beta-\delta e_\beta,\beta]_{{\mathcal}{x}}\}.\ ] ] and hence , @xmath599 and @xmath600 are the required traveling waves . if @xmath601 , then by lemma [ islatedequilibria ] we have @xmath602_{\mathcal}{x}\cup [ \beta-\delta e_\beta,\beta]_{\mathcal}{x}\}$ ] . consequently , for sufficiently large @xmath283 we can define @xmath603_{{\mathcal}{x}}\},\ , b_{s_k}:=\inf\{x\in{\mathbb { r}}:\phi_{-,s_k}(x)\in[\beta-\delta e_\beta,\beta]_{{\mathcal}{x}}\}.\ ] ] and hence , @xmath604 and @xmath605 are the required traveling waves . secondly , we show that there exists a subindex , still denoted by @xmath606 , such that @xmath607 in @xmath112 and @xmath608 . indeed , for each @xmath609 , there exists an integer @xmath610 such that @xmath611 . then @xmath612=q_2\circ q_{m_ks_k-2}\circ t_{m_kc_{s_k}}[\psi_{-,s_k}]\in q_1\circ q_1[{\mathcal}{c}_\beta].\ ] ] clearly , the compactness of @xmath218 implies that the set @xmath613 $ ] is precompact in @xmath112 . thus , there exists subsequence , still denoted by @xmath606 , and nonincreasing functions @xmath614 with @xmath615 and @xmath616 such that @xmath617 and @xmath618 in @xmath112 . also we claim that @xmath619 all exist . indeed , from we see that there exists @xmath620 such that @xmath621\to \psi_-$ ] . note that @xmath622\}_{k\ge 1}$ ] also has a convergent subsequence with the limit @xmath623 . and hence , by the uniqueness of limit we have @xmath624=\psi_-$ ] . note that @xmath625(k)=q_1[\phi(\cdot+k)](0)$ ] and @xmath626\}_{k\ge 1}$ ] has a convergent subsequence . it then follows that @xmath627 exist because @xmath628 is nonincreasing . similarly , @xmath629 exist . also , we have @xmath630 but @xmath631_{{\mathcal}{x}}\quad \text{and}\quad \psi_{+}(0)\not\in[[\beta-\delta e_\beta,\beta]_{{\mathcal}{x}}.\ ] ] consequently , by the monotonicity of @xmath632 , we have @xmath633_{{\mathcal}{x}},\forall x>0\quad \text{and}\quad \psi_{+}(x)\not\in[[\beta-\delta e_\beta,\beta]_{{\mathcal}{x}},\forall x<0.\ ] ] since @xmath628 and @xmath634 are the limits of the sequence of monotone functions with different translations , respectively , we can employ the same arguments as in the second step of the proof of theorem [ thdiscont ] to show that @xmath635 and @xmath636 are ordered . to prove that @xmath637 have convergent subsequences , we only need to prove that @xmath638 is bounded above and @xmath639 is bounded below because @xmath573 . assume , for the sake of contradiction , that some subsequence , still say @xmath640 , tends to @xmath641 . note that for each @xmath577 there exists @xmath642 such that the integer part of @xmath643 , denoted by @xmath644 , equals @xmath645 and @xmath646 . hence , @xmath647 as @xmath49 . it then follows that @xmath648 thus , using the first observation in , we have @xmath649&&\ge q_1[\psi_-(-\infty)]=q_1[\psi_-(-\infty)](0)=\lim_{x\to -\infty}q_1[\psi_-(\cdot+x)](0)\nonumber\\ & & = \lim_{x\to -\infty}q_1[\psi_-](x)=\lim_{x\to -\infty}\lim_{k\to \infty } ( q_{s_k})^{\langle{\frac}{1}{s_k}\rangle } [ \psi_{-,s_k}](x)\nonumber\\ & & = \lim_{x\to -\infty}\lim_{k\to \infty } \psi_{-,s_k}(x+\langle { \frac}{1}{s_k}\rangle c_{s_k})\ge \lim_{x\to -\infty}\lim_{y\to + \infty}\lim_{k\to \infty } \psi_{-,s_k}(y)\nonumber\\ & & = \lim_{y\to + \infty}\psi_-(y)=\psi_-(+\infty)\not\in [ 0,\delta e_0]]_{{\mathcal}{x}},\end{aligned}\ ] ] which contradicts the fact that @xmath650\ll \delta e_0 $ ] . similarly , if @xmath651 , then the second observation in implies that @xmath652&&\le q_1[\psi_+(+\infty)]=q_1[\psi_+(+\infty)](0)=\lim_{x\to + \infty}q_1[\psi_+(\cdot+x)](0)\nonumber\\ & & = \lim_{x\to + \infty}q_1[\psi_+](x)=\lim_{x\to + \infty}\lim_{k\to \infty } ( q_{s_k})^{\langle{\frac}{1}{s_k}\rangle } [ \psi_{+,s_k}](x)\nonumber\\ & & = \lim_{x\to + \infty}\lim_{k\to \infty } \psi_{+,s_k}(x+\langle { \frac}{1}{s_k}\rangle c_{+,s_k})\le \lim_{x\to + \infty}\lim_{y\to -\infty}\lim_{k\to \infty } \psi_{+,s_k}(y)\nonumber\\ & & = \lim_{y\to -\infty}\psi_+(y)=\psi_+(-\infty)\not\in [ [ \beta-\delta e_\beta,\beta]_{{\mathcal}{x}},\end{aligned}\ ] ] which contradicts the fact that @xmath653\gg \beta-\delta e_\beta$ ] . consequently , @xmath637 are bounded . finally , we show that either @xmath654 or @xmath655 established in the second step is a traveling wave connecting @xmath123 to @xmath124 . indeed , for any @xmath516 , there exists @xmath656 and @xmath657 such that @xmath658 . clearly , @xmath659 as @xmath408 . then we have @xmath660=\lim_{k\to\infty}q_{t+r_k}[\psi_{\pm , s_k}]=\lim_{k\to\infty } q_{m_ks_k}[\psi_{\pm , s_k}]=\lim_{k\to\infty } \psi_{\pm , s_k}(\cdot+m_kc_{\pm , s_k})\\ & & = \lim_{k\to\infty } \psi_{\pm , s_k}\left(\cdot+(t+r_k){\frac}{1}{s_k}c_{\pm , s_k}\right)=\psi_{\pm}(\cdot+c_\pm t),\end{aligned}\ ] ] where the last equality follows from proposition [ sequenceinc](2 ) . from the equality @xmath661=\psi_{\pm}(\cdot+ct),\forall t\ge0 $ ] , we see that @xmath662 are equilibria . recall that @xmath663 and @xmath664 . it then follows that @xmath665 , and there are only three possibilities for @xmath635 and @xmath636 : 1 . @xmath666 ; 2 . @xmath667 ; 3 . @xmath668 for some @xmath669 . since the time - one map @xmath218 satisfies ( a6 ) with @xmath521 , we can employ the same arguments as in the proof of lemma [ miniwavespeeddsf ] to exclude the possibility ( iii ) . thus , either ( i ) or ( ii ) holds , and hence , we complete the proof . in this case , time @xmath670 and habitat @xmath42 . let @xmath671 be an equilibrium of the semiflow @xmath672 . we start with the definition of traveling waves for this case . [ defcontdis ] @xmath673 with @xmath674 is said to be a traveling wave with speed @xmath222 of the continuous - time semiflow @xmath566 if @xmath567(i)=\psi(i+ct),\forall i\in{\mathbb { z}},t\ge 0 $ ] . clearly , @xmath54 is continuous if @xmath675 . for each @xmath516 , define @xmath676 by @xmath677(x)=q_s[\phi(\cdot+x)](0)$ ] . then it is easy to see the following result holds . @xmath678 has the following properties : 1 . @xmath679=\phi,\forall \phi\in { \mathcal}{b}$ ] . 2 . @xmath680=\tilde{q}_{t+s}[\phi],\ , \forall t , s\ge 0,\ , \phi\in { \mathcal}{b}$ ] . 3 . for fixed @xmath273 , if @xmath681 and @xmath682 in @xmath5 for any @xmath683 , then @xmath684(x)\to \tilde{q}_{t}[\phi](x)$ ] in @xmath5 . we combine the ideas in the proofs of theorems [ thdisdis ] and [ thcontcont ] to prove the following result for continuous - time semiflows in a discrete habitat @xmath4 . [ thcontdis ] let @xmath685 . assume that for each @xmath516 , the map @xmath518 satisfies ( a1 ) , ( a3)-(a5 ) with @xmath522 , and that the time - one map @xmath218 satisfies ( a6 ) with @xmath521 . then there exists @xmath222 such that @xmath47 admits a non - decreasing traveling wave with speed @xmath53 and connecting @xmath123 to @xmath124 . let @xmath686 be chosen as in , and . we proceeds with three steps . firstly , since for any @xmath577 the map @xmath578 satisfies assumptions ( a1)-(a5 ) , it then follows from the proof of theorems [ thdisdis ] and [ thcontcont ] that there exists @xmath570 such that @xmath687 admits two nondecreasing traveling waves @xmath688 with @xmath573 , that is , there exists countable subset @xmath689 such that @xmath690(x)=\tilde{\psi}_{\pm , s_k}(x+c_{\pm , s_k}),\quad \forall x\in { \mathbb { r}}\setminus \theta_k.\ ] ] furthermore , @xmath691 is left continuous and @xmath692 is right continuous with the following properties : @xmath693 but @xmath694_{{\mathcal}{x}}\quad \text{and}\quad \tilde{\psi}_{+,s_k}(0)\not\in[[\beta-\delta e_\beta,\beta]_{{\mathcal}{x}}.\ ] ] secondly , we show that for the above sequence @xmath606 , there exist a countable set @xmath695 and a subsequence , still denoted by @xmath606 , such that @xmath696 and that @xmath697 converges in @xmath5 for all @xmath698 . indeed , let @xmath699 . hence , @xmath700 is countable and @xmath690(x)=\tilde{\psi}_{\pm , s_k}(x+c_{\pm , s_k}),\quad \forall k\ge 1 , x\in { \mathbb { r}}\setminus \theta.\ ] ] from proposition [ setprop2 ] , we see that there exists another countably dense set @xmath701 such that @xmath702 . by the same arguments as in the proof of theorem [ thdisdis ] , we can show that @xmath703 and @xmath704 are well - defined and all @xmath705 exist . furthermore , @xmath706 and @xmath707 are ordered in @xmath5 and @xmath708 but @xmath709_{{\mathcal}{x}}\quad \text{and}\quad \tilde{\psi}_{+}(0)\not\in[[\beta-\delta e_\beta,\beta]_{{\mathcal}{x}}.\ ] ] further , @xmath710 exist for all @xmath711 . hence , it follows from theorem [ hellyth ] that there exists a countable subset @xmath712 of @xmath9 such that @xmath713 by similar arguments as in the second step of the proof of theorem [ thcontcont ] , we can show that @xmath637 are bounded . finally , we prove that either @xmath714 or @xmath715 is a nondecreasing traveling wave connecting @xmath123 to @xmath124 . indeed , from and proposition [ setprop1 ] , we see that there exists a countable subset @xmath364 of @xmath9 such that @xmath716 hence , for any @xmath717 and @xmath516 , we have @xmath718(x)&&=q_t[\tilde{\psi}_-(\cdot+x)](0)=\lim_{k\to \infty}q_{t+r_k}[\tilde{\psi}_{-,s_k}(\cdot+x)](0)\nonumber\\ & & = \lim_{k\to \infty}q_{m_k s_k}[\tilde{\psi}_{-,s_k}(\cdot+x)](0)=\lim_{k\to \infty}\tilde{q}_{m_ks_k}[\tilde{\psi}_{-,s_k}](x)\nonumber\\ & & = \lim_{k\to\infty } ( \tilde{q}_{s_k})^{m_k}[\tilde{\psi}_{-,s_k}](x)=\lim_{k\to \infty } \tilde{\psi}_{-,s_k}(x+m_kc_{-,s_k})\nonumber\\ & & = \lim_{k\to \infty } \tilde{\psi}_{-,s_k}(x+(t+r_k){\frac}{1}{s_k}c_{-,s_k}).\end{aligned}\ ] ] in the case where @xmath719 , we can choose @xmath720 such that @xmath721(i)=\lim_{k\to \infty } \tilde{\psi}_{-,s_k}(x+(t+r_k){\frac}{1}{s_k}c_{- , s_k})=\tilde{\psi}_-(x_0+i),\forall i\in\mathbb{z}.\ ] ] in the case where @xmath722 , we know that there exists a countable subset @xmath365 of @xmath9 such that @xmath723(x)=\lim_{k\to \infty } \tilde{\psi}_{-,s_k}(x+(t+r_k){\frac}{1}{s_k}c_{-,s_k})=\tilde{\psi}_-(x+c_-t),\quad \forall x\not\in \gamma_1 , x+c_-t\in\gamma_2.\ ] ] without loss of generality , we assume that @xmath724 . for any @xmath725 , we can choose @xmath726 and @xmath727 such that @xmath728 and @xmath729 is continuous at @xmath730 for all @xmath431 . now one can find @xmath731 and @xmath732 with @xmath733 such that @xmath734 and @xmath735 . note that @xmath736(0)=q_{t_0}[\tilde{\psi}_-(\cdot+x_0)](0)\ ] ] and @xmath737(0)=q_{t_0}[\tilde{\psi}_-(\cdot+x_0)](0).\ ] ] thus , @xmath729 is continuous in @xmath738 . and hence , again by proposition [ convergence ] and the equality , we have @xmath739(x)=\tilde{\psi}_-(x+c_-t)$ ] for all @xmath738 and @xmath740 . therefore , @xmath714 is a traveling wave connecting @xmath123 to some @xmath741 . similarly , we can construct the traveling wave @xmath715 connecting some @xmath742 . besides , @xmath590 and @xmath591 are ordered . now the rest of the proof is essentially the same as in the proof of theorem [ thcontcont ] . a typical example of evolution systems in a periodic habitat is @xmath743 where @xmath40 is a positive periodic function of @xmath738 . under the assumption that @xmath21 has exactly three ordered zeros @xmath744 and @xmath745 , xin @xcite employed perturbation methods to obtain the existence of spatially periodic traveling wave @xmath746 with @xmath747 and @xmath748 provided that @xmath40 is sufficiently closed to a positive constant in certain sense ( see also @xcite ) . for a general positive periodic function @xmath40 , the existence of such a traveling wave remains open . we will revisit this problem in subsection 6.3 . a map @xmath749 is said to be spatially periodic with a positive period @xmath750 if @xmath751 , where @xmath752 is the @xmath38-translation operator . similarly , a semiflow @xmath476 on @xmath378 is said to be spatially periodic with a positive period @xmath750 if @xmath753 for all @xmath740 . [ twpermedia ] 1 . an @xmath38-periodic function @xmath754 is said to be an @xmath38-periodic steady state of the map @xmath56 ( semiflow @xmath476 ) if @xmath755=\beta ( q_t[\beta]=\beta,\forall t\in { \mathcal}{t})$ ] . @xmath746 is said to be a spatially @xmath38-periodic traveling wave with speed @xmath53 of the semiflow @xmath476 if @xmath756(x)=v(x+ct , x)$ ] and @xmath757 is @xmath38-periodic in @xmath25 . besides , we say that @xmath758 connects @xmath123 to @xmath754 if @xmath759 and @xmath760 uniformly for @xmath761 . motivated by ( * ? * section 5 ) , we can regard a spatially periodic semiflow on @xmath478 as a spatially homogeneous semiflow on another phase space . for any positive @xmath762 , define @xmath763_{{\mathcal}{h}}:=\{l\in { \mathcal}{h}:0\le l\le h\}$ ] . we use @xmath764 to denote @xmath765_{{\mathcal}{h}},{\mathcal}{x})$ ] and @xmath766 to denote the set of all bounded functions from @xmath767 to @xmath764 . clearly , @xmath764 can be regarded as a subspace of @xmath766 . let @xmath768_{{\mathcal}{h}},{\mathcal}{x}^+)$ ] and @xmath769 be the set of all bounded functions from @xmath767 to @xmath770 . we equip @xmath764 with the norm @xmath771_{{\mathcal}{h}}\}$ ] and @xmath766 with the compact open topology . thus , @xmath764 is a banach lattice with the norm @xmath772 and the cone @xmath770 . let @xmath773 it is easy to see that @xmath774 for any @xmath499 , define @xmath775 by @xmath776_{{\mathcal}{h}}.\ ] ] then we have the following observation . [ setequivalent ] for any @xmath777 , there exists a unique @xmath778 such that @xmath779 . further , if @xmath780 , then @xmath781 is @xmath38-periodic . for any @xmath782 , we can find @xmath783 and @xmath784_{\mathcal}{h}$ ] such that @xmath785 . it is easy to see that such decomposition of @xmath25 is unique when @xmath786 and is in two possible ways when @xmath787 . more precisely , when @xmath787 , it can be decomposed into either @xmath788 or @xmath789 for some @xmath783 . note that @xmath790 . it then follows that @xmath791 is a well - defined function in @xmath57 . clearly , @xmath779 . if @xmath792 , then @xmath793 , which implies that @xmath781 is @xmath38-periodic . if we define @xmath794 by @xmath795 , then @xmath796 is a homeomorphism between @xmath57 and @xmath36 . let @xmath754 be a strongly positive @xmath38-periodic steady state of the semiflow @xmath47 . with a little abuse of notation , we use @xmath112 to denote the set @xmath797 . now we can define a semiflow @xmath798 on @xmath799 by @xmath800=f\circ q_t[\phi_f],\quad \forall f\in { \mathcal}{k}_{\tilde{\beta}},\ , t\in { \mathcal}{t}.\ ] ] clearly , @xmath801 , which implies that semiflows @xmath476 and @xmath798 are topologically conjugate . moreover , @xmath798 is spatially homogeneous and @xmath802 is its equilibrium . thus , we see that the semiflow @xmath476 on @xmath112 has a spatially @xmath38-periodic traveling wave if the semiflow @xmath803 on @xmath804 has a traveling wave . before stating the main result , we first introduce the bistability assumption . let @xmath805 be an @xmath38-periodic steady state of the semiflow @xmath476 . assume that @xmath123 is a trivial steady state . define @xmath806 as in definition [ defstabfix ] , we can define the strong stability of periodic steady states for a map @xmath56 in the space of periodic functions . [ defstabper ] a steady state @xmath807 is said to be strongly stable from below for the map @xmath808 if there exist a positive number @xmath809 and a strongly positive element @xmath810 such that @xmath811\gg \alpha - \eta e_\alpha^+,\,\forall \eta\in ( 0,\delta_\alpha^+].\ ] ] the strong instability from below is defined by reversing the inequality . similarly , we can define strong stability ( instability ) from above . we need the following bistability assumption on the spatially @xmath38-periodic map @xmath56 . 1 . ( it bistability ) @xmath123 and @xmath671 are two strongly stable @xmath38-periodic steady states from above and below , respectively , for @xmath808 , and the set of all intermediate @xmath38-periodic steady states are totally unordered in @xmath812 . we note that a sufficient condition for the non - ordering property of all intermediate @xmath38-periodic steady states is : @xmath813 is eventually strongly monotone and all intermediate fixed points are strongly unstable from both above and below . [ thmediaperi ] let @xmath411 . assume that for any @xmath516 , the map @xmath518 satisfies ( a2)-(a4 ) and the bistability assumption ( a5@xmath814 ) . further , assume that the map @xmath815 satisfies assumption ( a6 ) with @xmath57 and @xmath124 replaced by @xmath36 and @xmath802 , respectively . then the spatially @xmath38-periodic semiflow @xmath476 admits an @xmath38-periodic traveling wave @xmath816 . besides , @xmath817 is nondecreasing in @xmath818 and connecting @xmath123 to @xmath754 . let @xmath740 be fixed and @xmath819 be defined as in . then it is easy to see that @xmath819 satisfies ( a1)-(a5 ) with @xmath112 replaced by @xmath804 . from theorems [ thdisdis ] and [ thcontdis ] , we see that @xmath798 admits a traveling waves @xmath820 with @xmath821 connecting @xmath123 to @xmath802 . by the definitions of traveling waves in a discrete habitat ( see definitions [ defdisdis ] and [ defcontdis ] ) , we can find @xmath383 such that @xmath822 and @xmath823(ri)=u(ri+ct+x_0),\forall i\in \mathbb{z}$ ] . by lemma [ setequivalent ] , we can find @xmath824 such that @xmath825 and @xmath826 , and hence , @xmath827=\tilde{h_t}$ ] . by the topological conjugacy of @xmath518 and @xmath819 , we have @xmath567=h_t$ ] . note that @xmath828 . it then follows from lemma [ setequivalent ] that @xmath829 . if @xmath830 , then we obtain @xmath567=h_t\equiv h_0=\psi$ ] , which implies that @xmath54 is a traveling wave with speed zero . if @xmath675 , then we define @xmath831 . consequently , @xmath832(x)=q_t[h_0](x)=q_t[v(\cdot,\cdot)](x),\forall x\in{\mathcal}{h},t\ge 0.\ ] ] this completes the proof . to finish this section , we remark that the bistability structure can be obtained for equation under appropriate conditions so that the existence result in @xcite is improved ( see the details in subsection 6.3 ) . further , theorem [ thmediaperi ] with @xmath833 and @xmath12 can be used to rediscover the existence result in @xcite for one dimensional lattice equation under the bistability assumption . in assumption ( a4 ) of section 2 , we assume that @xmath834 is compact with respect to the compact open topology . in this section , we establish the existence of bistable waves under some weaker compactness assumptions . let @xmath27 be a fixed number . it is well known that the time-@xmath520 solution map of time - delayed reaction - diffusion equations such as @xmath835 is compact with respect to the compact open topology if and only if @xmath836 , where the phase space @xmath57 is chosen as @xmath837,{\mathbb { r}}))$ ] . the first purpose of this section is to show that our results are still valid for this kind of evolution equations by introducing an alternative assumption ( a4@xmath135 ) . in order to state this assumption , we need some notations for time - delayed evolution systems . let @xmath838 be a positive number , @xmath839 be a banach lattice with the positive cone @xmath840 having non - empty interior , @xmath841 , and @xmath842,{\mathcal}{f}_\beta)$ ] . for any @xmath623 , we can regard it as an element in @xmath843\times{\mathcal}{h},{\mathcal}{f}^+)$ ] . for any subset @xmath844 of @xmath845\times{\mathcal}{h}$ ] , we define @xmath846 as the restriction of @xmath72 on @xmath844 . 1 . ( _ compactness _ ) there exists @xmath847 $ ] such that 1 . @xmath127(\theta , x)=\phi(\theta+s , x)$ ] whenever @xmath848 . 2 . for any @xmath849 , the set @xmath402| _ { [ -s+\epsilon,0]\times { \mathcal}{h}}$ ] is precompact . 3 . for any subset @xmath850 with @xmath851 being precompact , the set @xmath852|_{[-s,0]\times { \mathcal}{h}}$ ] is precompact . this assumption was motivated by ( * ? ? ? * assumption ( a6@xmath135 ) ) . let us use equation to explain ( a4@xmath135 ) . for any @xmath836 , one can directly verify that the solution map @xmath518 satisfies ( a4 ) by rewriting as an integral form ( see , e.g. , @xcite ) ; and for any @xmath853 $ ] , one can show that @xmath518 satisfies ( a4@xmath135 ) ( i ) and ( ii ) by the same arguments . for ( a4@xmath135 ) ( iii ) , we provide a proof below . let @xmath854 , and for any @xmath855 $ ] , let @xmath856 be the time-@xmath520 map of the heat equation @xmath857 . then can be written as the following form : @xmath858 and hence , @xmath481(\theta , x)=u(t+\theta , x)$ ] . note that for any @xmath859 , @xmath860 with respect to the compact open topology as @xmath861 . it then follows from the triangular inequality and the absolute continuity of integrals that for any compact subset @xmath862 , the set @xmath863|_{[-t,0]\times { \mathcal}{h}_1}$ ] is equi - continuous , and hence , @xmath852|_{[-t,0]\times { \mathbb { r}}}$ ] is precompact in @xmath112 . [ delayeqcompactness ] let @xmath864 , be defined as in section 3 and @xmath865 . assume that @xmath126 satisfies ( a4@xmath135 ) . then there exists an integer @xmath866 such that @xmath867}(q\circ a_\xi)^{m_0}[{\mathcal}{c}_\beta]\subset{\mathcal}{c}_\beta$ ] is precompact when @xmath41 , and @xmath868}(\tilde{q}\circ a_\xi)^{m_0}[{\mathcal}{b}_\beta](x)\subset{\mathcal}{x}_\beta$ ] is precompact for any @xmath738 when @xmath42 . we only prove the case where @xmath41 since the proof for @xmath833 is essentially similar . let @xmath869 and @xmath870 be defined in ( a4@xmath135 ) . for such @xmath869 and @xmath870 , there exists @xmath871 such that @xmath872 $ ] . by assumption ( a4@xmath135)(i ) , we see that for any @xmath873 and @xmath874 , @xmath875(\theta , x)= \begin{cases } \phi_0(\theta+s,\xi x ) , & \theta+s\le 0\\ q[\phi_0(\xi\cdot)](\theta , x ) , & \theta+s>0 , \end{cases}\end{aligned}\ ] ] this implies that for any @xmath873 and @xmath876 , @xmath877}q\circ a_\xi[{\mathcal}{c}_\beta]| _ { [ -s+\epsilon,0]\times { \mathbb { r } } } \subset q[{\mathcal}{c}_\beta]| _ { [ -s+\epsilon,0]\times { \mathbb { r}}}.\ ] ] since @xmath402| _ { [ -s+\epsilon,0]\times { \mathbb { r}}}$ ] is precompact , as assumed in ( a4@xmath135(ii ) ) , it then follows that @xmath878}q\circ a_\xi[{\mathcal}{c}_\beta](0,\cdot)\subset c(\mathbb{r},{\mathcal}{y}_\beta)$ ] is precompact . by ( a4@xmath135)(iii ) and similar arguments as above , we have @xmath879(\theta , x)= \begin{cases } \phi_1^\xi(\theta+s,\xi x ) , & \theta+s\le 0\\ q[\phi_1^\xi(\xi\cdot)](\theta , x ) , & \theta+s>0 \end{cases } \nonumber\\ & = & \begin{cases } \phi_0(\theta+2s,\xi^2x ) , & \theta+2s\le 0\\ q[\phi_0(\xi\cdot)](\theta+s,\xi x ) , & 0 < \theta+2s\le s\\ q[\phi_1^\xi(\xi\cdot)](\theta , x ) , & \theta+s>0 , \end{cases}\end{aligned}\ ] ] this implies that @xmath878}(q\circ a_\xi)^2[{\mathcal}{c}_\beta]| _ { [ -2s+\epsilon,0]\times \mathbb{r}}$ ] is precompact . consequently , @xmath878}(q\circ a_\xi)^2[{\mathcal}{c}_\beta](0,\cdot)\subset c(\mathbb{r},{\mathcal}{y}_\beta)$ ] is compact . by induction , we have @xmath880(\theta , x)= \begin{cases } \phi_{m_0}^{\xi}(\theta+s,\xi x ) , & \theta+s\le 0\\ q[\phi_{m_0}^{\xi}(\xi\cdot)](\theta , x ) , & \theta+s>0 \end{cases } \nonumber\\ & & = \cdot\cdot\cdot\nonumber\\ & & = \begin{cases } q[\phi_0(\xi\cdot)](\theta+(m_0 + 1)s,\xi^{m_0 } x ) , & 0 < \theta+(m_0 + 1)s\le s\\ q[\phi_1^\xi(\xi\cdot)](\theta+m_0s,\xi^{m_0 - 1}x ) , & 0<\theta+m_0s\le s\\ \cdot\cdot\cdot\\ q[\phi_{m_0 - 1}^{\xi}(\xi\cdot)](\theta+s,\xi x),&0<\theta+s\le s\\ q[\phi_{m_0}^{\xi}](\theta , x),&\theta+s>0 . \end{cases}\end{aligned}\ ] ] this implies that @xmath878}(q\circ a_\xi)^{m_0}[{\mathcal}{c}_\beta]$ ] is precompact in @xmath112 . [ delay ] all results in theorems [ thdiscont]-[thcontdis ] and [ thmediaperi ] are valid if we replace ( a4 ) with ( a4@xmath135 ) . following the proof of theses theorems , we only need to modify the parts where we use the compactness assumption ( a4 ) . at these parts , by lemma [ delayeqcompactness ] we can easily complete the proof . note that the solution maps of the integro - differential equation @xmath881 satisfy neither ( a4 ) nor ( a4@xmath135 ) . the second purpose of this section is to modify our developed theory in such a way that it applies to these integro - differential systems . let @xmath8 denote the set of all nondecreasing functions from @xmath65 to @xmath5 and @xmath882 . we equip @xmath8 with the compact open topology . assume that @xmath56 maps @xmath883 to @xmath883 . let @xmath113 denote the set of fixed point of @xmath56 restricted on @xmath114 . suppose that @xmath123 and @xmath124 are in @xmath113 . we impose the following assumptions on @xmath56 : 1 . ( _ translation invariance _ ) @xmath125=q\circ t_y [ \phi ] , \forall \phi\in{\mathcal}{m}_\beta , y\in{\mathbb { r}}$ ] . ( _ continuity _ ) @xmath884 is continuous in the sense that if @xmath885 in @xmath883 , then @xmath886(x)\to q[\phi](x)$ ] in @xmath114 for almost all @xmath887 . ( _ monotonicity _ ) @xmath56 is order preserving in the sense that @xmath127\ge q[\psi]$ ] whenever @xmath90 in @xmath883 . ( _ weak compactness _ ) for any fixed @xmath738 , the set @xmath888(x)$ ] is precompact in @xmath114 . ( _ bistability _ ) fixed points @xmath123 and @xmath124 are strongly stable from above and below , respectively , for the map @xmath128 , and the set @xmath889 is totally unordered . 6 . ( _ counter - propagation _ ) for each @xmath134 , @xmath206 . comparing assumptions ( a1)-(a6 ) and ( b1)-(b6 ) , one can find that the assumptions of translation invariance , monotonicity , bistability and counter - propagation are the same . the difference lies in the assumptions of continuity and compactness . clearly , compactness assumption ( b4 ) is much weaker than ( a4 ) . [ disweakcomp ] let @xmath685 and assume that @xmath890 satisfies ( b1)-(b6 ) . then there exists @xmath498 and @xmath891 connecting @xmath123 to @xmath124 such that @xmath892(x)=\psi(x+c)$ ] for all @xmath738 . combining the proofs of theorems [ thdiscont ] and [ thdisdis ] , we can obtain the result . more precisely , one can repeat the proof of theorem [ thdiscont ] except for the parts where the compactness assumption ( a4 ) are used . for these parts , one use the idea in theorem [ thdisdis ] , where @xmath55 has the same compactness property as @xmath56 . in the rest of this section , we say @xmath47 is a semiflow on @xmath883 provided that @xmath893 ; @xmath894 ; and @xmath895(x)\to q_t[\phi](x)$ ] in @xmath114 for almost all @xmath738 whenever @xmath896 and @xmath77 in @xmath883 . [ contweakcomp ] let @xmath685 . assume that @xmath47 is a semiflow on @xmath883 , and for any @xmath516 , the map @xmath518 satisfies ( b1 ) and ( b3)-(b6 ) . then there exist @xmath498 and @xmath891 connecting @xmath123 to @xmath124 such that @xmath567(x)=\psi(x+ct)$ ] for all @xmath738 . as in the proof of theorem [ disweakcomp ] , we can prove the conclusion by combing the proofs of theorems [ thcontcont ] and [ thcontdis ] . similarly , we can define @xmath477-time periodic semiflows on @xmath883 and then obtain the following result . let @xmath685 . assume that @xmath47 is an @xmath477-time periodic semiflow on @xmath883 . let @xmath491 be a strongly positive periodic solution of @xmath47 restricted on @xmath114 . further , assume that the poincar map @xmath484 satisfies ( b1 ) and ( b3)-(b6 ) with @xmath493 . then there exist @xmath498 and @xmath897 with @xmath898 and @xmath899 such that @xmath567(x)=\psi(t , x+ct)$ ] for all @xmath738 . besides , @xmath900 and @xmath901 is @xmath477-periodic in @xmath740 . in this section , we apply the obtained abstract results to four kinds of monotone evolution systems : a time - periodic reaction - diffusion system , a parabolic system in a cylinder , a parabolic equation with variable diffusion , and a nonlocal and time - delayed reaction - diffusion equation . consider the time - periodic reaction - diffusion system @xmath902 where @xmath903 , @xmath904 with each @xmath905 and @xmath906 is @xmath477-periodic in @xmath740 ( i.e. , @xmath907 ) . the existence of periodic bistable traveling waves of with @xmath908 was proved in @xcite . here we generalize this result to the case @xmath909 . let @xmath910 . in order to apply theorem [ thtimeperi ] to system , we choose @xmath911 , and @xmath215 to be the set of all bounded functions from @xmath9 to @xmath912 . using the solution maps @xmath913 of the heat equation @xmath914 , we write as the following integral form : @xmath915 define @xmath481:=u(t;\phi),\forall \phi\in { \mathcal}{e}$ ] . let @xmath123 and @xmath387 be two fixed points of the poincar map @xmath484 in @xmath5 , and let @xmath113 be the set of all spatially homogeneous fixed points of @xmath484 in @xmath114 . we impose the following assumptions : 1 . the jacobian matrix @xmath916 is cooperative and irreducible for all @xmath740 and @xmath917 . the spatially homogeneous system @xmath918 is of bistable type , that is , @xmath123 and @xmath124 are two stable fixed points of @xmath484 in the sense that @xmath919)<0 $ ] and @xmath920)<0 $ ] , and any @xmath921 is a unstable in the sense that @xmath922)>0 $ ] , where @xmath923 is the stability modulus of the matrix @xmath63 defined by @xmath924 . assume that ( c1)-(c2 ) hold , and let @xmath925 be the periodic solution of @xmath918 with @xmath926 . then there exists @xmath222 such that admits a time - periodic traveling wave @xmath485 connecting @xmath123 to @xmath491 . it is easy to see that the discrete semiflow @xmath927 on @xmath112 satisfies ( a1)-(a5 ) with @xmath928 . next we show that ( a6 ) holds with @xmath928 . note that for any @xmath161 , @xmath929_{{\mathcal}{c}}\to [ \alpha,\beta]_{{\mathcal}{c}}$ ] performs a monostable dynamics , where @xmath152 is unstable and @xmath124 is stable . by the theory developed in @xcite , it follows that @xmath484 admits leftward and rightward spreading speeds @xmath189 and @xmath212 . since @xmath484 is reflectively invariant , we further have @xmath930 , which is called the spreading speed of this monostable subsystem . note that @xmath931_{{\mathcal}{c}}\to [ 0,\alpha]_{{\mathcal}{c}}$ ] also performs a monostable dynamics , where @xmath123 is stable and @xmath152 is unstable . similarly , this monostable subsystem also admits a spreading speed @xmath932 . let @xmath933 be the solution map of the linearized system of at the periodic solution @xmath934 $ ] : @xmath935 by a similar argument as in the proof of ( * ? ? ? * lemma 4.1 ) , we see that for each @xmath516 , there exists a strongly positive vector @xmath936 such that @xmath937\ge m_t[u]\quad \text{whenever}\quad u\in [ \alpha,\alpha+\eta]_{{\mathcal}{c}}\ ] ] and @xmath938\le m_t[u]\quad \text{whenever}\quad u\in [ \alpha-\eta,\alpha]_{{\mathcal}{c}}.\ ] ] let @xmath939 be the principle floquet multiplier of the following linear periodic cooperative and irreducible system @xmath940v.\ ] ] let @xmath941 be the solution of satisfying @xmath942 . it is easy to see that @xmath943 is the solution of linear periodic system . define @xmath944 . from ( * theorem 3.10 ) and inequalities - , we then have @xmath945 now we prove that @xmath946 . let @xmath947 . by the floquet theory , it then follows that there exists a positive @xmath477-periodic function @xmath948 such that @xmath949 is a solution of . in particular , we have @xmath950 dividing @xmath951 in both sides and integrating the above equality from @xmath123 to @xmath477 gives @xmath952 since the matrix @xmath953 is cooperative and @xmath954 is positive , we obtain @xmath955 this implies that @xmath956 and hence , @xmath946 . by ( * lemma 3.8 ) , we then have @xmath957 . thus , the assumption ( a6 ) with @xmath928 holds . consequently , theorem [ thtimeperi ] completes the proof . in this subsection , we consider the following system @xmath958 where @xmath959 are positively definite diagonal @xmath960 matrix , @xmath113 is diagonal matrix of smooth functions of @xmath961 , @xmath22 is a bounded and convex open subset in @xmath962 with smooth boundary @xmath963 , @xmath964 , and @xmath965 is the outer unit normal vector to @xmath966 . the existence of bistable traveling waves for with @xmath908 was obtained in @xcite . here we extend this result to the case @xmath967 . assume that @xmath968 satisfies the following two conditions : 1 . the jacobian matrix @xmath969 is cooperative and irreducible for all @xmath917 . @xmath21 is of bistable type in the sense that it has exactly three ordered zeros : @xmath970 and @xmath971 . assume that ( d1)-(d2 ) hold . then there exists @xmath222 such that system admits a traveling wave connecting @xmath123 to @xmath124 with speed @xmath53 . in order to employ theorem [ thcontcont ] , we choose @xmath972 and @xmath973 with the standard cones @xmath59 and @xmath974 , respectively . let @xmath975 be the green function of the linear equation @xmath976 then the solution of with initial value @xmath977 can be expressed as @xmath978 define @xmath979 . using the constant variation formula , we write subject to @xmath980 as an integral equation @xmath981(x , y)+\int_{0}^t t(t - s)f(u(s , x , y))ds.\ ] ] by the linear operators theory , we see that for any @xmath982 , system has a unique solution @xmath983 with @xmath984 , which exists globally on @xmath985 . define @xmath481:=u(t,\phi)$ ] . then @xmath672 is a subhomogeneous semiflow on @xmath112 ( see ( * ? ? ? * section 5.3 ) ) . also , assumption ( d1 ) assures that the semiflow @xmath47 restricted on @xmath114 is strongly monotone ( see @xcite ) . further , it is easy to see that @xmath986 , satisfies assumption ( a1)-(a4 ) . since the domain @xmath22 is convex , it follows from the result in @xcite that any non - constant steady state of the @xmath25-independent system @xmath987 is linearly unstable . this then implies that @xmath518 satisfies ( a5@xmath135 ) . now it remains to show that ( a6 ) holds for @xmath218 . for each @xmath25-independent steady state @xmath988 in @xmath989_{{\mathcal}{x}}$ ] , system performs a monostable dynamics on @xmath163_{{\mathcal}{c}}$ ] . to better understand the dynamics of this subsystem , we make a transform @xmath990 . then its dynamics is equivalent to that of the following system on @xmath991_{{\mathcal}{c}}$ ] : @xmath992 system has exactly two @xmath25-independent steady state @xmath993 and @xmath994 . by the theory developed in @xcite , it follows that has a leftward spreading speed @xmath995 in a strong sense . let @xmath189 be defined as in with @xmath996 . we then have @xmath997 . to verify ( a6 ) for @xmath218 , we first estimate the speed @xmath995 . consider the linearized system of at equilibrium @xmath998 : @xmath999 suppose @xmath1000 is a solution of , then @xmath1001 satisfies the @xmath1002-parameterized linear parabolic equation @xmath1003v , & y\in\omega , t>0 , \\ { \frac}{\partial v}{\partial \nu}=0 , & \text{on}\,(0,+\infty)\times \partial \omega . \end{cases}\ ] ] let @xmath1004 be the principle eigenvalue of the elliptic problem : @xmath1005v , & y\in\omega , \\ { \frac}{\partial v}{\partial \nu}=0 , & \text{on}\,\partial \omega . \end{cases}\ ] ] by the theory in ( * ? ? ? * section 3 ) , it follows that @xmath1006 , and @xmath1004 is convex . then it is easy to see from that @xmath1007 and @xmath1008 , and hence , @xmath1009 attains its infimum at some @xmath1010 . similarly , system performs a monostable dynamics on @xmath162_{{\mathcal}{c}}$ ] . to better understand the dynamics of this subsystem , we make a transform @xmath1011 . then its dynamics is equivalent to that of the following system on @xmath991_{{\mathcal}{c}}$ ] : @xmath1012 by the same arguments , such system have a rightward spreading speed @xmath1013 , and we have @xmath1014 . also , by the same procedure as above , we define @xmath1015 as the principle eigenvalue of the following elliptic problem : @xmath1016v , & y\in\omega , \\ { \frac}{\partial v}{\partial \nu}=0 , & \text{on}\,\partial \omega . \end{cases}\ ] ] it then follows that @xmath1017 , and @xmath1018 attains its infimum at some @xmath1019 . clearly , @xmath1020 . from assumption ( d2 ) , we see that @xmath1021 is a linearly unstable ( stable ) steady state of the @xmath25-independent system @xmath1022 more precisely , letting @xmath1023 be the principle eigenvalue of the following elliptic problem : @xmath1024 then @xmath1025 . obviously , equations and with @xmath1026 both become equation , and hence , @xmath1027 . with the information above , now we can show that @xmath218 satisfies ( a6 ) . let @xmath1028 . note that @xmath1029 . it then follows that @xmath1030\nonumber\\ & & \ge { \frac}{\mu_1+\mu_2}{\mu_1\mu_2 } \lambda^+(\theta \mu_1+(1-\theta)(-\mu_2))\nonumber\\ & & = { \frac}{\mu_1+\mu_2}{\mu_1\mu_2 } \lambda^+(0)={\frac}{\mu_1+\mu_2}{\mu_1\mu_2 } \lambda_0>0.\end{aligned}\ ] ] consequently , theorem [ thcontcont ] completes the proof . in this subsection , we study the existence of spatially periodic traveling waves of the parabolic equation @xmath1031 where @xmath1032 , and @xmath40 is a positive , @xmath1033-continuous , and @xmath38-periodic function on @xmath9 for some real number @xmath31 . for any @xmath1034)$ ] , equation admits a unique solution @xmath983 with @xmath984 . define @xmath1035)\to c({\mathbb { r}},[0,1])$ ] by @xmath481=u(t;\phi)$ ] . it then follows that @xmath47 is a continuous , compact and monotone semiflow on @xmath1036)$ ] equipped with the compact open topology . let @xmath1037)$ ] be the set of all continuous and @xmath38-periodic functions from @xmath9 to @xmath1038 $ ] . then the semiflow @xmath47 restricted on @xmath1037)$ ] is strongly monotone . choosing @xmath41 and @xmath12 in theorem [ thmediaperi ] , one can easily verify that @xmath47 satisfies assumptions ( a2)-(a4 ) . if admits the bistability structure , then proposition [ bistabilitysemiflow ] implies ( a5@xmath814 ) and a similar argument as in the previous section shows that ( a6 ) also holds . thus , we focus on finding sufficient conditions on @xmath40 under which admits the bistability structure . let @xmath1039 be an @xmath38-periodic steady state of . as in @xcite , we define @xmath1040 as the largest number such that there exists a function @xmath1041 which satisfies @xmath1042 we call @xmath1040 the principle eigenvalue of @xmath1039 , and @xmath72 the corresponding eigenfunction . we say @xmath1039 is linearly unstable if @xmath1043 , and linearly stable if @xmath1044 . define @xmath1045 with the @xmath1046-norm induced topology . we say @xmath1047 has the property ( p ) if every possible non - constant @xmath38-periodic steady state of with @xmath1048 is linearly unstable , that is , if the equation with @xmath1048 does not admit any non - constant @xmath38-periodic steady state @xmath1039 such that @xmath1049 . define @xmath1050 [ constantiny ] any positive constant function is in @xmath1051 . let @xmath1052 be given . if has no non - constant @xmath38-periodic steady state , we are done . let @xmath1039 be a non - constant @xmath38-periodic steady state of . we need to prove @xmath1053 . assume , for the sake of contradiction , that @xmath1054 . let @xmath72 be the positive eigenfunction associated with @xmath1055 . define @xmath1056 and @xmath1057 . it is easy to see that @xmath1058 for all @xmath25 and @xmath520 . let @xmath1059 and @xmath1060 . then we have @xmath1061 and @xmath1062 ^ 2+\phi^{-3}[2\bar{u}_x\bar{u}_{xxx } \phi^{2}-|\bar{u}_x|^2\phi\phi_{xx}].\ ] ] note that @xmath1063_x=\bar{d}\bar{u}_{xxx}+f'(\bar{u})\bar{u}_x.\ ] ] it then follows that @xmath1064\psi\right)\\ & & = -[\bar{d } \xi_{xx}+f'(\bar{u})\xi]+\lambda_1(\bar{u},\bar{d } ) \eta\\ & & = -2\bar{d}\phi^{-3}[\bar{u}_{xx}\phi-\bar{u}_x\phi_x]^2-\bar{d } \phi^{-3}[2\bar{u}_x\bar{u}_{xxx}\phi^{2}-|\bar{u}_x|^2\phi\phi_{xx } ] -f'(\bar{u})|\bar{u}_x|^2\phi^{-1}+\lambda_1(\bar{u},\bar{d})\eta\\ & & \le \bar{d}|\bar{u}_x|^2\phi^{-2}\phi_{xx}+f'(\bar{u})\phi^{-1}|\bar{u}_x|^2 -2f'(\bar{u})|\bar{u}_x|^2\phi^{-1}-2\bar{d}\phi^{-1}\bar{u}_x\bar{u}_{xxx}+\lambda_1(\bar{u},\bar{d})\eta\\ & & = \lambda_1(\bar{u},\bar{d})\xi+\lambda_1(\bar{u},\bar{d})\eta -2\bar{u}_x\phi^{-1}[f'(\bar{u})\bar{u}_x+\bar{d}\bar{u}_{xxx}]\\ & & = \lambda_1(\bar{u},\bar{d})[\xi+\eta].\end{aligned}\ ] ] hence , @xmath1065\psi\le 0 $ ] because @xmath1054 . since @xmath1039 is not a constant and @xmath1066 is @xmath38-periodic in @xmath738 , we can choose @xmath720 such that @xmath1067 , and hence , @xmath1068 . thus , @xmath1066 with @xmath1069 $ ] satisfies the following equation @xmath1070\psi\le 0,x\in(x_0,x_0+r),\\ \psi_x|_{x = x_0}=\psi_x|_{x = x_0+r}=0 , \end{cases}\ ] ] and @xmath1066 attains its maximum @xmath123 at @xmath1071 with @xmath1072 . by the strong maximum principle , we see that @xmath1073 , which implies that @xmath1074 is a constant . since @xmath1074 is @xmath38-periodic , it then follows that @xmath1075 , and hence , @xmath1039 is a constant , a contradiction . by the proof above , it follows that the conclusion of lemma [ constantiny ] is valid for any @xmath1076 . [ yopeninx ] @xmath1051 is open in @xmath1077 . clearly , lemma [ constantiny ] implies that @xmath1078 . let @xmath1079 be given . we need to show that @xmath1080 is an interior point of @xmath1051 . assume , for the sake of contradiction , that there is a sequence of points @xmath1081 such that @xmath1082 in @xmath1077 as @xmath557 . then with @xmath1083 admits a non - constant @xmath38-periodic steady state @xmath148 with the principle eigenvalue @xmath1084 . using the transformation @xmath1085 , we see that @xmath1086 is a periodic solution of the following ordinary differential system : @xmath1087 by elementary phase plane arguments , it then follows that @xmath1088 thus , the sequence of functions @xmath1089 is uniformly bounded and equicontinuous , and hence , @xmath1086 has a uniformly convergent subsequence , still denoted by @xmath1086 . let @xmath1090 be the limiting function of @xmath1086 . then @xmath1091 is an @xmath38-periodic steady state of with @xmath1092 . it is easy to see from that @xmath1091 is not the constant function @xmath123 or @xmath463 . let @xmath71 be the positive eigenfunction associated with @xmath1093 . then @xmath1094 dividing both sides of ( [ eigensequence ] ) by @xmath71 and integrating from @xmath123 to @xmath38 , we obtain @xmath1095 ^ 2}{\phi_n^2}dx+\int_0^r f'(u_n(x))dx=\lambda_1(u_n , d_n)r \le 0.\ ] ] since @xmath1076 and @xmath1096 , we see that @xmath1091 can not be the constant @xmath1097 . otherwise , the uniform convergence of @xmath148 to @xmath1097 implies that @xmath1098 for all @xmath1099 $ ] and sufficiently large @xmath180 , which contradicts ( [ negativeintegral ] ) . thus , @xmath1091 is a non - constant @xmath38-periodic function . since @xmath1079 , we have @xmath1100 . note that @xmath1101 in @xmath37 and @xmath1102 in @xmath1077 . by the variational characterization of the principal eigenvalue @xmath1093 ( see , e.g. , eq . ( 5.2 ) of @xcite ) , it then follows that @xmath1103 , a contradiction . the following counter - example shows that the parabolic equation ( [ application4 ] ) admits no bistability structure in the general case of periodic function @xmath40 . let either @xmath1104 , or @xmath1105 . then there exists a positive function @xmath1106 such that admits a pair of linearly stable , non - constant , and @xmath38-periodic steady states . we only consider the case where @xmath1104 since the other one can be obtained under appropriate scalings . our proof is based on the main result in ( * ? ? ? * theorem 3 ) . without loss of generality , we assume that @xmath1107 . in what follows , we use some notations of @xcite . let @xmath1108 be fixed and @xmath1109 be the step function on @xmath1110 $ ] defined by @xmath1111\cup(l,1],\\ 0 , & x\in ( -l , l ] . \end{cases}\ ] ] define @xmath1112\}\cup$ ] graph of @xmath1109 . by ( * ? ? ? * theorem 3 ) , it then follows that for any positive even function @xmath1113,{\mathbb { r}}^+)$ ] which is sufficiently closed to @xmath1109 ( in the sense that the distance between @xmath18 and the graph of @xmath53 is small enough ) , the following neumann boundary problem @xmath1114 admits an odd increasing steady state @xmath1115 which is linearly stable . that is , there exist @xmath1116 and @xmath1041 such that @xmath1117 in particular , we can choose @xmath53 such that @xmath1118 . since @xmath53 is even and @xmath21 is odd , we see that @xmath1119 is also a steady state , and @xmath1120 is the corresponding eigenvalue with the positive eigenfunction @xmath1121 . now we can construct a linearly stable @xmath1122-periodic steady state of . define two @xmath1122-periodic functions : @xmath1123\\ c(2-x ) , & x\in ( 1,3 ) \end{cases } \quad\text{and}\quad w_1(x)=\begin{cases } u_c(x),&x\in [ -1,1]\\ u_c(2-x ) , & x\in ( 1,3 ) . \end{cases}\ ] ] then @xmath1124 is a @xmath1122-periodic steady state of with @xmath1125 . let the positive @xmath1122-periodic function @xmath1126 be defined by @xmath1127\\ \phi(2-x ) , & x\in ( 1,3 ) . \end{cases}\ ] ] it follows that @xmath1120 and @xmath1128 solve the following eigenvalue problem @xmath1129 this implies that @xmath1130 is a linearly stable periodic steady state of with @xmath1125 . similarly , so is @xmath1131 . as a consequence of theorem [ thmediaperi ] , together with lemmas [ constantiny ] and [ yopeninx ] , we have the following result on the existence of bistable traveling waves for . [ thapplication4 ] let @xmath1132 be a given positive constant . then there exists @xmath1133 such that for any @xmath1106 with @xmath1134 , admits a spatially periodic traveling wave solution @xmath1135 with some speed @xmath498 and connecting @xmath123 to @xmath463 . besides , @xmath758 is nondecreasing in @xmath818 . we remark that theorem [ thapplication4 ] is a @xmath1046-perturbation result in @xmath1077 , and hence , it improves the existence result in ( * ? ? ? * theorem 3.1 ) , where the @xmath1136-perturbation is used for some @xmath1137 . let @xmath27 be a fixed real number . choose @xmath1138,\mathbb{r } ) , { \mathcal}{y}:=c(\mathbb{r},\mathbb{r})$ ] and @xmath1139,{\mathcal}{y})$ ] . we equip @xmath5 with the maximum norm , @xmath764 and @xmath57 with the similar norms as in . define @xmath1140 . let @xmath39 be the metric in @xmath1141 induced by the norm . we are interested in bistable traveling waves of the following nonlocal and time - delayed reaction - diffusion equation : @xmath1142 , \end{cases}\ ] ] where @xmath1143 is lipschitz continuous and for each @xmath740 , @xmath1144 is defined by @xmath1145,x\in\mathbb{r}.\ ] ] if the functional @xmath21 takes the form @xmath1146 , then becomes a local and time - delayed reaction - diffusion equation : @xmath1147 the bistable traveling waves of ( [ locale ] ) were studied in @xcite . if @xmath1148 , then becomes a nonlocal and time - delayed reaction - diffusion equation : @xmath1149 the existence , uniqueness and stability of bistable waves of ( [ nonlocale ] ) were established in @xcite . note that @xmath65 can be regarded as a subspace of @xmath5 , and the latter can also be regarded as a subspace of @xmath57 . define @xmath1150 by @xmath1151 and @xmath1152 by @xmath1153 . in order to obtain the existence of bistable waves for system , we impose the following assumptions on the functional @xmath21 : 1 . @xmath1154 are three equilibria and there are no other equilibria between @xmath123 and @xmath124 . the functional @xmath1155 is quasi - monotone in the sense that @xmath1156+h[f(\phi)-f(\psi)];{\mathcal}{y}_+)=0\quad \text{whenever}\ , \ , \phi\ge\psi \ , \ , \text{in } \ , \ , { \mathcal}{c}_\beta.\ ] ] 3 . equilibria @xmath123 and @xmath124 are stable , and @xmath152 is unstable in the sense that @xmath1157 and @xmath1158 . 4 . for each @xmath1159 , the derivative @xmath1160 of @xmath1161 can be represented as @xmath1162 where @xmath1163 is a positive borel measure on @xmath845 $ ] and @xmath1164)>0 $ ] for all small @xmath192 . 5 . for any small number @xmath192 , there exits a number @xmath1165 and a linear operator @xmath1166 such that @xmath1167 , as @xmath1168 and that @xmath1169 using the solution maps @xmath1170 generated by the heat equation @xmath1171 , we write system as the integral form @xmath1172 . \end{cases}\ ] ] note that traveling waves of system are those of system . it then remains to show admits a bistable traveling wave . under assumption ( e1)-(e5 ) , system admits a nondecreasing traveling wave @xmath1173 with @xmath1174 and @xmath1175 . from assumptions ( e1)-(e2 ) , we see that system generates a monotone semiflow @xmath47 on @xmath112 with @xmath216(\theta , x)=u_t(\theta , x;\phi ) , \quad \forall ( \theta , x)\in[-\tau,0]\times \mathbb{r},\ ] ] where @xmath1176 is the unique solution of system satisfying @xmath1177 . by similar arguments as in section 5 , it follows that @xmath518 satisfies ( a4 ) if @xmath836 and ( a4@xmath135 ) if @xmath855 $ ] . let @xmath1178 be the restriction of @xmath518 on @xmath114 . denote the derivative @xmath1179 $ ] of @xmath1178 by @xmath1180 , then @xmath1180 is the solution map of the following functional equation : @xmath1181 by assumptions ( e2 ) and ( e4 ) , it follows that system admits a principle eigenvalue @xmath1182 with an associated eigenfunctions @xmath1183 ( see ( * ? ? ? * theorem 5.5.1 ) ) . more precisely , @xmath1184=e^{s_0t}v_0 $ ] . furthermore , * corollary 5.5.2 ) implies that @xmath1185 since @xmath1186 . therefore , there exists @xmath1187 such that @xmath1188&&=\bar{q}_t[0]+{\text{d}}\bar{q}_t[0][\delta v_0]+o(\delta^2)\\ & & = \delta \bar{m}_{0,t}[v_0]+o(\delta^2)\\ & & = \delta e^{s_0t}v_0+o(\delta^2)\\ & & = \delta v_0+\delta [ e^{s_0t}-1]v_0+o(\delta^2)\ll \delta v_0,\quad \forall \delta\in(0,\delta_0(t)].\end{aligned}\ ] ] similarly , there exists @xmath1189 and @xmath1190 such that @xmath1191\gg \beta-\delta v_\beta , \quad \forall \delta\in(0,\delta_\beta(t)]\ ] ] and @xmath1192\gg \alpha+\delta v_\alpha,\quad \bar{q}_t[\alpha-\delta v_\alpha]\ll \alpha-\delta v_\alpha,\quad \forall \delta\in(0,\delta_\alpha(t)].\ ] ] till now , it remains to show ( a6 ) is also true . indeed , we see from ( * ? ? ? * theorem 2.17 ) that the solution semiflows @xmath566 restricted on @xmath162_{{\mathcal}{c}}$ ] and @xmath163_{{\mathcal}{c}}$ ] admit a spreading speed @xmath932 and @xmath1193 , respectively . let @xmath1194 be the solution maps of the linear system @xmath1195 . \end{cases}\ ] ] then assumption ( e5 ) guarantees that @xmath481\ge m^\epsilon_t[\phi]$ ] when @xmath1196 , where @xmath1197 is defined in ( e5 ) . we see from ( * ? ? ? * theorem 3.10 ) that @xmath1198 and @xmath1199 , where @xmath243 is positive number determined by the linearized system of at @xmath1200 , and hence , ( a6 ) holds . consequently , theorem [ delay ] completes the proof . [ stabilityremark ] at this moment we are unable to present a general result on the uniqueness and global attractivity of bistable waves under the current abstract setting . however , one may use the convergence theorem for monotone semiflows ( see ( * ? ? ? * theorem 2.2.4 ) ) and the similar arguments as in the proof of ( * ? ? ? * theorem 10.2.1 ) and ( * ? ? ? * theorem 3.1 ) to obtain the global attractivity ( and hence , uniqueness ) of bistable waves for four examples in this section . in this appendix , we present certain properties of banach lattices and countable subsets in @xmath9 , and some convergence results for sequences of monotone functions , including an abstract variant of helly s theorem . 1 . let @xmath72 be a monotone function in @xmath57 . if @xmath1208 nondecreasingly tends to @xmath1209 and @xmath1210 , then @xmath1211 . the similar result holds if @xmath1212 nonincreasingly tends to @xmath1213 . 2 . assume that @xmath1214 are continuous and @xmath1215 in @xmath57 . if @xmath1216 uniformly for @xmath25 in any bounded subset of @xmath2 , then @xmath1217 in @xmath57 . since @xmath364 is countable and @xmath1223 , there must exist a sequence @xmath1224 such that @xmath1225 . note that @xmath1226 is countable . then we see that there exists @xmath1227 such that @xmath1228 . this means that @xmath1229 , and hence , @xmath1230 . define @xmath1231 . we then see that @xmath365 is countable and dense in @xmath65 , and @xmath1232 . [ convergence ] assume that @xmath1233 are nondecreasing and the set @xmath18 is dense in @xmath65 . if @xmath1234 , @xmath1235 is continuous on @xmath18 and @xmath1236 for every @xmath1237 , then @xmath1238 for every @xmath1237 . let @xmath1237 be fixed . for any @xmath1239 , since @xmath1240 is dense in @xmath65 , we can choose @xmath1241 and @xmath1242 . clearly , @xmath1243 , @xmath1244 . thus , there exists an integer @xmath1245 such that @xmath1246 . since @xmath1247 we have @xmath1248 it then follows that @xmath1249 for all @xmath1250 . now the pointwise convergence of @xmath1251 in @xmath18 and the continuity of @xmath21 on @xmath18 complete the proof . to end this section , we prove a convergence theorem for sequences of monotone functions from @xmath9 to the special banach lattice @xmath1252 defined in section 2 , which is a variant of helly s theorem ( * ? ? ? * p.165 ) for sequences of monotone functions from @xmath9 to @xmath9 . 1 . for any @xmath1237 , @xmath1255 is convergent in @xmath5 . 2 . there exists a countable set @xmath1256 such that for any @xmath1257 , the limits @xmath1258 exist in @xmath5 , where @xmath1259 and @xmath1260 with @xmath1261 . due to assumption ( ii ) , we can define @xmath1263 by @xmath1264 we first show that the discontinuous points of @xmath21 are at most countable . define the sets @xmath1265 and @xmath1266 for any @xmath1267 , there exists @xmath1268 and @xmath1269 such that @xmath1270 . recall that @xmath63 is compact , so there is a countable dense subset @xmath1271 . it then follows that there must be @xmath1272 such that @xmath1273 . therefore , @xmath1274 since for each fixed @xmath1275 and @xmath25 , @xmath1276 is a nondecreasing function from @xmath1277 to @xmath65 , we know that @xmath1278 is at most countable , and hence so is the set @xmath844 . now we can prove the conclusion . assume that @xmath1279 is a continuous point of @xmath21 . for any @xmath117 , choose @xmath1280 and @xmath1281 . then we have @xmath1282 which , together with proposition [ basicprop](2 ) , implies that @xmath1283 in the other hand , by and the triangular inequality we have @xmath1284 now the pointwise convergence of @xmath1251 in @xmath18 and the continuity of @xmath21 at @xmath869 complete the proof . * acknowledgment . * j. fang s research is supported in part by the nsf of china ( grant 10771045 ) and the collaborative research groups program at hit . zhao s research is supported in part by the nserc of canada and the mitacs of canada . zhao , spatial dynamics of some evolution systems in biology , _ recent progress on reaction - diffusion systems and viscosity solutions _ , y. du , h. ishii and w .- y . lin , eds . , 332 - 363 , world scientific , 2009 .

this paper is devoted to the study of traveling waves for monotone evolution systems of bistable type . under an abstract setting , we establish the existence of bistable traveling waves for discrete and continuous - time monotone semiflows . this result is then extended to the cases of periodic habitat and weak compactness , respectively . we also apply the developed theory to four classes of evolution systems . * keywords : * monotone semiflows , traveling waves , bistable dynamics , periodic habitat . * ams msc 2010 * : 37c65 , 35c07 , 35k55 , 35b40 .

following the ideas , motivations , and objectives of @xcite , where we prove that the structure of lines with binary coplanarity relation is a sufficient system of primitive notions for grassmann spaces and polar grassmann spaces , we use similar methods to prove the same for spine spaces . as previously , the key role play cliques of coplanarity which are contained in strong subspaces . the structure of strong subspaces in spine spaces is much more complex but pretty well known if we take a look into @xcite , @xcite , @xcite , and @xcite . the major difference is that we have to deal with three types of lines here as strong subspaces can be projective , semi - affine or affine spaces . a point - line structure @xmath0 , where the elements of @xmath1 are called _ points _ , the elements of @xmath2 are called _ lines _ , and where @xmath3 , is said to be a _ partial linear space _ , or a _ point - line space _ , if two distinct lines share at most one point and every line is of size ( cardinality ) at least 2 ( cf . @xcite ) . a _ subspace _ of @xmath4 is any set @xmath5 with the property that every line which shares with @xmath6 two or more points is entirely contained in @xmath6 . we say that a subspace @xmath6 of @xmath4 is _ strong _ if any two points in @xmath6 are collinear . if @xmath1 is strong then @xmath4 is said to be a _ linear space_. a _ plane _ in @xmath4 is a strong subspace @xmath7 of @xmath8 with the property that the restriction of @xmath4 to @xmath7 is a projective or semi - affine plane . for lines @xmath9 we say that they are _ coplanar _ and write @xmath10 for a subspace @xmath6 of @xmath4 we write @xmath11 let us fix a subset @xmath12 . consider the structure @xmath13 where @xmath14 which is a fragment of @xmath4 . note that @xmath8 is a partial linear space . the incidence relation in @xmath8 is again @xmath15 , inherited from @xmath4 , but limited to the new point set and line set . following a standard convention we call the points and lines of @xmath8 _ proper _ , and those points and lines of @xmath8 that are not in @xmath16 , @xmath17 respectively are said to be _ improper_. the set @xmath18 will be called the _ horizon of @xmath8_. if @xmath19 is a proper or improper point in a subspace @xmath6 of @xmath8 , then we write @xmath20 if @xmath7 is a plane in @xmath8 , then @xmath21 is a @xmath22-clique . we call such cliques _ flats_. if @xmath6 is a strong subspace of @xmath8 and @xmath19 is a point in @xmath6 , then @xmath23 is a different kind of a @xmath22-clique which we call a _ semibundle_. let @xmath24 be a vector space of dimension @xmath25 with @xmath26 . the set of all subspaces of @xmath24 will be written as @xmath27 and the set of all @xmath28-dimensional subspaces ( or @xmath28-subspaces in short ) as @xmath29 . by a _ @xmath28-pencil _ we call the set of the form @xmath30 where @xmath31 , @xmath32 , and @xmath33 . the family of all such @xmath28-pencils will be denoted by @xmath34 . a _ grassmann space _ ( also known as a _ space of pencils _ or a _ projective grassmannian _ ) is a point - line space @xmath35 with @xmath28-subspaces of @xmath24 as points and @xmath28-pencils as lines ( see @xcite , @xcite for a more general definition , see also @xcite ) . for @xmath36 it is a partial linear space . for @xmath37 and @xmath38 it is a projective space . so we assume that @xmath39 it is known that there are two classes of maximal strong subspaces in @xmath8 : _ stars _ of the form @xmath40 where @xmath31 , and _ tops _ of the form @xmath41_k = \{u\in\sub_k(v)\colon u\subset b\},\ ] ] where @xmath32 . although non - maximal stars @xmath42_k$ ] and non - maximal tops @xmath43_k$ ] , for some @xmath44 , make sense but in this paper when we say ` a star ' or ` a top ' we mean a maximal strong subspace . it is trivial that every line , a @xmath28-pencil @xmath45 , of @xmath8 can be uniquely extended to the star @xmath46 and to the top @xmath47 . a _ spine space _ is a fragment of a grassmann space chosen so that it consists of subspaces of @xmath24 which meet a fixed subspace in a specified way . the concept of spine spaces was introduced in @xcite and developed in @xcite , @xcite , @xcite , @xcite . let @xmath48 be a fixed subspace of @xmath24 and let @xmath49 be an integer with @xmath50 from the points of the grassmann space @xmath51 we take those which as subspaces of @xmath24 meet @xmath48 in dimension @xmath49 , that is : @xmath52 as new lines we take those lines of @xmath51 which have at least two new points : @xmath53 the point - line structure : @xmath54 will be called a _ spine space_. this is a gamma space . specifically , depending on @xmath55 and @xmath56 it can be : a projective space , a slit space ( cf . @xcite , @xcite ) , an affine space or the _ space of linear complements _ ( cf . @xcite , @xcite ) . as @xmath8 is a fragment of the grassmann space @xmath51 we can distinguishe a set @xmath18 of improper points in @xmath8 , i.e. a horizon . a line @xmath57 of @xmath51 either , is entirely contained in @xmath18 , meets @xmath18 in all except one points , meets @xmath18 in precisely one point , or is disjoint to @xmath18 . only in the last two cases new lines arise : they will be called _ affine _ and _ projective _ , respectively . note that for affine lines we can define parallelism in a natural way : those lines are parallel which meet in @xmath18 . the class of affine lines is denoted by @xmath58 and projective lines fall into two disjoint classes @xmath59 and @xmath60 . for details see table [ tab : lines ] . .the classification of lines in a spine space ( k , m , v , w ) . [ cols="<,^,^",options="header " , ] a line of @xmath8 can be in at most two maximal strong subspaces of different type : a star and a top . [ fact : spineintersections ] a projective star and a projective top are either disjoint or they share a point . in remaining cases , a star and a top are either disjoint or they share a line . two stars ( or two tops ) are either disjoint or they share a point . [ lem : spineplanes ] three pairwise coplanar and concurrent , or parallel , lines not all on a plane span a star or a top . there are three planes and it suffices to note by [ fact : spineintersections ] that no two of them can be of distinct type , i.e. they all are of type star or top . consequently , all these lines lie in one maximal strong subspace . let @xmath8 be a spine space . the goal now is to show that the set of lines equipped with coplanarity relation @xmath22 is a sufficient system of primitive notions for @xmath8 . as it was mentioned before , we can speak about the horizon in case of spine spaces . from this point of view a vertex @xmath19 of a semibundle @xmath23 can be either proper or improper . thus the name _ proper _ and _ improper semibundle_. we omit the adjective when we mean a semibundle in general . let @xmath61 be a maximal @xmath22-clique which is not a flat . so , there are three pairwise distinct lines in @xmath61 not all on a plane . they all meet in a point @xmath19 , possibly improper . moreover , for any @xmath62 we have @xmath63 . now , let @xmath70 be the grassmann space embracing @xmath8 , and let @xmath71 be a maximal collinearity clique in @xmath70 containing @xmath65 . set @xmath72 . it is clear that @xmath73 . as @xmath70 is a gamma space @xmath71 is a maximal strong subspace of @xmath70 . therefore , @xmath6 is a maximal strong subspace of @xmath8 . note that @xmath74 . take a point @xmath75 distinct from @xmath19 . there is a , possibly affine , line @xmath76 contained in @xmath71 . since @xmath71 is , up to an isomorphism , a projective space , all the lines through @xmath19 in @xmath71 are pairwise coplanar , so @xmath77 . hence @xmath78 and consequently @xmath79 . we have actually shown that @xmath80 which means that @xmath61 is a semibundle . the reasoning in [ lem : max - copla - cliques ] is universal in that it remains valid for general grassmann spaces discussed in @xcite and for polar spine spaces discussed later in this paper . it is worth to point out here that lemma 1.3 in @xcite states the same as [ lem : max - copla - cliques ] for general grassmann spaces but its proof fails to be complete . as indispensable as the property of the family of maximal @xmath22-cliques provided by [ lem : max - copla - cliques ] is the characterization of this family in terms of lines and coplanarity , that is an elementary definition of maximal @xmath22-cliques within @xmath81 . we can assume that all stars as projective spaces have dimension @xmath82 and all tops have dimension @xmath83 . without loss of generality we can assume that @xmath84 . for lines @xmath85 we define @xmath86 note that for @xmath87 the relation @xmath88 is empty , while for @xmath89 it says that the lines @xmath90 form a @xmath22-clique , they are not in a pencil , and they could not be contained in the intersection of two @xmath22-cliques . in grassmann spaces and spine spaces , two distinct @xmath22-cliques share at most one line . this is because two strong subspaces share at most a line . it is different in polar grassmann spaces , where two star semibundles can have a lot more in common ( cf . @xcite ) . note that in grassmann spaces and spine spaces 3 lines are enough in . in case of polar grassmann spaces however @xmath82 lines are required . to explain this let us take @xmath91 lines @xmath92 , which are pairwise coplanar , but not all lie on a plane . assume that these lines are contained in a star @xmath1 of dimension @xmath82 and @xmath93 . then , in considered polar geometries there is another star @xmath94 which contains all the lines @xmath92 . we can find two lines @xmath95 such that @xmath96 , @xmath97 , @xmath98 , and @xmath99 but @xmath100 . it means that @xmath88 fails to be true for @xmath101 . in view of [ lem : spinecliques ] the set @xmath104 consists of two families : semibundles and flats . by [ fact : spineintersections ] the intersection of a flat with a semibundle in a spine space is the empty set , a line , or a pencil of lines . two distinct maximal @xmath22-cliques of the same type are disjoint or share a line . hence we get the following loosely speaking , a plane is affine if it contains two disjoint pencils . a pencil @xmath115 lies on an affine plane iff there are two distinct pencils @xmath107 such that @xmath116 , @xmath117 and @xmath114 . we say that a line @xmath118 lies on an affine plane iff there is a pencil @xmath115 such that @xmath119 and @xmath120 lies on an affine plane . if a pencil @xmath115 does not lie on an affine plane and every @xmath119 lies on an affine plane , then we call @xmath120 a _ punctured _ pencil . a pencil is punctured iff its vertex is an improper point and it lies on a punctured plane . a maximal @xmath22-clique is _ proper _ if it contains no improper or punctured pencil . let us recall that our goal is to reconstruct the point universe of a spine space given a line universe equipped with coplanarity relation . the idea is to use vertices of semibudles to do that . this means that only proper @xmath22-cliques are of our concern . note that a proper maximal @xmath22-clique together with pencils of lines it contains carries the structure of a projective space . the geometrical dimension of a proper flat is always 2 whereas a proper semibundle @xmath23 has dimension one less then the dimension of @xmath6 . this lets us distinguish proper flats from proper semibundles if we assume that @xmath121 that is , in view of and table [ tab : subs ] , respectively @xmath122 [ lem : semibundle - flaty ] the family @xmath126 defined in @xmath81 coincides with the family of all proper top semibundles , the family of all proper star semibundles or the union of these two families depending on whether tops , stars or all of them are at least 4 dimensional projective or semi - affine spaces . by there are lines @xmath134 and @xmath135 , which are coplanar . note that @xmath136 and @xmath137 as we assume that @xmath131 . let @xmath138 be a plane containing @xmath139 . then @xmath140 and @xmath141 . in view of [ fact : spineintersections ] it means that @xmath132 are both of the same type and @xmath138 is of different type . there is another pair of coplanar lines @xmath142 such that @xmath143 and @xmath144 , since @xmath130 . let @xmath145 be the plane spanned by @xmath142 . note that @xmath146 are planes of the same type . as @xmath147 , by [ fact : spineintersections ] we get that @xmath148 or @xmath149 . if @xmath148 , then @xmath150 which yields a contradiction as @xmath151 are of the same type . the inverse of [ lem : zlepsemibundle1 ] is not true in general . let @xmath152 be a semi - affine , but not affine , star . then @xmath152 is an @xmath153-star ( cf . @xcite , @xcite , @xcite ) . take a projective line @xmath154 . the line @xmath57 is an @xmath153-line and the unique top - extension of @xmath57 is an @xmath153-top @xmath155 , that is a projective space . in case @xmath156 is an @xmath157-star , that is also a projective space , there is no line in @xmath158 , provided by [ fact : spineintersections ] . therefore we can not find a line in @xmath156 , which is coplanar with @xmath57 . however , two affine lines in @xmath159 are enough to fix this problem . so , it means that @xmath152 can not be a punctured projective space as in such there is only one affine line through a given proper point . in view of table [ tab : subs ] punctured projective spaces arise in a spine space as stars when @xmath160 or as tops when @xmath161 . note that either , all or none of the stars , and respectively , all or none of the tops , are punctured projective spaces . for this reason we assume that @xmath162 which in view of table [ tab : subs ] reads as follows @xmath163 by [ fact : spineintersections ] we get @xmath131 . without loss of generality assume that @xmath151 are stars . if @xmath152 is a projective space ( i.e. it is an @xmath157-star ) , then we take two distinct projective @xmath157-lines @xmath164 . each of @xmath165 can be extended to the semi - affine @xmath157-top @xmath166 , respectively . we have @xmath167 for @xmath129 , so @xmath156 and @xmath168 share a line @xmath169 . moreover @xmath170 , hence @xmath171 , and consequently @xmath130 . assume that @xmath152 is a semi - affine space ( i.e. it is an @xmath153-star ) . there are two distinct affine lines @xmath172 as @xmath152 is not a punctured projective space . as in the first case we extend @xmath165 to the semi - affine tops and we get our claim . for a proper semibundle @xmath177 we write @xmath178 we will show that it is the bundle of all lines through the point determined by the semibundle @xmath61 . thanks to and all stars or all tops , no matter if they are @xmath153 or @xmath157 , are at least 4 dimensional and are not punctured . this is essential here . the left - to - right inclusion is immediate by [ lem : zlepsemibundle ] . to show the right - to - left inclusion let @xmath57 be a line through @xmath19 . by there is a maximal strong subspace @xmath155 of the same type as @xmath6 which contains @xmath57 and by it is not a punctured projective space . then @xmath181 . again by [ lem : zlepsemibundle ] we get @xmath182 which makes the proof complete . in fact [ lem : allbundles ] says that @xmath183 is the bundle of all lines through @xmath19 . we can partition the line set of @xmath8 by @xmath175 , so that the equivalence classes will be the points of @xmath8 . note that points @xmath184 are collinear iff @xmath185 . this suffices to state the following theorem . [ thm : main ] let @xmath186 be a spine space satisfying and , and let @xmath2 be its line set . then @xmath186 and the structure of its lines together with coplanarity relation @xmath81 are definitionally equivalent .

a spine space can be considered as fragment of a projective grassmann space . we prove that the structure of lines together with binary coplanarity relation is a sufficient system of primitive notions for such a geometry . mathematics subject classification ( 2010 ) : 51a45 , 51a15 . + keywords : grassmann space , projective space , spine space , coplanarity

extending the variational methods and the geometric measure theory from the euclidean to the wiener space has recently attracted a lot of attention . in particular , the theory of functions of bounded variation in infinite dimensional spaces started with the works by fukushima and hino @xcite . since then , the fine properties of @xmath2 functions and sets of finite perimeter have been investigated in @xcite . we point out that this theory is closely related to older works by m. ledoux and p. malliavin @xcite . in the euclidean setting it is well - known that the perimeter can be approximated by means of more regular functionals of the form @xmath3 when @xmath4 tends to zero , in the sense of @xmath1-convergence with respect to the strong @xmath5-topology @xcite . an important ingradient in this proof is the compact embedding of @xmath2 in @xmath5 . a natural question is whether a similar approximation property holds in the infinite dimensional case . the main goal of this paper is answering to this question by computing the @xmath6-limit , as @xmath7 , of the allen - cahn - type functionals ( see section [ notation ] for precise definitions ) @xmath8 in the wiener space there are two possible definitions of gradient , and consequently two different notions of sobolev spaces , functions of bounded variation and perimeters @xcite . in one definition the compact embedding of @xmath9 in @xmath10 still holds ( * ? ? ? 5.3 ) and the @xmath6-limit of @xmath11 is , as expected , the perimeter up to a multiplicative constant . we do not reproduce here the proof of this fact , since it is very similar to the euclidean one . a more interesting situation arises when we consider the other definition of gradient , which gives rise to a more invariant notion of perimeter and is therefore commonly used in the literature @xcite . in this case , the compact embedding of @xmath9 in @xmath10 does not hold anymore . in particular sequences with uniformly bounded @xmath11-energy are not generally compact in the ( strong ) @xmath12-topology , even though they are bounded in @xmath13 , and hence compact with respect to the weak @xmath13-topology . this suggests that the right topology for considering the @xmath6-convergence should rather be the weak @xmath13-topology . a major difference with the finite dimensional case is the fact that the perimeter function defined by @xmath14 + \infty \qquad & \textrm{otherwise } \end{array}\right.\ ] ] is no longer lower semicontinuous in this topology , and therefore can not be the @xmath6-limit of the functionals @xmath11 . the problem is that the sets of finite perimeter are not closed under weak convergence of the characteristic functions . however , it is possible to compute the relaxation @xmath15 of @xmath16 ( theorem [ relax ] ) , which reads : @xmath17 + \infty \qquad & \textrm{otherwise . } \end{array}\right.\ ] ] such functional is quite familiar to people studying log sobolev and isoperimetric inequalities in wiener spaces @xcite . our main result is to show that the @xmath6-limit of @xmath11 , with respect to the weak @xmath13-topology , is a multiple of @xmath15 ( theorem [ modmortmal ] ) . the proof relies on the interplay between symmetrization , semicontinuity and isoperimetry . the plan of the paper is the following . in section [ notation ] we recall some basic facts about wiener spaces and functions of bounded variation . in section [ rapehr ] we give the main properties of the ehrhard symmetrizations . we also prove a plya - szeg inequality and a bernstein - type result in the wiener space ( propositions [ ehrfunc ] and [ probern ] ) , which we believe to be interesting in themselves . in section [ secrelax ] , we use the ehrhard symmetrization to compute the relaxation of the perimeter ( theorem [ relax ] ) . finally , in section [ mmm ] we compute the @xmath6-limit of the functionals @xmath11 ( theorem [ modmortmal ] ) and discuss some consequences of this result . * acknowledgements : * the authors wish to thank michele miranda for valuable discussions . the first author would like also to thank the scuola normale di pisa for the kind hospitality , and luigi ambrosio for the invitation and the interest in this work . a clear and comprehensive reference on the wiener space is the book by bogachev @xcite ( see also @xcite ) . we follow here closely the notation of @xcite . let @xmath18 be a separable banach space and let @xmath19 be its dual . we say that @xmath18 is a wiener space if it is endowed with a non - degenerate centered gaussian probability measure @xmath20 . that amounts to say that @xmath20 is a probability measure for which @xmath21 is a centered gaussian measure on @xmath22 for every @xmath23 . the non - degeneracy hypothesis means that @xmath20 is not concentrated on any proper subspace of @xmath18 . as a consequence of fernique s theorem ( 2.8.5 ) , for every @xmath23 , the function @xmath24 is in @xmath25 . let @xmath26 be the closure of @xmath27 in @xmath13 ; the space @xmath26 is usually called the reproducing kernel of @xmath20 . let @xmath28 , the operator from @xmath26 to @xmath18 , be the adjoint of @xmath29 that is , for @xmath30 , @xmath31 where the integral is to be intended in the bochner sense . it can be seen that @xmath28 is a compact and injective operator . we will let @xmath32 . we denote by @xmath33 the space @xmath34 . this space is called the cameron - martin space . it is a separable hilbert space with the scalar product given by @xmath35_h={\langle { \hat{h}}_1 , { \hat{h}}_2 \rangle}_{{l^2_{\gamma}(x)}}\ ] ] if @xmath36 . we will denote by @xmath37 the norm in @xmath33 . the space @xmath33 is a dense subspace of @xmath18 , with compact embedding , and @xmath38 if @xmath18 is of infinite dimension . for @xmath39 we denote by @xmath40 the projection from @xmath18 to @xmath41 given by @xmath42 we will also denote it by @xmath43 when specifying the points @xmath44 is unnecessary . two elements @xmath45 and @xmath46 of @xmath19 will be called orthonormal if the corresponding @xmath47 are orthonormal in @xmath33 . we will fix in the following an orthonormal base of @xmath33 given by @xmath48 . we also denote by @xmath49 and @xmath50 , so that @xmath51 . the map @xmath43 induces the decomposition @xmath52 , with @xmath53 gaussian measures on @xmath54 respectively . let @xmath55 be in @xmath26 then the image measure of @xmath20 under the map @xmath56 is a gaussian in @xmath41 . if the @xmath57 are orthonormal , then such measure is the standard gaussian measure on @xmath41 . given @xmath58 , we will consider the canonical cylindrical approximation @xmath59 given by @xmath60 notice that @xmath61 is a cylindrical functions depending only on the first @xmath62 variables , and @xmath61 converges to @xmath63 in @xmath13 . we will denote by @xmath64 the space of cylindrical @xmath65 bounded functions that is the functions of the form @xmath66 with @xmath67 a @xmath65 bounded function from @xmath41 to @xmath22 . we denote by @xmath68 the space generated by all functions of the form @xmath69 , with @xmath70 and @xmath71 . we now give the definitions of gradients , sobolev spaces functions of bounded variation . given @xmath72 and @xmath73 , we define @xmath74 whenever the limit exists , and @xmath75 we define @xmath76 , the gradient of @xmath63 by @xmath77 and the divergence of @xmath78 by @xmath79_h.\ ] ] the operator @xmath80 is the adjoint of the gradient so that for every @xmath81 and every @xmath82 , the following integration by parts holds : @xmath83_h d{\gamma}.\ ] ] the @xmath84 operator is thus closable in @xmath13 and we will denote by @xmath85 its closure in @xmath13 . from this , formula still holds for @xmath86 and @xmath82 . following @xcite , given @xmath87 we say that @xmath88 if @xmath89 we will also denote by @xmath90 the total variation of @xmath63 . if @xmath91 is the characteristic function of a set @xmath92 we will denote @xmath93 its total variation and say that @xmath92 is of finite perimeter if @xmath94 is finite . as shown in @xcite we have the following properties of @xmath9 functions . [ defbv ] let @xmath88 then the following properties hold : * @xmath95 is a countably additive measure on x with finite total variation and values in @xmath33 ( we will note the space of these measures by @xmath96 ) , such that for every @xmath97 we have : @xmath98 where @xmath99_h$ ] . * @xmath100 . let @xmath101 be a cylindrical function then @xmath88 if and only if @xmath102 . we then have @xmath103 [ procoarea ] if @xmath88 then for every borel set @xmath104 , @xmath105 in proposition [ ehrfunc ] , we will need the following extension of proposition [ procoarea ] . [ coairg ] for every function @xmath88 and every non - negative borel function @xmath106 , @xmath107 where @xmath108 . the proof of this lemma mimic the standard proof in the euclidean case ( * ? ? ? * th.2.2 ) . by ( * ch.1,th.7 ) we can write @xmath106 as @xmath109 where the @xmath110 are borel sets . using the coarea formula , we then get @xmath111 in @xcite it is also shown that sets with finite gaussian perimeter can be approximated by smooth cylindrical sets . [ denscyl ] let @xmath112 be a set of finite gaussian perimeter then there exists smooth sets @xmath113 such that @xmath114 converges in @xmath10 to @xmath92 and @xmath115 converges to @xmath94 when @xmath62 tends to infinity . note that , for half - spaces , the perimeter can be exactly computed ( * ? ? ? * cor . 3.11 ) . [ perhalf ] let @xmath73 and @xmath116 then the half - space @xmath117 has perimeter @xmath118 the ehrhard symmetrization has been introduced by ehrhard in @xcite for studying the isoperimetric inequality in a gaussian setting . we recall the definition and the main properties of such symmetrization . we define the functions @xmath119 and @xmath120 by @xmath121 we then let @xmath122 . notice that @xmath123 is the volume of the half - space @xmath124 and that @xmath125 is the perimeter of a half - space of volume @xmath126 . [ lemfin ] let @xmath127 , with @xmath128 , and suppose that there exist @xmath129 such that @xmath130 then @xmath131 . assume by contradiction @xmath132 , and let @xmath133 be such that @xmath134 . we shall bound from below by a positive constant the quantity @xmath135 thus contradicting the inclusion @xmath136 letting @xmath137 be a unitary vector in @xmath33 orthogonal to @xmath138 , we can write @xmath139 with @xmath140 . up to exchanging @xmath137 with @xmath141 , we can also assume that @xmath142 . we then have @xmath143 and thus @xmath144 . let us first suppose that @xmath145 , then @xmath146 as @xmath147 and @xmath148 are orthogonal we have @xmath149 and thus @xmath150 hence , for @xmath151 , @xmath152 if now @xmath153 , we can assume that @xmath154 is such that @xmath155 . let us start by computing the fourier transform of @xmath156 . denoting by @xmath157 the fourier transform of a measure @xmath158 ( see ( * ? ? ? * sec . 1.2 ) ) and letting @xmath159 , for every @xmath160 we have @xmath161}d{\gamma}(x)\\ & = \int_x e^{i [ ( z_1+z_2{\lambda } ) { \hat{h}}_1(x)+z_2\beta { \hat{h}}(x)]}d{\gamma}(x)\\ & = \int_{{\mathbb r}^2 } e^{i [ ( z_1+z_2{\lambda})x_1+z_2\beta x_2]}d{\gamma}_2(x_1,x_2)\\ & = \widetilde{{\gamma}_2}(z_1+{\lambda}z_2,\beta z_2)\\ & = e^{-\frac{1}{2 } [ ( z_1+{\lambda}z_2)^2+\beta^2 z_2 ^ 2]}\\ & = e^{-\frac{1}{2 } [ z_1 ^ 2+z_2 ^ 2 + 2{\lambda}z_1 z_2]}.\end{aligned}\ ] ] thus , if we set @xmath162 , we have @xmath163 . it follows that @xmath164 is a centered gaussian measure with density @xmath165 and thus @xmath166 } dz.\ ] ] we now compute @xmath167 } dz_1 dz_2\\ & \ge \frac{1}{2\pi}\sqrt{\frac{3}{4}}\int_{-\infty}^{c_1}\int_{c_2}^{+\infty}e^{-\frac{1}{2}z_1 ^ 2}e^{-\frac{1}{2}z_2 ^ 2 } e^{{\lambda}z_1 z_2 } dz_1 dz_2.\end{aligned}\ ] ] finally , when @xmath168 , we can bound @xmath169 from below by @xmath170 , and when @xmath171 we can bound it form below by @xmath172 so that we can always bound from below @xmath173 by a positive constant . we now define the ehrhard symmetrization . let @xmath112 and let @xmath174 . the ehrhard symmetral of @xmath92 along the first @xmath62 variables is defined as ( see figure [ ehrhardsym ] ) : @xmath175 the interest of this symmetrization is that it decreases the gaussian perimeter , while keeping the volume fixed . [ priso ] let @xmath92 be a set of finite perimeter and @xmath176 be an ehrhard symmetral of @xmath92 , then @xmath177 @xmath178 and @xmath179 in particular , we have the isoperimetric inequality @xmath180 with equality if and only if @xmath92 is a half - space . for the proof we refer to @xcite , and to @xcite for the extension to infinite dimensions . we can also prove a stronger result which is a kind of bernstein theorem in this setting . [ probern ] the half - spaces are the only local minimizers of the gaussian perimeter with volume constraint . let @xmath181 be a local minimizer of the ( gaussian ) perimeter and let @xmath182 . this means that , for every @xmath183 and every set @xmath16 of finite perimeter , with @xmath184 and @xmath185 ( where @xmath186 denotes the ball of radius @xmath28 centered at @xmath187 ) , we have @xmath188 if @xmath92 is not an half space then , by proposition [ priso ] , there exists @xmath133 such that @xmath189 let @xmath190 be such that @xmath191 we have that @xmath190 tends to @xmath192 when @xmath28 goes to infinity and @xmath193 tends to @xmath194 . letting @xmath195 we get @xmath196 where we used various time the inequality ( see @xcite ) @xmath197 and where @xmath198 is a function which goes to zero when @xmath28 goes to infinity . we thus found a contradiction . in the euclidean setting , half - spaces are the only local minimizers of the perimeter only in dimension lower than 8 ( see @xcite ) . notice also that if we drop the volume constraint , half spaces are no longer local minimizers for the gaussian perimeter , since there are no nonempty local minimizers . in the sequel we will also need another transformation which from a finite dimensional function gives an ehrhard symmetric set whose sections have volume prescribed by the original function . more precisely : given a measurable function @xmath199 $ ] , we define its ehrhard set @xmath200 by @xmath201 given a measurable cylindrical function @xmath202 $ ] depending only on the first @xmath62 variables , that is , @xmath203 for some @xmath199 $ ] , we set @xmath204 the link between ehrhard sets and ehrhard symmetrization is the following : let @xmath92 be a set of finite perimeter and @xmath176 be its ehrhard symmetrization with respect to the first @xmath205 variables , then @xmath206 in the next proposition we compute the perimeter of ehrhard sets . it slightly extends a result in @xcite . [ egalite ] let @xmath207 with @xmath208 , then @xmath209 where @xmath210 and @xmath211 is the radon - nikodym decomposition of @xmath95 . by ( * 4.3 ) the result holds for @xmath212 . we will show by approximation that the same holds for @xmath207 . let @xmath213 , then we can find sets @xmath214 such that @xmath215 and @xmath216 as @xmath217 , and all the @xmath214 have smooth boundary and are ehrhard symmetric . thus , for every @xmath218 , there exists a smooth function @xmath219 such that @xmath220 , @xmath221 , @xmath222 in @xmath223 , and @xmath224 since , by proposition [ dual ] , the functional @xmath225 is lower semicontinuous in @xmath223 , we get @xmath226 the other inequality follows as in @xcite . let @xmath227 and observe that @xmath228 and @xmath229 . by volpert theorem ( 3.108 ) there exists a set @xmath230 such that for every @xmath231 , @xmath232 exists and is not equal to zero , where @xmath233 denotes the last coordinate of the unit external normal to @xmath234 . by ( * lemma 4.4 ) , @xmath235-almost every @xmath231 is a point of approximate differentiability for @xmath63 . by lemma 4.5 and 4.6 of @xcite we then have @xmath236 as @xmath237 , we find that @xmath238 and thus @xmath239 . the last transformation that we consider is the analog of the schwarz symmetrization in the gaussian setting , and was first introduced by ehrhard in @xcite . let @xmath240 be a measurable function and let @xmath241 be fixed . we define the @xmath62-dimensional ehrhard symmetrization @xmath242 of @xmath63 as follows : * for all @xmath243 we let @xmath244 be the ehrhard symmetrization of @xmath108 with respect to the first @xmath62 variables ; * we let @xmath245 . as implies @xmath246 for all @xmath243 , from the layer cake formula it follows that , if @xmath247 , then @xmath248 and @xmath249 indeed , we have @xmath250 [ lemlam ] let @xmath251 belonging to @xmath252 , then @xmath253 the proof is a straightforward adaptation of the analogous proof for the schwarz symmetrization ( * ? ? ? recalling with @xmath254 , we have only to show that @xmath255 again by the layer cake formula we have @xmath256 thus would follow from the same inequality for sets , that is , @xmath257 let @xmath258 and assume that @xmath259 then by definition of the ehrhard symmetrization we have @xmath260 and therefore @xmath261 this inequality also holds if @xmath262 so that finally @xmath263 which gives . as for the schwarz symmetrization , a plya - szeg principle holds for the ehrhard symmetrization . [ ehrfunc ] let @xmath264 , let @xmath241 and let @xmath242 be the @xmath62-dimensional ehrhard symmetrization of @xmath63 . then @xmath265 and @xmath266 moreover , if @xmath267 and equality holds in , then @xmath268 and @xmath148 can be chosen to be a unitary vector . in ( * ? ? 3.1 ) , inequality is proven for lipschitz functions , in finite dimensions . we extend it by approximation to sobolev functions . we can assume @xmath269 , since we have @xmath270 , where @xmath271 denote the positive and negative part of @xmath63 and @xmath242 , respectively . let @xmath272 be positive functions converging to @xmath63 in @xmath85 , then by , @xmath273 converges to @xmath242 in @xmath13 and thus by the lower semicontinuity of the @xmath85 norm we have @xmath274 we now turn to the equality case for one - dimensional symmetrizations . for this we closely follow @xcite and give an alternative proof of , based on ideas of brothers and ziemer @xcite for the schwarz symmetrization . let @xmath275 and @xmath276 . by the coarea formula , for all @xmath243 we have @xmath277 hence @xmath278 since @xmath242 is a function depending only on one variable , arguing as in @xcite we get @xmath279 as @xmath242 is monotone we have that @xmath280 is constant on @xmath281 . observe also that , being @xmath242 one - dimensional , @xmath282 has a well defined meaning . we thus find : @xmath283 which implies , recalling , @xmath284 let us note that as in ( * ? ? ? 4.2 ) , using with @xmath285 we find @xmath286 and thus for almost every @xmath287 , @xmath288 this shows that for almost every @xmath289 , @xmath290 for @xmath291-almost every @xmath292 and thus @xmath293 by , , and , we eventually get @xmath294 \\ & \displaystyle = \int_{{\mathbb r } } \frac{p_{{\gamma}}(\{u^*>t\})^2 } { \left(\frac{p_{\gamma}(\{u^*>t\})}{|\nabla_h u^*|_{\{u^*=t\}}}\right)}dt&\\[18pt ] \\ & \displaystyle \le \int_{{\mathbb r } } \frac{p_{{\gamma}}(\{u > t\})^2}{\int_{x } \frac{1}{{|\nabla_h u|_h } } d|d_{\gamma}\chi_{e_t}|(x)}\ ; dt \\[14pt ] \\ & \displaystyle \le \int_{{\mathbb r } } \int_{x } { |\nabla_h u|_h } \ ; d|d_{\gamma}\chi_{e_t}|(x)\ ; dt \\[14pt ] & = \displaystyle \int_{x } { |\nabla_h u|_h}^2 d{\gamma}\ , . \end{array}\ ] ] as a consequence , if equality holds in , then equality holds in the gaussian isoperimetric inequality , that is , @xmath295 this implies that almost every level - set of @xmath63 is a half - space , i.e. for almost every @xmath243 there exists @xmath296 such that @xmath297 , and without loss of generality we can assume that @xmath298 . such half - spaces being nested , by lemma [ lemfin ] we have that @xmath299 does not depend on @xmath300 and thus @xmath301 . we notice that the fact that equality in implies that @xmath63 is one - dimensional is a specific feature of the gaussian setting , and the analogous statement does not hold for the schwarz symmetrization in the euclidean case @xcite . indeed , this property is a consequence of the fact that gaussian measures , differently from the lebesgue measure , are not invariant under translations . in this section we compute the relaxation of the perimeter functional @xmath302 + \infty \qquad & \textrm{otherwise } \end{array}\right.\ ] ] with respect to the weak @xmath13-topology . the fact that @xmath16 is not lower semicontinuous can be easily checked by taking the sequence @xmath303 . indeed , the characteristic functions of these sets weakly converge to the constant function @xmath304 , which is not a characteristic function , while the perimeter of @xmath214 is constantly equal to @xmath305 . we will show that the relaxation of @xmath16 is equal to @xmath306 + \infty \qquad & \textrm{otherwise } \end{array}\right.\ ] ] where @xmath307 with @xmath308 . observe that the functional @xmath15 already appears in the seminal work of bakry and ledoux @xcite and in the earlier work of bobkov @xcite in the context of log - sobolev inequalities . this functional has been also studied in @xcite . see also ( * ? ? ? * remark 4.3 ) where it appears in a setting closer to ours . let us first recall the definition of the lower semicontinous envelope of a function ( see @xcite for more details ) . let @xmath18 be a topological vector space . for every function @xmath309 , its lower semicontinuous envelope ( or relaxed function ) is the greatest lower semicontinuous function that lies below @xmath16 . when @xmath18 is a metric space , the following caracterization holds . let @xmath18 be a metric space . for every function @xmath310 , and every @xmath292 , the relaxed function @xmath15 is given by @xmath311 we now show a representation formula for @xmath15 which is reminiscent of the definition of the total variation and of the nonparametric area functional ( see @xcite ) . we start with a preliminary result . [ lemdual ] let @xmath312 with @xmath313 , let @xmath314 , and define @xmath315 where @xmath316 . there holds @xmath317_h + \int_x g\,\xi \ , d{\gamma } : \ ; { |\phi|_h}^2+|\xi|^2\le 1\ { \rm a.e.\ in\ } x\right\}.\ ] ] the proof is adapted from @xcite . notice first that , for @xmath318 , the function @xmath319 defines a norm on the product space @xmath320 . moreover , if we let @xmath321 , then the convex conjugate of @xmath322 is @xmath323 . we divide the proof into three steps . _ let @xmath324_h + \int_x g\sqrt{1-{|\phi|_h}^2 } \ , d{\gamma}:\;{|\phi|_h}\le 1 \ { \rm a.e.\ in\ } x\right\}.\ ] ] we will show that @xmath325 by definition of convex conjugate , it is readily checked that @xmath326 . we thus turn to the other inequality . by definition of the bochner integral , for every @xmath327 , there exists @xmath328 and @xmath329 with @xmath330 disjoints borel sets and @xmath331 $ ] such that if we set @xmath332 then @xmath333 . analogously there exists @xmath334 such that setting @xmath335 we have @xmath336 . by the observation at the beginning of the proof and the triangle inequality we get @xmath337 for every @xmath338 , by definition of convex conjugate , there exists @xmath339 with @xmath340 such that @xmath341_h + \eta_i \sqrt{1-{|\xi_i|_h}^2 } + \delta.\ ] ] from this , setting @xmath342 we have @xmath343_h + \eta_i \sqrt{1-{|\xi_i|_h}^2 } d{\gamma}+3\delta\\ & = \int_x [ \phi , h]_h + \tilde{g}\sqrt{1-{|\phi|_h}^2 } d{\gamma}+ 3 \delta . \end{aligned}\ ] ] since @xmath344 we find @xmath345 since @xmath346 is arbitrary we have @xmath347 . _ step 2 . _ the proof proceeds exactly as in @xcite and we only sketch it . recalling , it remains to show that @xmath348 one inequality is easily obtained , since @xmath349_h d{\gamma}+\int_x \phi\cdot d\mu^s+\int_x g(x)\sqrt{1-{|\phi|_h}^2 } d{\gamma}\\ & \le \left(\sup_\phi \int_x [ \phi , h]_h d{\gamma}+\int_x g(x)\sqrt{1-{|\phi|_h}^2 } d{\gamma}\right ) + \int_x |d\mu^s|\\ & = m(g , h { \gamma})+ |\mu^s|(x ) . \end{aligned}\ ] ] for the opposite inequality , let @xmath350 be fixed then there exists @xmath351 and @xmath352 such that @xmath353_h d{\gamma}+ \int_x g(x)\sqrt{1-{|\phi_1|_h}^2 } d{\gamma}+ \delta\\ taking @xmath119 equal to @xmath352 on a sufficiently small neighborhood of the support of @xmath354 and equal to @xmath351 outside this neighborhood , we get @xmath355_h d{\gamma}+ \int_x g(x)\sqrt{1-{|\phi|_h}^2 } d{\gamma}+ \int_x [ \phi , d\mu^s]_h + c\delta\\ & \le m(g , h { \gamma}+ \mu^s)+c\delta\end{aligned}\ ] ] which gives the opposite inequality . _ in order to conclude the proof , it is enough to notice that for every @xmath356 , with @xmath357 , we have @xmath358_h + \int_x g\,\xi \ , d{\gamma } : \ ; { |\phi|_h}^2+|\xi|^2\le 1\ { \rm a.e.\ in\ } x\right\ } \\ & & \quad = \int_x [ \phi , d\mu]_h + \int_x g\sqrt{1-{|\phi|_h}^2 } \ , d{\gamma}.\end{aligned}\ ] ] [ dual ] let @xmath359 then @xmath360 we apply lemma [ lemdual ] with @xmath361 and @xmath362 . since @xmath158 is tight @xcite , the space @xmath68 is dense in @xmath363 so that we can restrict the supremum in to smooth cylindrical functions @xmath364 . since @xmath365 is concave , the duality formula is not sufficient to prove that @xmath15 is lower semicontinuous for the weak @xmath13-topology . it shows however the lower - semicontinuity of @xmath15 in the strong @xmath13-topology . we now prove that @xmath15 is the lower semicontinuous envelope of @xmath16 . [ relax ] @xmath15 is the relaxation of @xmath16 in the weak @xmath13-topology . let us first notice that @xmath16 takes finite values only on functions of the closed unit ball of @xmath13 which is metrizable for the weak convergence . therefore the relaxation and the sequential relaxation in the weak topology of @xmath13 coincide . let @xmath366 be a sequence of sets weakly converging in @xmath13 to @xmath359 , with uniformly bounded perimeter . we shall show that @xmath367 notice that , by weak convergence , we necessarily have @xmath208 a.e . on @xmath18 . for all @xmath368 and @xmath369 , we let @xmath370 be the ehrhard symmetral of @xmath214 with respect to the first @xmath371 variables . recalling the notation of section [ rapehr ] , we have @xmath372 as @xmath373 and @xmath374 depends only on the first @xmath371 variables , by the compact embedding of @xmath375 into @xmath376 we can extract a subsequence from @xmath374 which converges strongly to @xmath377 . from this we get that @xmath378 tends strongly to @xmath379 . by the lower semicontinuity of the perimeter we then have @xmath380 for every @xmath381 , with @xmath382 depending only of the @xmath383 first variables , there holds @xmath384 which implies that the sequence @xmath385 tends weakly to @xmath63 . in order to conclude the proof it remains to show that @xmath386 notice that , by proposition [ egalite ] , there holds @xmath387 for every @xmath82 and @xmath388 , depending on the first @xmath371 variables and such that the range of @xmath119 is included in @xmath389 , by proposition [ dual ] , we have @xmath390 taking the supremum in @xmath364 and recalling , we then get @xmath391 repeating the same argument with @xmath392 instead of @xmath63 , we obtain that @xmath393 is nondecreasing in @xmath371 therefore there exists @xmath394 such that @xmath395 assume by contradiction that @xmath396 . then there exists @xmath350 such that @xmath397 for all @xmath371 , hence there exist @xmath398 , @xmath82 and @xmath388 , depending only on the first @xmath399 variables , such that @xmath400 but for @xmath401 we have @xmath402 which leads to a contradiction . theorem [ relax ] provides an example of a nonconvex functional , namely @xmath15 , which is lower semicontinuous for the weak @xmath13-topology . we also know that semicontinuity does not holds for general functional of the form @xmath403 since if we take for instance @xmath404 with @xmath106 such that @xmath405 and @xmath406 , then , letting @xmath407 , we have @xmath408 weakly in @xmath13 , so that @xmath409 one could wonder what are the right hypotheses for a functional of this form to be lower semicontinuous with respect to the weak topology . let us briefly recall the definition of @xmath6-convergence . we refer to @xcite for a comprehensive treatment of the subject . let @xmath18 be a topological space , and let @xmath410 be a sequence of functions . the @xmath6-lower limit and the @xmath6-upper limit of the sequence @xmath411 is defined as @xmath412 where @xmath413 denotes the set of all open neighbourhoods of @xmath126 in @xmath18 . when the @xmath6-lower limit and the @xmath6-upper limit coincide , we say that the sequence @xmath411 @xmath6-converges . let now @xmath418 be a double - well potential with minima in @xmath419 , that is , @xmath420 for all @xmath289 , and @xmath421 iff @xmath422 . we also assume @xmath423 for some @xmath424 and @xmath243 . a typical example of such potential is @xmath425 . * for all @xmath431 , letting @xmath432 , we have @xmath433 ; * @xmath434 for all @xmath431 , which implies that the @xmath6-limit is concentrated on the functions @xmath435 such that @xmath436 for a.e . @xmath292 . since the restricted domain is contained in the unit ball of @xmath13 , which is metrizable for the weak @xmath13-topology , by theorem [ thmetric ] the @xmath6-limit and the sequential @xmath6-limit of @xmath11 coincide . let @xmath437 be such that @xmath438 and @xmath439 for some @xmath424 . then @xmath440 , which gives a uniform bound on @xmath441 recalling that @xmath442 . as a consequence , there exists a weakly converging subsequence , still denoted by @xmath443 . letting @xmath63 be its weak limit , from @xmath438 we get @xmath208 . using the coarea formula , we obtain the estimate @xmath444 fix now @xmath350 . from the fact that @xmath445 as @xmath7 , it follows that , for every sequence @xmath446 $ ] , then functions @xmath447 tend weakly to @xmath63 in @xmath13 . for every @xmath426 let us choose @xmath446 $ ] such that @xmath448 then , by theorem [ relax ] we have @xmath449 since @xmath346 is arbitrary we get the @xmath6-liminf inequality . the @xmath6-limsup is done similarly to the ( euclidean ) finite dimensional case @xcite . since @xmath15 is the relaxation of @xmath16 in the weak @xmath13-topology and since we can approximate sets of finite perimeter by smooth cylindrical sets by proposition [ denscyl ] , for every @xmath88 with @xmath208 there exists a sequence @xmath214 of smooth cylindrical sets with @xmath366 converging weakly to @xmath63 and such that @xmath450 tends to @xmath451 . this shows that we can restrict ourselves to smooth cylindrical sets for computing the @xmath6-limsup of @xmath11 . let @xmath174 and @xmath452 , where @xmath453 is a smooth set with finite gaussian perimeter , and let @xmath454 where @xmath455 is the usual distance function from @xmath456 in @xmath41 . notice that @xmath457 moreover @xmath458 is differentiable almost everywhere with @xmath459 . let @xmath327 , @xmath460 \cup [ 1-\delta,1]\}$ ] and define @xmath461\rightarrow { \mathbb r}$ ] as @xmath462 finally let @xmath463 be the usual truncated one - dimensional transition profile defined as @xmath464 observe that @xmath465 is a lipschitz function which verifies @xmath466 . we then set @xmath467 we finally have @xmath468 + w_\delta\left(\eta_\delta\left(\frac{d}{{\varepsilon}}\right)\right ) \right ) \frac{|\nabla d|}{{\varepsilon}}\,d{\gamma}_m\\ & = \int_0^{h_{\delta}(1 ) } \left(\frac{{\eta_\delta'}^2(t)}{2}+ w_\delta(\eta_\delta(t))\right)p_{\gamma_m}(\{d>{\varepsilon}t\})\,dt . \end{aligned}\ ] ] the proof is completed since for every @xmath469 $ ] , @xmath470 tends to @xmath471 as @xmath7 , and @xmath472 thus we have @xmath473 which gives the desired inequality letting @xmath474 and @xmath475 . as in the euclidean case , a similar result can be proven for the volume constrained problems . in this case , the proof of the @xmath6-liminf is exactly the same as in theorem [ modmortmal ] , and the @xmath6-limsup is also very similar . the only difference comes from the fact that we have to adapt the recovery sequence to have the right volume , and this can be done as in @xcite by slightly translating @xmath463 . let @xmath476 $ ] and @xmath443 be a minimizer of @xmath477 then @xmath478 for some @xmath479 with @xmath480 and some @xmath481 minimizer of the one - dimensional problem @xmath482 in particular , @xmath481 ( strongly ) converges to the characteristic function of a half - line . for every @xmath86 , by proposition [ ehrfunc ] , we have @xmath483 and @xmath484 , with equality only if @xmath63 is of the form @xmath301 for some @xmath485 with @xmath486 . using that @xmath148 is the limit in @xmath13 of linear functions of the form @xmath487 , it is readily seen that @xmath488 , and thus we get @xmath489 therefore problem reduces to the one - dimensional problem . using the compact embedding of @xmath490 in @xmath491 ( see ( * ? ? ? * th . 4.10 ) ) and the direct method of the calculus of variations , we get that has a minimizer . moreover , by the @xmath6-convergence of the one - dimensional functionals in the strong @xmath491-topology towards the a multiple of the perimeter ( which can be obtained exactly as in the classical modica - mortola theorem since compact embedding of @xmath492 in @xmath493 holds ) , we find that every sequence of minimizers @xmath481 of has a subsequence strongly converging towards the characteristic of the half - line of measure @xmath62 . moreover , when this holds the two minimizers coincides . finally , if @xmath443 is a sequence in @xmath85 satisfying @xmath498 for some @xmath424 , then @xmath443 has a subsequence strongly converging to @xmath499 in @xmath13 , where @xmath92 is the common minimizer of and . we first notice that the problem always has a solution . indeed , arguing as in @xcite , if @xmath214 is a minimizing sequence for , it has a subsequence weakly converging to some @xmath88 . by the lower semicontinuity of the total variation and the coarea formula we then have @xmath500 and thus the sets @xmath501 minimize @xmath502 for almost every @xmath300 . as @xmath15 is the relaxation of the perimeter we have that the minimum values in and are the same and thus any minimizer of @xmath502 is also a minimizer of @xmath503 . this shows that if uniqueness does not hold in then it does not hold in , too . now , if @xmath63 is a minimizer of @xmath503 , applying the coarea formula once again we get @xmath504 as above , this implies that @xmath501 solves for almost every @xmath300 . therefore , if the minimizer of @xmath503 is not a characteristic function , then uniqueness does not hold neither in nor in . this proves the first part of the proposition . the second statement easily follows from theorem [ modmortmal ] . indeed , as the functionals @xmath505 @xmath6-converge to @xmath503 in the weak @xmath13-topology , for every sequence @xmath443 bounded in energy , there exists a subsequence weakly converging to @xmath499 ( where @xmath92 is the unique minimizer of and ) . however , by the lower semicontinuity of the norm , @xmath506 thus @xmath441 converges to @xmath507 , which implies the strong convergence of @xmath443 . @xmath2 functions in a hilbert space with respect to a gaussian measure _ , rend . lincei , to appear . , _ functions of bounded variation and free discontinuity problems _ , oxford science publications , 2000 . , _ surface measures and convergence of the ornstein - 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we characterize the relaxation of the perimeter in an infinite dimensional wiener space , with respect to the weak @xmath0-topology . we also show that the rescaled allen - cahn functionals approximate this relaxed functional in the sense of @xmath1-convergence .

the recently discovered iron - based superconductors@xcite rfe@xmath9o@xmath10f@xmath11 ( r = rare earth , @xmath9=as , p ) with a transition temperature @xmath12 exceeding 50k@xcite have attracted much attention . the f nondoped compound lafeaso exhibits the structural transition from tetragonal ( p4/nmm ) to orthorhombic ( cmma ) phase at a transition temperature @xmath13155k and stripe - type antiferromagnetic order at @xmath14 with a magnetic moment @xmath15@xcite at low temperature . with increasing f doping , the system becomes metallic and the antiferromagnetic order disappears@xcite , and then , the superconductivity emerges for @xmath16 with @xmath17 . rare - earth substitution compounds exhibit superconducting transition with higher @xmath12@xcite . specific features of the systems are two - dimensionality of the conducting fe@xmath18as@xmath18 plane and the orbital degrees of freedom in fe@xmath19 ( 3@xmath20)@xcite . the pairing symmetry together with the mechanism of the superconductivity is one of the most significant issues . the nmr knight shift measurements revealed that the superconductivity of the systems is the spin - singlet pairing@xcite . fully gapped superconducting states have been predicted by various experiments such as the penetration depth@xcite , the specific heat@xcite , the angle resolved photoemission spectroscopy ( arpes)@xcite and the impurity effect on @xmath12@xcite . in contrast to the above mentioned experiments , the nmr relaxation rate shows the power low behavior @xmath21 below @xmath22@xcite , suggesting the nodal or highly anisotropic gap structure . the other nmr measurements@xcite , however , revealed @xmath23 below @xmath12 and there is still controversy . theoretically , the first principle calculations have predicted that the nondoped system is metallic with two or three concentric hole fermi surfaces around the @xmath24 point ( @xmath25 ) and two elliptical electron fermi surfaces around the @xmath26 point ( @xmath27)@xcite . mazin _ et al . _ suggested that the spin - singlet extended @xmath4-wave pairing whose order parameter changes its sign between the hole pockets and the electron pockets ( @xmath5-wave ) is favored due to the antiferromagnetic spin fluctuations@xcite . according to the weak coupling approaches based on multi - orbital hubbard models@xcite , the @xmath5-wave pairing or the @xmath28-wave pairing is expected to emerge . it is shown that the @xmath5-wave pairing is realized also in the strong coupling region by the mean field study based on the @xmath29-@xmath30-@xmath31 model@xcite and the exact diagonalization study based on the one - dimensional two - band hubbard model@xcite . generally speaking , the details of the band structure and the fermi surface are crucial for determining the pairing symmetry . in the 5-band hubbard model originally introduced by kuroki _ _ , the energy bands obtained by reproduce those obtained by the density functional calculation very well@xcite . in this model , however , the spatial extensions of the fe @xmath32 like wannier orbitals are different from each other@xcite and the resulting intra - orbital terms of the on - site coulomb interaction are strongly orbital dependent@xcite . in addition , since the model explicitly includes the transfer integrals up to the fifth nearest neighbor sites@xcite , one should take the off - site coulomb interaction , which is considered to be about @xmath33 between the nearest neighbor sites , into account to ensure the consistency of the model@xcite . on the other hand , in the effective model which includes both the fe @xmath32 orbitals and the as @xmath34 orbitals , so called @xmath0-@xmath1 model , the spatial extensions and the differences of those between the orbitals will be considerably reduced@xcite . due to these facts , in the @xmath0-@xmath1 model , it is expected that the intra - orbital terms of the on - site coulomb interaction for each orbitals have almost the same values and the off - site coulomb interaction are negligible . therefore , theoretical studies based on the @xmath0-@xmath1 model , are highly desired . in the previous papers@xcite , we have investigated the electronic states of the fe@xmath35as@xmath35 plane in iron - based superconductors on the basis of the two - dimensional 16-band @xmath0-@xmath1 model which includes the coulomb interaction on a fe site : the intra- and inter - orbital direct terms @xmath2 and @xmath3 , the hund s coupling @xmath36 and the pair - transfer @xmath37 . using the random phase approximation ( rpa ) , we have found that , for a larger value of @xmath36 , the most favorable pairing symmetry is @xmath5-wave , while , for a smaller value of @xmath36 , it is @xmath28-wave . the present paper is a full paper to our previous papers@xcite with some numerical improvements@xcite . in the present paper , we investigate the superconductivity in the wider parameter space by treating @xmath2 , @xmath3 , @xmath36 and @xmath37 as independent parameters in contrast to the previous study under the condition that @xmath38 and @xmath39 based on the two - dimensional 16-band @xmath0-@xmath1 model . solving the superconducting gap equation with the pairing interaction obtained by using the rpa , we find that two kinds of the @xmath4-wave superconducting states appear . as above mentioned , the @xmath5-wave superconducting state emerges near the incommensurate spin density wave ( isdw ) with @xmath40 phase . in addition , for @xmath6 , the @xmath4-wave superconducting state appears around the ferro - orbital ordered phase . the order parameter for this @xmath4-wave state dose not change its sign in @xmath41 space . we refer to this @xmath4-wave state as the @xmath7-wave state , hereafter . first of all , we perform the density functional calculation for lafeaso with the generalized gradient approximation of perdew , burke and ernzerhof@xcite by using the wien2k package@xcite , where the lattice parameters ( @xmath42 , @xmath43 ) and the internal coordinates ( @xmath44 , @xmath45 ) are experimentally determined@xcite . the crystal structure of fe@xmath35as@xmath35 layer is shown in fig . [ crystal ] ( a ) . since as atoms are tetrahedrally arranged around a fe atom , there are two distinct fe and as sites in the crystallographic unit cell ( see figs . [ crystal ] ( a ) , ( b ) ) . considering these facts , we then derive the two - dimensional 16-band @xmath0-@xmath1 model@xcite , where @xmath32 orbitals ( @xmath46 , @xmath47 , @xmath28 , @xmath48 , @xmath49 ) of two fe atoms ( fe@xmath50=@xmath51 , fe@xmath52=@xmath53 ) and @xmath34 orbitals ( @xmath54 , @xmath55 , @xmath56 ) of two as atoms are explicitly included . we note that @xmath57 axes are directed along second nearest fe - fe bonds ( see fig . [ crystal ] ( b ) ) . as@xmath35 layer . small and large balls represent fe and as atoms , respectively . the solid line represents the unit cell . it is noted that as@xmath50 and as@xmath52 denote the as atoms on the upper side and on the lower side of the fe@xmath35as@xmath35 layer , respectively[crystal].,width=264 ] as@xmath35 layer . small and large balls represent fe and as atoms , respectively . the solid line represents the unit cell . it is noted that as@xmath50 and as@xmath52 denote the as atoms on the upper side and on the lower side of the fe@xmath35as@xmath35 layer , respectively[crystal].,width=264 ] the total hamiltonian of the @xmath0-@xmath1 model is given by @xmath58 where @xmath59 and @xmath60 are the noninteracting and interacting parts of the hamiltonian , respectively . the noninteracting part of the @xmath0-@xmath1 model is given by the following tight - binding hamiltonian , @xmath61 where @xmath62 is the annihilation operator for fe-@xmath32 electrons with spin @xmath63 in the orbital @xmath64 at the site @xmath65 and @xmath66 is the annihilation operator for as-@xmath34 electrons with spin @xmath63 in the orbital @xmath67 at the site @xmath65 . in eq . ( [ d - p ] ) , the transfer integrals @xmath68 , @xmath69 , @xmath70 and the atomic energies @xmath71 , @xmath72 are determined so as to fit both the energy and the weights of orbitals for each band obtained from the tight - binding approximation to those from the density functional calculation@xcite . similar models have been used by the other authors@xcite but the model parameters are different from ours . the doping concentration @xmath73 corresponds to the number of electrons per unit cell @xmath74 in the present model . now we consider the effect of the coulomb interaction on fe site . the interacting part of the hamiltonian is given as follows , @xmath75 where @xmath2 and @xmath3 are the intra- and inter - orbital direct terms , respectively , and @xmath36 and @xmath37 are the hund s coupling and the pair - transfer , respectively . for the isolated atoms , the relations between coulomb matrix elements @xmath38 and @xmath39 are derived due to the rotational invariance of the coulomb interaction and the reality of the wave functions , respectively@xcite . for the atoms in the crystal , however , the relation is not satisfied generally due to the crystallographic effects and the many body effects due to the coulomb interaction and the electron - phonon coupling which will be discussed later . therefore , we treat @xmath2 , @xmath3 , @xmath36 and @xmath37 as independent parameters in the present paper . within the rpa@xcite , the spin susceptibility @xmath76 and the charge - orbital susceptibility @xmath77 are given in the @xmath78 matrix representation as follows@xcite , @xmath79 with the noninteracting susceptibility @xmath80 where @xmath81 , @xmath82 ( = 1 - 16 ) are band indexes , @xmath83 , @xmath84 ( @xmath85@xmath86 ) represent two fe sites , @xmath64 represents fe 3@xmath0 orbitals , @xmath87 is the eigenvector which diagonalizes @xmath59 eq . ( [ d - p ] ) , @xmath88 is the corresponding eigenenergy of band @xmath81 with wave vector @xmath41 and @xmath89 is the fermi distribution function . in eqs . ( 4 ) and ( 5 ) , the interaction matrix @xmath90 ( @xmath91 ) is given by @xmath92 in the weak coupling regime , the superconducting gap equation is given by@xcite @xmath93 where @xmath94 is the gap function and @xmath95 is the effective pairing interaction @xcite . within the rpa@xcite , @xmath95 is given in the @xmath78 matrix , @xmath96 where @xmath97 for the spin - singlet state and @xmath98 for the spin - triplet state . the gap equation ( [ eq_gap ] ) is solved to obtain the gap function @xmath99 with the eigenvalue @xmath100 . at @xmath101 , the largest eigenvalue @xmath100 becomes unity . in the present paper , we only focus on the case with @xmath102 , where the superconductivity is observed in the compounds@xcite . for simplicity , we set @xmath102 and @xmath103 in the present study . we use @xmath104 @xmath41 points in the numerical calculations for eqs . ( [ eq_chis])-([eq_veff_s ] ) , and also use the fast fourier transformation ( fft ) to solve the gap equation eq . ( [ eq_gap ] ) . here and hereafter , we measure the energy in units of ev . .tight - binding parameters ( in units of ev ) for the @xmath0-@xmath1 hamiltonian eq . ( [ d - p ] ) . it is noted that we define the @xmath0-@xmath1 hopping and the in - plane @xmath1-@xmath1 hopping parameters along @xmath73-axis . [ table - d - p ] [ cols="^,^,^",options="header " , ] -@xmath1 model eq . ( [ d - p ] ) ( solid line ) and that obtained from the density functional calculation ( cross ) for @xmath102 . ( b ) fermi surface obtained from the @xmath0-@xmath1 model for @xmath102 . the solid and dashed lines show the fermi surfaces which have mainly @xmath105 and @xmath47 orbital character , respectively . ( c ) the dos obtained from the @xmath0-@xmath1 model for @xmath102 . upper panel : total dos , middle panel : partial dos of the fe @xmath32 orbitals , lower panel : partial dos of the as @xmath34 orbitals . the inset of the middle panel shows the dos near the fermi level . we note that the fermi level set to 0 on the energy axis . [ fs],width=151 ] -@xmath1 model eq . ( [ d - p ] ) ( solid line ) and that obtained from the density functional calculation ( cross ) for @xmath102 . ( b ) fermi surface obtained from the @xmath0-@xmath1 model for @xmath102 . the solid and dashed lines show the fermi surfaces which have mainly @xmath105 and @xmath47 orbital character , respectively . ( c ) the dos obtained from the @xmath0-@xmath1 model for @xmath102 . upper panel : total dos , middle panel : partial dos of the fe @xmath32 orbitals , lower panel : partial dos of the as @xmath34 orbitals . the inset of the middle panel shows the dos near the fermi level . we note that the fermi level set to 0 on the energy axis . [ fs],width=139 ] -@xmath1 model eq . ( [ d - p ] ) ( solid line ) and that obtained from the density functional calculation ( cross ) for @xmath102 . ( b ) fermi surface obtained from the @xmath0-@xmath1 model for @xmath102 . the solid and dashed lines show the fermi surfaces which have mainly @xmath105 and @xmath47 orbital character , respectively . ( c ) the dos obtained from the @xmath0-@xmath1 model for @xmath102 . upper panel : total dos , middle panel : partial dos of the fe @xmath32 orbitals , lower panel : partial dos of the as @xmath34 orbitals . the inset of the middle panel shows the dos near the fermi level . we note that the fermi level set to 0 on the energy axis . [ fs],width=207 ] we show the band structure obtained from the @xmath0-@xmath1 tight - binding hamiltonian eq . ( [ d - p ] ) , where the tight - binding parameters are listed in table [ table - d - p]@xcite , together with that obtained from the density functional calculation in the fig . [ fs ] ( a ) . the result of our density functional calculation is similar to that previously reported by the other authors@xcite . it is found that the former reproduces the latter very well . we note that the weights of orbitals also agree very well with each other ( not shown ) . due to the weak crystalline electric field from the as@xmath106 ions tetrahedrally arranged around a fe atom and the strong hybridization between the fe @xmath32 orbitals , the resulting energy bands have very complicated structure . the fermi surface for the @xmath0-@xmath1 tight - binding hamiltonian is shown in fig . [ fs ] ( b ) , where we can see nearly circular hole pockets around the @xmath24 point and elliptical electron pockets around the @xmath26 point . these results are consistent with the previous first principle calculations@xcite . the density of states ( dos ) obtained by the @xmath0-@xmath1 tight - binding hamiltonian eq . ( [ d - p ] ) is shown in fig . [ fs ] ( c ) . it is found that the dominant contribution near the fermi level comes from fe @xmath32 orbitals and the contribution of as @xmath34 orbitals is small but is not negligible . we show the partial dos of fe @xmath32 orbitals and that of as @xmath34 orbitals in the middle panel and the lower panel of fig . [ fs ] ( c ) , respectively . the @xmath48 , @xmath49 and @xmath47 states comprise the large part of the dos near the fermi level , while , the @xmath46 , @xmath28 states occupy the small one and are comparable with the @xmath54 , @xmath107 and @xmath108 states . the @xmath48 and @xmath49 states at the fermi level are larger than the @xmath47 ones and this corresponds to the fact that the electron pockets have @xmath48 , @xmath49 and @xmath47 orbital characters , while , the hole pockets have only @xmath48 and @xmath49 orbital characters . however , the @xmath47 states have large values just below the fermi level as shown in the inset of the middle panel of fig . [ fs ] ( c ) . this is due to the hole band near the @xmath24-point just below the fermi level . therefore , it is anticipated that the @xmath48 , @xmath49 and @xmath47 orbitals play significant roles to determine the magnetic , orbital and superconducting properties . in this subsection , we concentrate our attention on the case with @xmath8 . we set the typical parameters as @xmath109 , @xmath110 and @xmath111 , where the condition for the superconducting transition @xmath112 is satisfied as mentioned below ( see fig . [ chi ] ( d ) ) . the several components of the spin susceptibility @xmath113 given in eq . ( [ eq_chis ] ) are plotted in fig . [ chi ] ( a ) . the spin susceptibility is enhanced due to the effect of the coulomb interaction . it is found that the most dominant component is the @xmath47 diagonal component and the incommensurate peaks around the @xmath26 point are observed as reflecting the nesting between the hole pockets and the electron pockets . as mentioned before , the hole band which has mainly @xmath47 orbital character exists just below ( @xmath114ev ) the fermi level and contributes to the large value of the dos ( see fig . [ fs ] ( c ) . therefore , the @xmath47 diagonal component of @xmath113 becomes most dominant at finite temperature @xmath115ev ( @xmath116ev ) . the result is consistent with the rpa results based on the 5-band hubbard model @xcite . the several components of the charge - orbital susceptibility @xmath117 given in eq . ( [ eq_chic ] ) are plotted in fig . [ chi ] ( b ) . in contrast to the case with the spin susceptibility , the off - diagonal component of @xmath118 which corresponds to the transverse orbital susceptibility becomes most dominant and shows peaks around the @xmath26 point together with those at the @xmath24 point . it is noted that for @xmath8 the spin fluctuations dominate over the charge - orbital fluctuations as shown in figs . [ chi ] ( a ) and ( b ) . the several components of the effective pairing interaction @xmath119 for the spin - singlet state given in eq . ( [ eq_veff_s ] ) are plotted in fig . [ chi ] ( c ) . since the largest eigenvalue @xmath100 is always spin - singlet state in the present study , we show the effective pairing interaction only for the spin - singlet state . since in the case for @xmath109 , @xmath110 and @xmath111 , the spin fluctuations dominate over the orbital fluctuations as mentioned above , the structures of @xmath119 are similar to those of the spin susceptibility . substituting @xmath119 into the gap equation eq . ( [ eq_gap ] ) , we obtain the gap function @xmath120 with the eigenvalue @xmath100 . in fig . [ chi ] ( d ) , the eigenvalues @xmath100 for various pairing symmetries are plotted as functions of @xmath2 for fixed values of @xmath3 , @xmath36 , @xmath37 . with increasing @xmath2 , @xmath100 monotonically increases and finally becomes unity at a critical value @xmath121 above which the superconducting state is realized . for @xmath110 and @xmath122 the largest eigenvalue @xmath100 is for the @xmath4-wave symmetry and @xmath123 . the second largest eigenvalue is for @xmath28-wave symmetry and the eigenvalue for the @xmath28-wave symmetry increases as @xmath36 increases for @xmath8 . in this subsection , we concentrate our attention on the case with @xmath6 . we set the typical parameters as @xmath124 , @xmath125 and @xmath122 , where the condition for the superconducting transition @xmath112 is satisfied as mentioned below ( see fig . [ chi-2 ] ( d ) ) . the several components of the spin susceptibility @xmath113 given in eq . ( [ eq_chis ] ) are plotted in fig . [ chi-2 ] ( a ) . in contrast to the case with @xmath8 ( see fig . [ chi ] ( a ) ) , the off - diagonal element @xmath118 is most dominant owing to the inter - orbital direct term @xmath126 . the several components of the charge - orbital susceptibility @xmath117 given in eq . ( [ eq_chic ] ) are plotted in fig . [ chi-2 ] ( b ) . in contrast to the case with the spin susceptibility , the diagonal component of @xmath47 becomes most dominant and shows peaks around the @xmath24 point . it is noted that for @xmath6 the charge - orbital fluctuations , which corresponds to the fluctuations near the ferro - orbital ordered state realized in the large @xmath3 regime as mentioned later ( see fig . [ phasediagram ] ) , dominate over the spin fluctuations as shown in figs . [ chi-2 ] ( a ) and ( b ) . the several components of the effective pairing interaction @xmath119 for the spin - singlet state given in eq . ( [ eq_veff_s ] ) are plotted in fig . [ chi-2 ] ( c ) . since for @xmath124 , @xmath125 and @xmath111 , the charge - orbital fluctuations are larger than the spin fluctuations , the diagonal components of @xmath119 are always negative in @xmath127 space . substituting @xmath119 into the gap equation eq . ( [ eq_gap ] ) , we obtain the gap function @xmath120 with the eigenvalue @xmath100 . in fig . [ chi-2 ] ( d ) , the eigenvalues @xmath100 for various pairing symmetries are plotted as functions of @xmath3 for fixed values of @xmath2 , @xmath36 and @xmath37 . with increasing @xmath3 , @xmath100 monotonically increases and finally becomes unity at a critical value @xmath128 above which the superconducting state is realized . similar to the case of @xmath8 , the largest eigenvalue @xmath100 is for the @xmath4-wave symmetry but the superconducting gap structure is significantly different from that for @xmath8 as shown below . first , we discuss the gap functions in the case with @xmath8 . [ fig_gap_1 ] shows the diagonal components of the gap function @xmath129 for @xmath109 , @xmath110 @xmath122 [ fig_gap_1 ] ( a)-(d ) show the gap functions in the orbital representation and figs . [ fig_gap_1 ] ( e)-(h ) show those in the band representation . we note that the energy bands are numbered as descending energy . it is found that the gap function has the @xmath4-wave symmetry and the most dominant component is the @xmath47 diagonal component . we find that the gap functions in the band representation have different signs between the electron pockets and the hole pockets without any nodes on the fermi surfaces ( @xmath5-wave symmetry)@xcite . it is noted that the diagonal components of the gap function in the orbital representation , except for the @xmath28 component , also change those signs in @xmath41 space . the absolute values of the gap functions on the fermi surfaces are almost isotropic but largely depend on the energy bands ; those on the electron pockets of the 13th and 14th bands are twice or more larger than those on the hole pockets of the 11th and 12th bands . this is because the @xmath47 component , which has dominant contribution in @xmath130 as shown in fig . [ chi ] ( a ) , for the 13th and 14th bands is larger than that for the 11th and 12th bands . we note that the 10th band ( hole band ) with the largest @xmath47 component has the largest absolute value of the gap function , although the fermi level is just above the 10th band and does not cross it for @xmath102 . next , we discuss the gap functions in the case with @xmath6 . fig . [ fig_gap_2 ] shows the diagonal components of the gap function @xmath129 for @xmath124 , @xmath125 @xmath122 [ fig_gap_2 ] ( a)-(d ) shows the gap functions in the orbital representation and figs . [ fig_gap_2 ] ( e)-(h ) show those in the band representation . the diagonal components of the gap function in the orbital representation have no sign change in the @xmath41 space due to the diagonal components of @xmath131 as shown in fig . [ chi-2 ] ( c ) . we call this @xmath4-wave state as the @xmath7-wave state . the gap function in the band representation , however , has sign change between the fermi surfaces and line nodes on the 14th band fermi surface . these facts reflect that the sign change of the gap function in the orbital representation between the @xmath47 diagonal component and the other orbital diagonal components . the 11th band and 12th band fermi surface has mainly @xmath48 and @xmath49 orbital character , while , the 13th band fermi surface has mainly @xmath47 orbital character . therefore , the gap function has different sign between the hole pockets and the 13th band electron pocket . the 14th band electron pocket has mainly @xmath48 and @xmath49 orbital character away from the brillouin zone boundary , while @xmath47 orbital character on the 14th band electron pocket is comparable with @xmath48 and @xmath49 one near the brillouin zone boundary . thus , the gap function on the 14th band electron pockets has plus sign near the zone boundary and minus sign away from the zone boundary . by the simple mean field analysis of the pair transfer term of the interacting part of the hamiltonian eq ( [ eq_h_int ] ) , @xmath132 it is shown that the pair transfer @xmath133 favors the sign change between the diagonal components of the gap function in the orbital representation . in fact , we also examine the case with @xmath134 and we find that the @xmath7-wave state without sing change between the orbitals is realized for @xmath6 . we show the gap function for @xmath124 , @xmath135 , @xmath136 , @xmath134 in fig . [ fig_gap_3 ] . it is found that the all diagonal components of the gap function in the orbital representation have the same sign and those in the band representation have no sign change between all fermi surfaces . therefore , it is considered that the sign change of the gap function between the the @xmath47 diagonal component and the others is due to the pair transfer @xmath37@xcite . it is helpful for understanding the difference between the @xmath5-wave state and the @xmath7-wave state in more detail to consider the gap function in the real space . for @xmath5-wave state , the on - site pairing is comparable with the nearest neighbor and/or the next nearest neighbor one . on the other hand , for the @xmath7-wave state , the on - site pairing is dominant and the off - site pairings are negligibly small as compared to the on - site pairing . here we discuss the reason why the on - site part of the gap function for @xmath5-wave state is large ( especially in the @xmath47 diagonal component ) even though the most dominant component of the effective interaction is always repulsive in @xmath127 space ( see fig . [ chi ] ( c ) ) , and then the on - site effective interaction is repulsive . when we perform the fourier transformation of the gap equation eq . ( [ eq_gap ] ) , the on - site part of the left hand side is proportional to the on - site gap function , while that of the right hand side is given by the product of the on - site effective interaction ( @xmath137 ) and the on - site anomalous green s function which is proportional to the @xmath127 summation of the gap - function times the single - particle spectral weight times the thermal factor . in the case with the @xmath47 diagonal component , the on - site gap function is negative as the negative contribution of the gap function in @xmath127 space is much larger than the positive one as shown in fig . [ fig_gap_1 ] ( a ) . on the other hand , the on - site anomalous green s function becomes positive as the single - particle spectral weight of the @xmath47 hole band is very large around the @xmath24 point where the gap function is positive as compared to that of the electron band around the @xmath26-point where the gap function is negative . then , the gap equation can be satisfied with the large value of the on - site gap function against the repulsive on - site effective interaction . when the doping @xmath73 increases , the fermi level rises apart from the @xmath47 hole band , and then the effect of the hole band decreases resulting in the decrease in the on - site gap function as well as the decrease in the superconducting transition temperature ( not shown ) . such doping dependence of the on - site gap function has recently been observed in the 5-band hubbard model@xcite . on the contrary , in the @xmath7-wave state , the on - site pairing is always dominant almost independent of the doping @xmath73 . -@xmath2 plane for @xmath138 at @xmath102 , @xmath103 . the solid and dotted lines show the @xmath7-wave and the @xmath5-wave superconducting instabilities , respectively . the dashed and dot - dashed lines show instabilities towards the incommensurate spin density wave and the ferro - orbital order , respectively . [ phasediagram],width=283 ] the phase diagram on @xmath3-@xmath2 plane for @xmath122 is shown in fig . [ phasediagram ] , where the magnetic and charge - orbital instability is determined by @xmath139 and @xmath140 , respectively and the superconducting instability is determined by @xmath141 as mentioned before . the isdw with @xmath40 appears in the large @xmath2 region , while , the ferro - orbital order appears for @xmath6@xcite [ see also figs . [ chi ] ( a ) , ( b ) and figs [ chi-2 ] ( a ) , ( b ) ] . it is noted that on the phase boundary where the charge - orbital instability takes place , the longitudinal orbital susceptibility @xmath142 diverges , while , the charge susceptibility @xmath143 dose not . the @xmath5-wave pairing is realized near the isdw due to the spin fluctuations , while , the @xmath7-wave pairing is realized near the ferro - orbital ordered phase due to the charge - orbital fluctuations , where we regard the superconducting states as the @xmath7-wave states if @xmath48 , @xmath49 and @xmath47 diagonal components of the gap function have no sign change in @xmath41 space and as the @xmath5-wave states if not . the way to determine whether the superconducting state is the @xmath5-wave state or the @xmath7-wave state is not unique . this is because the @xmath5-wave and the @xmath7-wave state are same symmetry ( a@xmath144 ) and the change between @xmath7-wave and the @xmath5-wave state is crossover . in fact , as @xmath2 increases , the on - site paring decreases , while , the off - site pairing increases continuously . at @xmath145 , the nodes appear around the @xmath26-point for the @xmath47 diagonal component and those approaches the @xmath24-point as @xmath2 increases . it is noted that we also obtain the phase diagram on @xmath3-@xmath2 plane for @xmath146 and find that the phase diagram is essentially the same as that for @xmath122 except that the magnetic and the @xmath5-wave superconducting instabilities are slightly enhanced by the larger value of the hund s coupling @xmath36 . in this subsection , we discuss the effects of the electron - phonon coupling . by performing the group theoretical analysis for lafeaso , it is found that there are 14 kinds of the optical phonon modes at the @xmath147 point : 2@xmath148 + 2@xmath149 + 4@xmath150 + 3@xmath151 + 3@xmath152 . here , we concentrate on the @xmath148 mode in which la and as ions oscillate along the c - axis . the @xmath148 phonon dose not break the symmetry of the orbital and the resulting electron - phonon coupling matrix @xmath153 is diagonal in the orbital representation . within the rpa , the charge - orbital susceptibility @xmath154 including the effects of both the electron - electron and the electron - phonon coupling is obtained by replacing @xmath2 with @xmath155 and @xmath156 with @xmath157 in eqs . ( [ eq_chic ] ) and ( [ eq - u ] ) , where @xmath158 , @xmath159 is the frequency of the @xmath148 phonon and we neglect the orbital- and @xmath127-dependence of the electron phonon interaction . it is found that the inter - orbital direct term @xmath3 which enhances the orbital fluctuations is harder to be reduced by the electron - phonon coupling than the intra - orbital direct term @xmath2 . as a result , the orbital fluctuations are relatively enhanced by the electron - phonon coupling as compared to the spin fluctuations . in summary , we have investigated the pairing symmetry of the two - dimensional 16-band @xmath0-@xmath1 model by using the rpa and have obtained the phase diagram including the magnetic and orbital orders and the superconductivity . for @xmath8 , the @xmath5-wave superconductivity is realized near the isdw with @xmath160 phase . on the other hand , for @xmath6 , the @xmath7-wave superconductivity appears near the ferro - orbital ordered phase . the @xmath5-wave pairing is mediated by the spin fluctuations , while that the @xmath7-wave pairing is mediated by the orbital fluctuations . for @xmath8 , the gap function for the @xmath5-wave pairing changes its sign between the hole pockets and the electron pockets and the most dominant contribution of the gap function is the @xmath47 orbital diagonal component . this is qualitatively consistent with the results based on the 5-band hubbard model@xcite . however , the @xmath47 diagonal component of the gap function in our 16-band @xmath0-@xmath1 model have much larger value than the other matrix elements in comparison with the results based on the 5-band hubbard model@xcite . this may be because the outer hole fermi surface which has mainly @xmath48 and @xmath49 orbital character obtained by the @xmath0-@xmath1 model is almost circular , but that obtained by the 5-band hubbard model is diamond shape@xcite . therefore , the nesting effect which enhances the spin fluctuations becomes weak in our @xmath0-@xmath1 model , and the resulting components of @xmath120 related to @xmath48 , @xmath49 orbitals have smaller values . for @xmath6 , the gap function in the orbital representation for the @xmath7-wave pairing dose not change its sign in @xmath41 space . in other words , the on - site pairing is much larger than the off - site pairing in the real space . this is similar to the conventional phonon - mediated superconductivity . however , the gap functions have different signs between orbitals in contrast to the conventional phonon - mediated superconductivity . we have shown that this sign change of the gap functions between orbitals is due to the effect of the pair transfer interaction @xmath37@xcite . it is noted that the @xmath7-wave state has been observed also in the one - dimensional 2-band hubbard model in the same parameter region with @xmath6 @xcite . it seems that the both @xmath5-wave and the @xmath7-wave states with full superconducting gaps are consistent with various experiments such as , the nmr relaxation rate , the knight shift , the arpes , the magnetic penetration depth measurements , although the sign of the gap function has not been directly observed there . however , according to the recent theoretical studies of the nonmagnetic impurity effects@xcite , anderson s theorem is violated for the @xmath5-wave superconductivity in contrast to the experimental results of very weak @xmath12-suppression in fe site substitution@xcite and neutron irradiation@xcite . since it can be considered that the impurity potential by the fe - site substitution is diagonal and local in the orbital basis according to the first principle calculation@xcite , it is expected that the @xmath7-wave state observed in the present study is more robust against the nonmagnetic impurity effects than the @xmath5-wave state . in addition to the coulomb interaction , we have also discussed the effects of the coupling @xmath161 between the electron and the @xmath148 phonon within the rpa . it has been found that the @xmath7-wave pairing realized in the unrealistic parameter region with @xmath6 for @xmath162 is enhanced due to the effect of @xmath161 and can be expanded over the realistic parameter region with @xmath8 for a realistic value of @xmath161 . in the first principle calculations for iron - based superconductors in conjunction with the migdal - eliashberg theory , the electron - phonon coupling is found to be too small to obtain high @xmath12 observed in experiments@xcite . the effect of the coulomb interaction , however , has not been discussed there . in the present study , the cooperative effect of the coulomb interaction and the electron - phonon coupling is crucial for the enhancement of the orbital fluctuations which induce the @xmath7-wave superconductivity . recently , the large isotope effects on the transition temperatures for both the sdw and the superconductivity have been observed@xcite . this experimental result implies that not only the coulomb interaction but also the electron - phonon coupling plays crucial effects on the electronic states for iron - based superconductors . in early theoretical studies for the copper oxide superconductors , the effect of the coulomb interaction between the @xmath0 and @xmath1 electrons @xmath163 was studied by several authors@xcite . according to the rpa study based on the @xmath0-@xmath1 model with the single @xmath47 orbital , @xmath163 enhances the charge fluctuations with @xmath164 and the @xmath4-wave superconductivity is realized due to the effect of charge fluctuations@xcite . in addition , the @xmath165-expansion approaches ( @xmath166 is the spin - orbital degeneracy ) revealed that the strong correlation effect enhances the charge fluctuations together with the @xmath4-wave superconductivity@xcite . therefore , it is expected that , in the present @xmath0-@xmath1 model with multi @xmath0 orbitals , @xmath163 enhances the charge - 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we study superconductivity in the two - dimensional 16-band @xmath0-@xmath1 model extracted from a tight - binding fit to the band structure of lafeaso , using the random phase approximation . when the intraorbital repulsion @xmath2 is larger than the interorbital one @xmath3 , an extended @xmath4-wave ( @xmath5-wave ) pairing with sign reversal of order parameter is mediated by antiferromagnetic spin fluctuations , while when @xmath6 another kind of @xmath4-wave ( @xmath7-wave ) pairing without sign reversal is mediated by ferro - orbital fluctuations . the @xmath7-wave pairing is enhanced due to the electron - phonon coupling and then can be expanded over the realistic parameter region with @xmath8 .

let @xmath6 be a domain with @xmath7 . we denote by @xmath8 the ( weak ) @xmath9 partial derivative of a locally integrable function @xmath10 , and by @xmath11 the ( weak ) gradient . then for @xmath12 we define the sobolev space as @xmath13 with the norm @xmath14 for @xmath15 , and @xmath16 it is a fundamental property of sobolev spaces that smooth functions defined in @xmath1 are dense in @xmath17 for any domain @xmath0 when @xmath15 . if each function in @xmath17 is the restriction of a function in @xmath18 one can then obviously use global smooth functions to approximate functions in @xmath19 . this is in particular the case for lipschitz domains . moreover , if @xmath1 satisfies the so - called segment condition " , then one has that @xmath5 is dense in @xmath3 ; see e.g. @xcite for references . in the planar setting , lewis proved in @xcite that @xmath20 is dense in @xmath17 for @xmath21 provided that @xmath1 is a jordan domain . more recently , in @xcite it was shown by giacomini and trebeschi that , for bounded simply connected planar domains , @xmath22 is dense in @xmath3 for all @xmath23 . motivated by the results above , koskela and zhang proved in @xcite that for any bounded simply connected domain and any @xmath4 , @xmath2 is dense in @xmath3 , and @xmath20 is dense in @xmath3 when @xmath1 is jordan . in this paper , we extend the main idea in @xcite so as to handle both multiply connected and higher dimensional settings . it turns out that simply connectivity ( or trivial topology ) is not sufficient for approximation results in higher dimensions . [ example ] given @xmath24 , there is a bounded domain @xmath25 homeomorphic to the unit ball via a locally bi - lipschitz homeomorphism , such that @xmath26 is not dense in @xmath3 for any @xmath27 . recall that @xmath28 is locally bi - lipschitz if for every compact set @xmath29 there exists @xmath30 such that for all @xmath31 @xmath32 the above example shows that the planar setting is very special . the crucial point is that a simply connected planar domain is conformally equivalent ( by the riemann mapping theorem ) to the unit disk , and conformal equivalence is in general much more restrictive than topological equivalence . one could then ask if the planar approximation results extend to hold for those spatial domains that are conformally equivalent to the unit ball . this is trivially the case since the liouville theorem implies that such a domain is necessarily a ball or a half - space . a bit of thought reveals that bi - lipschitz equivalence is also sufficient . our results below imply that bi - lipschitz equivalence can be relaxed to quasiconformal equivalence to the unit ball or even to quasiconformal equivalence to a uniform domain , a natural class of domains in the study of ( quasi)conformal geometry . in order to state our main result , we need to introduce some terminology . let @xmath33 be a domain . then the associated quasihyperbolic distance between two points @xmath34 is defined as @xmath35 where the infimum is taken over all the rectifiable curves @xmath36 connecting @xmath37 and @xmath38 . a curve attaining this infimum is called a quasihyperbolic geodesic connecting @xmath37 and @xmath38 . the distance between two sets is also defined in a similar manner . moreover , a domain @xmath1 is called _ @xmath39-gromov hyperbolic with respect to the quasihyperbolic metric _ , if for all @xmath40 and any corresponding quasihyperbolic geodesics @xmath41 , we have @xmath42 for any @xmath43 . for the existence of quasihyperbolic geodesics we refer to ( * ? ? ? * proposition 2.8 ) . for applications , it is usually easier to apply one of the equivalent definitions , see lemma [ eqdef ] below . recall that a set @xmath44 is called _ quasiconvex _ if there exists a constant @xmath45 such that any pair of points @xmath46 can be connected to each other with a rectifiable curve @xmath47 whose length satisfies @xmath48 . [ mainthm ] if @xmath0 is a bounded domain that is @xmath39-gromov hyperbolic with respect to the quasihyperbolic metric , then for any @xmath49 , @xmath2 is dense in @xmath3 . moreover , if @xmath1 is also either jordan or quasiconvex , we have that @xmath5 is dense in @xmath3 . each finitely connected planar domain is gromov hyperbolic with respect to the quasihyperbolic metric . therefore we recover the main theorem in @xcite . furthermore , domains which are quasiconformally equivalent to uniform domains , especially the ones quasiconformally equivalent to a ball , are gromov hyperbolic domains . see @xcite for these results . theorem [ mainthm ] also gives consequences for @xmath50 , the banach space of functions in @xmath51 with bounded variation . indeed , given @xmath52 we have a sequence of functions @xmath53 ( or smooth in @xmath1 ) that converges to @xmath10 in @xmath51 and so that the @xmath54-energy of @xmath55 @xmath56 satisfies @xmath57 based on theorem [ mainthm ] , we may further assume that @xmath58 when @xmath1 is bounded and gromov hyperbolic , and even that each @xmath59 is the restriction of a global smooth function when @xmath1 is jordan or quasiconvex . we refer the reader to @xcite for further information on the theory of @xmath54-functions . the paper is organized as follows . in section 2 we give some preliminaries . after this we decompose a bounded domain @xmath1 ( which is @xmath39-gromov hyperbolic with respect to the quasihyperbolic metric ) into several parts via lemma [ eqdef ] , and then construct a corresponding partition of unity . in @xcite conformal mappings and planar geometry were applied to obtain the desired composition . in our setting , we can not rely on mappings nor on simple geometry . instead of this we employ two characterizing properties of gromov hyperbolicity : the ball - separation condition and the gehring - hayman inequality ; see lemma [ eqdef ] below . the proof of theorem [ mainthm ] is given in section 3 , and finally in the last section we discuss the necessity of geometric conditions . the notation in this paper is quite standard . when we make estimates , we often write the constants as positive real numbers @xmath60 with the parenthesis including all the parameters on which the constant depends . the constant @xmath60 may vary between appearances , even within a chain of inequalities . by @xmath61 we mean that @xmath62 for some constant @xmath63 . also @xmath64 means @xmath65 with @xmath66 , and similar to @xmath67 . the euclidean distance between two sets @xmath68 is denoted by @xmath69 . we call a _ dyadic cube _ in @xmath70 any set @xmath71\times \cdots \times[m_n2^{-k},\ , ( m_n+ 1)2^{-k}],\ ] ] where @xmath72 . we denote by @xmath73 the side length of the cube @xmath74 , and by @xmath75 the length of a curve @xmath76 . given a cube @xmath74 and @xmath77 , by @xmath78 we mean the cube concentric with @xmath74 , with sides parallel to the axes , and with length @xmath79 . for a set @xmath80 , we denote by @xmath81 its interior , @xmath82 its boundary , and @xmath83 its closure . notation @xmath84 means that the set @xmath85 is compactly contained in @xmath86 . in this section , we first recall some lemmas related to gromov hyperbolic domains , and then decompose our domain into two main parts . at the end of this section we construct a corresponding partition of unity . define the _ inner distance with respect to @xmath1 _ between @xmath87 by setting @xmath88 where the infimum runs over all curves joining @xmath89 and @xmath90 in @xmath91 the ball centered at @xmath89 with radius @xmath92 respect to the inner distance is denoted by @xmath93 . let @xmath94 @xmath7 be a bounded domain that is @xmath39-gromov with respect to the quasihyperbolic metric . recall that @xmath39-gromov hyperbolicity can equivalently be defined as follows ; see @xcite and @xcite . [ eqdef ] a domain @xmath0 is @xmath39-gromov hyperbolic with respect to the quasihyperbolic metric if and only if it has the following two properties : 1 . @xmath95-ball - separation condition : there exists a constant @xmath96 such that , for any @xmath97 , any quasihyperbolic geodesic @xmath98 joining @xmath89 and @xmath90 , and every @xmath99 , the ball @xmath100 satisfies @xmath101 for any curve @xmath36 connecting @xmath89 and @xmath90 . @xmath102-gehring - hayman condition : for any @xmath103 , the euclidean length of each quasihyperbolic geodesic connecting @xmath89 and @xmath90 is no more than @xmath104 . here all the constants depend only on each other and @xmath105 . the above gehring - hayman condition was proven for simply connected planar domains in @xcite and the ball - separation condition in @xcite , respectively . recall that every open proper subset of @xmath70 admits a whitney decomposition . a standard reference for this is ( * ? ? ? * chapter vi ) . [ whitney ] let @xmath33 be a domain . then it admits a _ whitney decomposition _ , that is , there exists a collection @xmath106 of countably many dyadic ( closed ) cubes such that \(i ) @xmath107 and @xmath108 for all @xmath109 with @xmath110 ; \(ii ) @xmath111 ; \(iii ) @xmath112 whenever @xmath113 . the lemmas above allow us to establish the following key lemma . [ uniform bound ] suppose @xmath114 and @xmath115 are whitney cubes of @xmath1 satisfying @xmath116 for some constant @xmath117 . moreover assume that they can be joined by a chain of whitney cubes , whose edge lengths are larger than @xmath118 . then there exists a sequence of no more than @xmath119 whitney cubes of @xmath1 , of edge lengths comparable to @xmath120 , such that their union connects @xmath114 and @xmath115 . especially we have @xmath121 the @xmath102-gehring - hayman condition together with the assumption @xmath122 gives a quasihyperbolic geodesic @xmath76 connecting @xmath114 and @xmath115 such that @xmath123 since @xmath124 , the diameters of the whitney cubes intersecting @xmath76 are uniformly bounded from above by a multiple of @xmath120 . moreover , for every whitney cube @xmath74 with @xmath125 , by the @xmath95-ball - separation condition and the definition of whitney cubes , any other curve connecting @xmath114 and @xmath115 must intersect @xmath126 . on the other hand , by our assumption , there exists a sequence of cubes connecting @xmath114 and @xmath115 with edge lengths not less than @xmath127 . it follows that @xmath128 to conclude , for all @xmath129 , @xmath130 with the constant only depending on @xmath105 , @xmath131 , and @xmath95 . since @xmath132 the number of whitney cubes intersecting @xmath76 must be bounded by a constant depending only on @xmath95 , @xmath102 , @xmath105 and @xmath131 . fix a bounded domain @xmath1 which is @xmath39-hyperbolic as in lemma [ eqdef ] with the associated constants @xmath95 and @xmath102 . for any constant @xmath133 and any euclidean cube or internal metric ball @xmath74 centered at @xmath89 , we introduce the notation @xmath134 this is a ( relatively ) closed inner metric ball inside @xmath1 . let @xmath135 be large enough such that there is at least one whitney cube in @xmath1 whose edge length is larger than @xmath136 . let @xmath137 be the collection of all whitney cubes of @xmath1 , and @xmath138 be one of the largest ones . then define @xmath139 to be the path - component of @xmath140 with @xmath141 , see figure [ fig : omega_m0 ] . is the path - component of the union of cubes of side - length at least @xmath136 that contains @xmath142 . in order to have the properties listed in lemma [ core part ] for the subdomain @xmath143 we will cut out those parts from @xmath139 whose connection to @xmath142 is blocked by dilated boundary cubes.,scaledwidth=90.0% ] define @xmath144 to be the collection of the whitney cubes in @xmath137 that are contained in @xmath139 . also let @xmath145 and @xmath146 notice that , by definition , any whitney cube @xmath147 satisfies @xmath148 and thus there are at most finitely many of them since @xmath1 is bounded . up to relabeling all the @xmath149 s in @xmath137 we may assume that all the cubes in @xmath150 are ordered consecutively from @xmath151 to some finite number @xmath152 . recall the constant @xmath95 in lemma [ eqdef ] . we next refine @xmath139 according to the @xmath95-ball separation condition in order to obtain the desired set @xmath143 . it is constructed via an induction argument according to the cubes in @xmath150 . first for each cube @xmath153 , we define @xmath154 . let @xmath155 be large enough such that @xmath156 . for each @xmath157 let @xmath158 ( which might be empty ) be the union of all the path - components of @xmath159 not containing @xmath142 . roughly speaking , the set @xmath158 is the collection of points in @xmath1 whose connection to @xmath142 is blocked by @xmath160 . as any curve joining @xmath142 and some point outside @xmath139 has to pass through @xmath161 , the @xmath95-separation condition allows us to conclude that @xmath162 suppose that there exists @xmath163 such that @xmath164 and @xmath165 . then by the path - connectedness of @xmath166 and the definition of @xmath158 we conclude that @xmath167 now let us define @xmath168 and @xmath169 we also define @xmath170 we claim that @xmath171 indeed comparing to we have three cases . first of all if @xmath172 with @xmath173 , then @xmath174 this with gives us @xmath175 , and consequently @xmath176 by . therefore any curve from @xmath90 to @xmath142 needs to pass through @xmath177 by the definition of @xmath178 and the @xmath95-ball - separation condition , and then by definition @xmath179 . secondly if @xmath180 with @xmath173 , then again @xmath174 by the deduction above we similarly conclude that @xmath179 . at last suppose @xmath181 . then it belongs to some cube @xmath74 originally in @xmath144 but not in @xmath182 . therefore @xmath183 however @xmath74 is connected , and by the argument of we also conclude that @xmath179 . all in all we have shown . if @xmath184 , then we just let @xmath185 and accordingly define @xmath186 and so on . otherwise , we apply the procedure above , with @xmath114 replaced by @xmath115 and @xmath178 replaced by @xmath187 , to obtain these sets ( and collections ) . we repeat this process for every @xmath157 with @xmath188 . by iteration we finally obtain a set @xmath189 . notice that any whitney cube in @xmath190 intersecting @xmath191 is contained in @xmath192 for some @xmath193 . thus it has edge length comparable to @xmath136 with the constant only depending on @xmath105 and @xmath95 . hence there exists a constant @xmath194 such that @xmath195 whenever @xmath196 . the deduction above together with the fact that @xmath197 also gives @xmath198 moreover @xmath199 consists of cubes from @xmath150 . to conclude , we obtain the following lemma . [ core part ] let @xmath1 be a bounded domain which is @xmath39-gromov hyperbolic with respect to the quasihyperbolic metric , @xmath200 be the collection of whitney cubes of @xmath1 and @xmath142 be one of the largest whitney cubes . then there exists a sequence of sets @xmath201 such that by setting @xmath202 by letting @xmath154 for each @xmath203 and by finally defining @xmath158 ( which might be empty ) to be the union of all the path - components of @xmath159 not containing @xmath142 , we have the following properties . 1 . each @xmath143 consists of finitely many whitney cubes and any two of them can be joined by a chain of whitney cubes in @xmath1 of edge lengths not less than @xmath136 . moreover @xmath204 and there exists a constant @xmath205 such that @xmath206 and @xmath195 for any @xmath196 . 2 . for every whitney cube @xmath207 we have @xmath208 . we call such a cube a _ boundary cube of @xmath143_. 3 . there exists a subcollection @xmath209 of @xmath210 such that for each @xmath211 and @xmath212 @xmath213 moreover @xmath214 covers all the boundary cubes of @xmath143 . we have @xmath215 the property 3 ) above turns out to be crucial later and it may fail for @xmath210 ; this is the reason for introducing the subcollection @xmath209 of @xmath210 . obtained after the iterative procedure from sets @xmath216 still contains the cube @xmath142.,scaledwidth=90.0% ] in this subsection we first decompose @xmath217 into two main parts @xmath218 and @xmath219 , and then make further decompositions of them . first of all let @xmath220 secondly , we denote by @xmath219 the rest of @xmath1 , that is , @xmath221 notice that by lemma [ core part ] we have @xmath222 and @xmath223 where the set @xmath158 is defined in lemma [ core part ] . by abuse of notation , we also denote by @xmath218 and @xmath219 their closures with respect to the topology of @xmath1 , respectively . observe that the boundary of @xmath224 in @xmath1 is porous and hence of lebesgue measure zero , @xmath225 and each @xmath226 therefore we have @xmath227 we decompose @xmath218 further . recall that @xmath228 let @xmath229 for each @xmath212 . for simplicity we again assume that @xmath230 with some @xmath231 . we claim that for each fixed @xmath232 , @xmath233 where @xmath234 means the cardinality of the corresponding set . indeed , if @xmath235 , then @xmath236 by the definition of @xmath232 . then follows by the fact that @xmath237 with a constant independent of @xmath238 . define @xmath239 , and inductively for @xmath240 set @xmath241 notice that @xmath242 may well be disconnected , or even empty . we replace every @xmath242 by its closure with respect to the topology of @xmath1 , and still use the notation @xmath242 . notice that after all these changes , @xmath243 still satisfy all the corresponding properties above ; especially @xmath244 by for each @xmath242 @xmath245 and the corresponding @xmath246 satisfy @xmath247 similar reasons also give the fact that @xmath248 at last we remark that for any @xmath249 @xmath250 by . moreover by the definition of @xmath242 we have @xmath251 recall that @xmath142 is one of the largest whitney cubes contained in @xmath143 , and for each @xmath212 we have @xmath252 and @xmath229 . consist of all the path - components of @xmath219 for which the set @xmath160 centered at @xmath157 blocks all curves going from the path - component to the cube @xmath142.,scaledwidth=90.0% ] to decompose the last part @xmath219 , we introduce the following notation . recall the definition of @xmath158 in lemma [ core part ] and define @xmath253 see figure [ fig : tjrj ] . certainly @xmath254 could be empty . we replace @xmath254 by its closure with respect to the topology of @xmath1 and still denote it by @xmath254 . notice that by lemma [ core part ] @xmath255 fix @xmath254 and suppose that @xmath256 . we claim that @xmath257 . indeed , if @xmath258 , then @xmath259 and by the path - connectedness of @xmath260 any point @xmath261 can be connected to @xmath262 by a path in @xmath263 if @xmath264 can be connected to @xmath142 via a path in @xmath159 , then @xmath90 can be connected to @xmath142 via a path in @xmath159 , which leads to a contradiction to the definition of @xmath158 , which contains @xmath254 . then our claim follows . if @xmath264 can not be connected to @xmath142 via any path in @xmath159 , then @xmath265 , and by lemma [ core part ] we know that @xmath266 the claim follows from the definition of @xmath232 . therefore by the proof of and the definition of @xmath267 we conclude that , for each fixed ( non - empty ) @xmath254 , @xmath268 also note that if @xmath269 , then the path - component of @xmath254 containing @xmath90 is a subset of @xmath270 by the definition of @xmath254 . we define @xmath271 , and for @xmath240 set @xmath272 we also refer by @xmath273 to its closure with respect to the topology of @xmath1 . according to for each fixed non - empty @xmath273 @xmath274 similarly for each fixed @xmath242 @xmath275 to conclude from the subsections above , whenever @xmath276 with @xmath207 , @xmath277 or @xmath278 we always have the corresponding @xmath279 satisfying @xmath247 this fact with lemma [ core part ] allows us to apply lemma [ uniform bound ] later . we construct a partition of unity in this subsection . to this end , let us introduce the following notation . for a set @xmath280 , we define @xmath281 [ partition ] with all the notations above , there are functions @xmath282 , @xmath283 and @xmath284 with @xmath285 such that : 1 . the function @xmath282 is lipschitz in @xmath1 , compactly supported in @xmath143 , @xmath286 , and @xmath287 . 2 . for each @xmath238 , we have @xmath288 . the support of @xmath283 is relatively closed in @xmath1 and contained in @xmath289 , @xmath290 , and @xmath291 . 3 . for each @xmath238 , we have @xmath292 . the support of @xmath284 is relatively closed in @xmath1 and contained in @xmath293 , @xmath294 , and @xmath295 . 4 . @xmath296 for any @xmath297 . first of all we construct cut - off functions for each of our sets via the distance functions with respect to the inner metric . the function @xmath283 can be defined as @xmath298 and similarly @xmath299 the function @xmath282 is defined by @xmath300 it is obvious that these functions satisfy @xmath301 for every @xmath297 . note that by the essence of , , and we have for each @xmath207 @xmath302 and also for each @xmath212 @xmath303 hence by the decomposition of @xmath1 we conclude that for any @xmath304 @xmath305 therefore , by dividing @xmath306 by @xmath307 , respectively , we obtain the desired partition of unity . the new functions , still denoted by @xmath306 , satisfy the desired gradient control as @xmath307 is bounded from below and above . notice that @xmath308 moreover the proof of also shows that there are uniformly finitely many @xmath260 such that @xmath260 intersects @xmath309 . thus @xmath310 we conclude that @xmath311 fix @xmath312 . also fix @xmath313 with @xmath4 . we may assume that @xmath10 is smooth and bounded since bounded smooth functions are dense in @xmath3 ; e.g. see the proof of ( * ? ? ? * lemma 2.6 ) . we may further assume that @xmath314 . recall that @xmath315 . define @xmath316 to be the union of those whitney cubes @xmath317 for which there exists a chain of no more than @xmath318 whitney cubes joining @xmath74 to some cube in @xmath319 here the constant @xmath318 that depends on @xmath320 will be determined later . then the quasihyperbolic distance from @xmath74 to @xmath321 is uniformly bounded if @xmath322 observe that , for any whitney cube @xmath323 we have @xmath324 with a constant depending on @xmath320 . also notice that lemma [ core part ] implies @xmath325 . thus for @xmath135 large enough we have @xmath326 notice that @xmath327 since @xmath143 is compact and @xmath10 is smooth . we define a function @xmath328 on @xmath1 by setting @xmath329 where @xmath330 , @xmath331 and @xmath332 are the functions in lemma [ partition ] and @xmath333 is the integral average over @xmath207 . it is obvious that @xmath334 by our construction , since by boundedness of @xmath1 we only have finitely many @xmath207 and lemma [ partition ] gives the estimates on the derivatives . moreover we have @xmath335 by our assumption , lemma [ partition ] and the definition of @xmath328 . hence @xmath336 by . consequently , by the definition of @xmath337 and lemma [ partition ] , we only need to show that @xmath338 we will show this via the poincar inequality , lemma [ uniform bound ] and lemma [ partition ] . we write @xmath339 for the union of the cubes given by lemma [ uniform bound ] for each pair @xmath340 . recall that @xmath341 then for any @xmath207 with the associated average @xmath342 , by , , lemma [ uniform bound ] , lemma [ partition ] and the poincar inequality we obtain that @xmath343|^p \ , dx + \sum_{\substack{s_k\subset e_m\\ q_j \cap \mathcal n_{m,\,\omega}(s_k ) \neq\emptyset } } \int_{q_j } |\nabla [ ( a_k - a_j ) \phi_k(x ) ] |^p \ , dx \\ \lesssim & \int_{q_j } |\nabla u|^p + |u(x)-a_j|^p 2^{mp}\ , dx + \sum_{\substack{s_k\subset e_m\\ q_j \cap \mathcal n_{m,\,\omega}(s_k ) \neq\emptyset } } \int_{q_j } |a_k - a_j|^p 2^{mp } \ , dx \\ \lesssim & \int_{q_j } |\nabla u|^p\ , dx + \sum_{\substack{s_k\subset e_m\\ q_j \cap \mathcal n_{m,\,\omega}(s_k ) \neq\emptyset } } 2^{-m(n - p ) } 2^{-m(p - n ) } \int_{g(q_j,\ , q_k)}|\nabla u|^p \,dx\\ \lesssim & \int_{q_j } |\nabla u(x)|^p \ , dx+ \sum_{\substack{s_k\subset e_m\\ q_j \cap\mathcal n_{m,\,\omega}(s_k ) \neq\emptyset } } \int_{g(q_j,\ , q_k)}|\nabla u|^p \,dx . \end{aligned}\ ] ] notice that by lemma [ uniform bound ] there are uniformly finitely many cubes contained in the chain @xmath344 connecting @xmath157 and @xmath264 if @xmath345 . on the other hand recall that @xmath346 is compactly supported in @xmath143 . then for each @xmath242 , lemma [ uniform bound ] , , , , and the poincar inequality give @xmath347|^p \ , dx \\ & \qquad + \sum_{\substack{t_k\subset f_m\\ \mathcal n_{m,\,\omega}(s_j)\cap \mathcal n_{m,\,\omega}(t_k ) \neq\emptyset } } \int_{s_j } |\nabla [ ( a_k - a_j ) \varphi_k(x ) ] |^p \ , dx\\ \lesssim & \sum_{\substack{s_k\subset e_m\\ \mathcal n_{m,\,\omega}(s_j)\cap \mathcal n_{m,\,\omega}(s_k ) \neq\emptyset } } \int_{s_j } |a_k - a_j|^p 2^{mp } \ , dx + \sum_{\substack{t_k\subset f_m\\ \mathcal n_{m,\,\omega}(s_j)\cap \mathcal n_{m,\,\omega}(t_k ) \neq\emptyset } } \int_{s_j } |a_k - a_j|^p 2^{mp } \ , dx\\ \lesssim & \sum_{\substack{s_k\subset e_m\\ \mathcal n_{m,\,\omega}(s_j)\cap \mathcal n_{m,\,\omega}(s_k ) \neq\emptyset } } 2^{-m(p - n ) } 2^{-m(n - p)}\int_{g(q_j,\ , q_k ) } |\nabla u|^p \ , dx\\ & \qquad + \sum_{\substack{t_k\subset f_m\\ \mathcal n_{m,\,\omega}(s_j)\cap \mathcal n_{m,\,\omega}(t_k ) \neq\emptyset } } 2^{-m(p - n ) } 2^{-m(n - p)}\int_{g(q_j,\ , q_{k } ) } |\nabla u|^p \ , dx\\ \lesssim & \sum_{\substack{s_k\subset e_m\\ \mathcal n_{m,\,\omega}(s_j)\cap \mathcal n_{m,\,\omega}(s_k ) \neq\emptyset } } \int_{g(r_j,\ , r_k ) } |\nabla u|^p \ , dx+\sum_{\substack{t_k\subset f_m\\ n_{m,\,\omega}(s_j)\cap \mathcal n_{m,\,\omega}(t_k ) \neq\emptyset } } \int_{g(q_j,\ , q_{k } ) } |\nabla u|^p \ , dx . \end{aligned}\ ] ] the calculation for @xmath273 is almost the same . indeed by , , and the poincar inequality @xmath348|^p \ , dx \\ & \qquad + \sum_{\substack{t_k\subset f_m\\ \mathcal n_{m,\,\omega}(t_j)\cap \mathcal n_{m,\,\omega}(t_k ) \neq\emptyset } } \int_{t_j } |\nabla [ ( a_k - a_j ) \phi_i(x ) ] |^p \ , dx \\ \lesssim & \sum_{\substack{s_k\subset e_m\\ \mathcal n_{m,\,\omega}(t_j)\cap \mathcal n_{m,\,\omega}(s_k ) \neq\emptyset } } \int_{t_j } |a_k - a_j|^p 2^{mp } \ , dx + \sum_{\substack{t_k\subset f_m\\ \mathcal n_{m,\,\omega}(t_j)\cap \mathcal n_{m,\,\omega}(t_k ) \neq\emptyset } } \int_{t_j } |a_k - a_j|^p 2^{mp } \ , dx \\ \lesssim & \sum_{\substack{s_k\subset e_m\\ \mathcal n_{m,\,\omega}(t_j)\cap \mathcal n_{m,\,\omega}(s_k)\neq\emptyset } } 2^{-m(p - n ) } 2^{-m(n - p)}\int_{g(q_{j},\ , q_j ) } |\nabla u|^p \ , dx \\ & \qquad + \sum_{\substack{t_k\subset f_m\\ \mathcal n_{m,\,\omega}(t_j)\cap \mathcal n_{m,\,\omega}(t_k)\neq\emptyset } } 2^{-m(p - n ) } 2^{-m(n - p)}\int_{g(q_{j},\ , q_j ) } |\nabla u|^p \ , dx\\ \lesssim & \sum_{\substack{s_k\subset e_m\\ \mathcal n_{m,\,\omega}(t_j)\cap \mathcal n_{m,\,\omega}(s_k)\neq\emptyset } } \int_{g(q_{j},\ , q_k ) } |\nabla u|^p \ , dx + \sum_{\substack{t_k\subset f_m\\ \mathcal n_{m,\,\omega}(t_j)\cap \mathcal n_{m,\,\omega}(t_k)\neq\emptyset } } \int_{g(q_{j},\ , q_k ) } |\nabla u|^p \ , dx . \end{aligned}\ ] ] by lemma [ uniform bound ] , there is a constant @xmath349 such that , for any chain of cubes @xmath350 used above the number of cubes involved is uniformly bounded from above by @xmath351 . this gives us the constant @xmath318 in the definition of @xmath316 . sum over all the @xmath157 s , @xmath242 s and @xmath273 s above . notice that , since the number of whitney cubes in any chain @xmath339 above is always uniformly bounded by lemma [ uniform bound ] , the whitney cubes involved in our sums have uniformly finite overlaps . additionally all the cubes in these chains are contained in @xmath316 . thus we obtain and conclude the first part of the theorem . when @xmath1 is quasiconvex , we immediately have that @xmath5 is dense in @xmath3 since every function in @xmath2 can then be extended to a global lipschitz function ; by applying suitable cut - off functions and via a diagonal argument we obtain the approximation by smooth functions . the argument for the jordan domain case is similar to the proof of ( * ? ? ? * corollary 1.2 ) . recall that for any two non - empty subsets @xmath352 and @xmath353 of @xmath70 , the _ hausdorff distance _ @xmath354 is defined as @xmath355 when @xmath1 is jordan , we can construct a sequence of lipschitz domains @xmath356 approaching @xmath1 in hausdorff distance such that @xmath357 and @xmath358 for each @xmath359 . for example , by the morse - sard theorem we may define @xmath360 via the boundary of a suitable lower level set of @xmath361 , where @xmath361 is a smooth function obtained by applying suitable mollifiers and a partition of unity for @xmath362 to the distance function @xmath363 . now fix @xmath364 and choose @xmath365 such that @xmath366 . then , by the definition of @xmath367 the @xmath368-separation condition with respect to @xmath360 holds for our original cubes in @xmath143 . similarly for points with inner distance smaller than a multiple of @xmath136 in @xmath143 , the @xmath369-gehring - hayman condition with respect to @xmath360 still holds . moreover , the original whitney cubes contained in @xmath143 are also whitney - type for @xmath360 up to a multiplicative constant @xmath370 in lemma [ whitney ] . therefore we may repeat all the arguments above similarly to extend the function @xmath328 from @xmath143 to @xmath371 , with @xmath372 since each @xmath360 is a lipschitz domain , we may extend @xmath373 to a global sobolev function , and then by applying suitable mollifiers and via a diagonal argument we obtain the approximation by global smooth functions . when @xmath374 , unlike in the planar case , simply connectivity does not guarantee that @xmath2 be dense in @xmath3 for @xmath4 . indeed , given @xmath21 there exists a simply connected bounded domain @xmath375 such that even @xmath3 is not dense in @xmath26 for @xmath376 . towards this , let us recall the definition of removable sets . a closed set @xmath377 with lebesgue measure zero is said to be _ removable for @xmath378 _ if @xmath379 in the sense of sets . in ( * theorem a ) , for any @xmath380 , koskela gave an example of a compact set @xmath377 which is removable for @xmath381 but not for @xmath382 with @xmath376 . we give a related planar example for every @xmath21 . by taking the union of a suitable collection of scaled and translated copies @xmath387 of the above compact sets corresponding to an increasing sequence of @xmath388 tending to a fixed @xmath389 we obtain the following corollary . we first consider the case where @xmath394 . by * proposition 2.1 , theorem 2.2 , theorem 2.3 ) it suffices to construct a cantor set @xmath395 $ ] of positive length so that , by letting @xmath396 be the complementary intervals of @xmath384 on @xmath397 $ ] and @xmath398 the @xmath151-dimensional hausdorff measure , @xmath399 while @xmath384 is @xmath400-porous for all @xmath401 . recall that @xmath395 $ ] is @xmath400-porous if for @xmath398-almost every point @xmath402 there is a sequence of numbers @xmath403 and a constant @xmath404 such that @xmath405 as @xmath406 , and each interval @xmath407 $ ] contains an interval @xmath408\setminus e$ ] with @xmath409 . towards this construction , we let @xmath410 be a small constant to be determined momentarily . out set @xmath384 is obtained via the following cantor construction . at the @xmath411-th step with @xmath412 we delete an open interval of length @xmath413 from the middle of each of the remaining @xmath414 closed intervals with equal length , respectively . then e is defined as the intersection of all these closed intervals , and @xmath365 is chosen such that @xmath415 thus @xmath384 has positive length , and it is not difficult to check that @xmath384 has the desired properties . when @xmath416 we similarly construct @xmath384 by removing intervals of length @xmath417 with sufficiently small ( and fixed ) @xmath365 at @xmath411-th step . then by the proof of ( * ? ? ? * theorem a ) and ( * ? ? ? * theorem 3.1 ) , @xmath384 is not @xmath400-removable for any @xmath418 . the removability of @xmath384 for @xmath419 comes from ( * ? ? ? * proposition 2.1 ) again . * step 1 : the construction of @xmath384 . * the set @xmath384 is defined as a product set @xmath421 , where @xmath422 is a cantor set of hausdorff dimension less than @xmath151 and @xmath423 is a cantor set with positive lebesgue measure , called a _ fat cantor set_. let us start with the construction of @xmath424 . given a sequence @xmath425 with @xmath426 , we build a symmetric cantor set with @xmath427 as the contraction ratio at step @xmath411 . more precisely , define @xmath428 where @xmath429 $ ] and @xmath430 with @xmath431 are defined iteratively as follows : when @xmath432 $ ] has been defined , let @xmath433 $ ] and @xmath434 $ ] . this is well - defined as @xmath435 . then we set @xmath436 for the fat cantor set @xmath437 , likewise we associate it with a sequence of positive real numbers @xmath438 such that @xmath439 where @xmath440 , and @xmath427 are from the previous paragraph . clearly @xmath441 as @xmath442 . the numbers @xmath443 denote the lengths of the disjoint open intervals removed from the unit interval . to be more specific , we define the approximating sequence @xmath444 with respect to @xmath443 in the following way . let @xmath445 $ ] . then iteratively , at step @xmath105 , we replace one of the largest remaining intervals @xmath446 $ ] of the set @xmath447 by the set @xmath448 \cup \left[a+r + \beta_n , b\right ] , \quad \text{where } \quad r = \frac12\left(b - a-\sum_{i = n}^\infty \beta_i\right)\ ] ] and obtain @xmath449 in this way . we claim that , there is always one interval in @xmath450 that has length strictly larger than @xmath451 . if so , then @xmath437 is well - defined . indeed when @xmath452 our claim follows immediately . then under the induction assumption that there is an interval @xmath453\subset f_{n-1}$ ] satisfying @xmath454 , we further have that at the @xmath105-th step by there is an interval with length @xmath455 where the last inequality comes from the induction assumption . therefore the largest interval in @xmath450 has length strictly larger than @xmath451 . thus our claim follows . moreover the length of the largest remaining interval in @xmath449 goes to zero as @xmath456 by . thus @xmath437 is a topological cantor set . the fact that @xmath457 comes from . + * step 2 : the unremovability of @xmath384 for @xmath458 . * fix @xmath459 and a set @xmath460 from the step 1 , with the sequence @xmath461 to be determined later . let @xmath386 . we define a function @xmath462 such that it can not be extended to @xmath463 . to do this , we first construct a function @xmath464 with @xmath465 . let @xmath466 if @xmath467 and @xmath468 if @xmath469 . for each @xmath470 further define @xmath471 where @xmath472 then for @xmath470 , @xmath473 is a cantor step function with respect to @xmath89 if we extend it continuously . in figure [ fig : strips ] we give an example of such a function . next we define @xmath473 for @xmath474 . for @xmath475 ^ 2\setminus e$ ] and @xmath476 we also set @xmath471 where @xmath472 then for fixed @xmath477 , on the horizontal line @xmath478 we have already defined the function @xmath10 up to finitely many open intervals . we then simply define @xmath10 as an affine function on each remaining interval so that it is continuous on this line . then @xmath10 is defined in @xmath479 , and the set @xmath480 has lipschitz boundary . we claim that @xmath10 is also continuous in @xmath479 . indeed if @xmath481 , then by definition @xmath10 is locally constant and hence certainly continuous . for the remaining case where @xmath482 , there is an open interval @xmath483 such that @xmath484 , @xmath485 , and for every @xmath486 the function @xmath487 is lipschitz with the constant depending only on @xmath488 ( as @xmath384 is already fixed ) . then for any such an @xmath89 , in the vertical direction @xmath10 is also continuous since the affine - extension is done with respect to domains where @xmath10 is locally constant and whose boundary is a @xmath151-lipschitz graph . consequently , @xmath10 is even locally lipschitz . hence @xmath10 is a continuous function . for @xmath489 . on the @xmath89-axis the function @xmath10 is defined as the cantor step function . the constant regions are extended to the complement as shown by the gray areas . for any horizontal line there are finitely many open intervals where the function is not defined by the previous extension . on each such interval we extend it as an affine function . we then estimate @xmath490 separately on each horizontal strip @xmath491.,scaledwidth=90.0% ] we next estimate the sobolev - norm of @xmath10 . first up to a suitable translation we consider @xmath10 in a strip @xmath491 which is defined as @xmath492,\ ] ] and is a part of @xmath479 ( up to a suitable translation ) . also recall that @xmath493 then each @xmath491 minus the triangles where the function is defined as constant has at most @xmath414 connected components @xmath494 . up to another translation , each component @xmath494 equals @xmath495 and up to adding a constant the function @xmath10 restricted on it is defined as @xmath496 see figure [ fig : stripf ] . thus @xmath497 in the strip @xmath491 . since each of the @xmath414 components @xmath494 have width and height comparable to @xmath498 , we get @xmath499 consists of @xmath414 connected components outside the regions where @xmath500 was initially defined as constant . such a component @xmath494 is drawn here in the case @xmath489 . also the choice of the coordinates used in the estimate is indicated in the figure.,scaledwidth=45.0% ] let us recall that @xmath501 . this implies that we only have copies of @xmath491 in @xmath502 with @xmath503 . consequently we have @xmath504 by hlder s inequality and the fact that @xmath384 is compact , it suffices to check the non - removability for the case @xmath505 . choose @xmath427 in such a way that @xmath506 for all @xmath411 large enough . that is , @xmath507 observe that @xmath508 with the constant independent of @xmath411 . with this choice @xmath509 therefore by we conclude that @xmath510 . by letting @xmath511 where @xmath512 has support in @xmath513 ^ 2 $ ] and satisfies @xmath514 for @xmath515 ^ 2 $ ] , we have @xmath516 . however @xmath517 can not be extended to a function in @xmath518 indeed , by the sobolev embedding theorem for @xmath519 the precise representative of an extension @xmath520 would continuous , while by definition the extension of @xmath517 is a cantor function ( multiplied by a smooth function ) when restricted to @xmath478 for @xmath521 with @xmath522 . this would contradict the fact that the precise representative of a sobolev function is absolutely continuous along almost every line parallel to the coordinate axes ; see @xcite . + * step 3 : the removability of @xmath384 for @xmath523 . * we claim that , for the set @xmath384 defined above , for every two points @xmath524 there is a curve @xmath525 such that @xmath526 if so , then by ( * ? ? ? * theorem 1.1 ) ( or by @xcite ) , we conclude that any function in @xmath527 can be extended to @xmath463 . since the lebesgue measure of @xmath384 is zero , it follows that @xmath528 and hence @xmath384 is removable for @xmath463 . now let us show the claim . we only consider the case where @xmath529 ^ 2 $ ] , as the general case can be easily reduced to it . for any @xmath529 ^ 2\setminus e$ ] , we write @xmath530 and @xmath531 . first we may assume that @xmath532 . indeed if @xmath533 then @xmath534 . then there is a removed interval @xmath535 $ ] ( in the construction of @xmath424 ) containing @xmath536 . find a point @xmath537 such that @xmath538 the existence of such an @xmath89 follows from the triangle inequality . let @xmath539 . next since @xmath437 is topologically a cantor set , as @xmath533 one can find a point @xmath540 such that @xmath541 and @xmath542 . then the curve consisting of the two line segments @xmath543 $ ] and @xmath544 $ ] satisfies @xmath545\cup [ w_1,\,z_1 ' ] } \dist(z,\,e)^{\frac 1 { 1-q}}\ , ds(z ) \lesssim \int_{0}^{|z_1-z_2| } t^{\frac 1 { 1-q}}\ , dt+|z_1-z_2| ^{\frac { 2-q } { 1-q}}\lesssim |z_1-z_2| ^{\frac { q-2 } { q-1}},\ ] ] with the constant depending only on @xmath400 . we may also apply a similar argument for @xmath38 . thus our assumption is legitimate . under such an assumption we are going to construct the curve connecting @xmath546 . recall that @xmath508 and @xmath547 . then there is a natural number @xmath105 such that @xmath548 . notice that there is an interval @xmath549 such that @xmath550 with the constant depend only on @xmath389 by the cantor construction . denote by @xmath551 the middle point of such an interval . let @xmath552\cup[(x_0,\,y_1),\,(x_0,\,y_2)]\cup [ ( x_0,\,y_2),\,z_2]$ ] be the curve joining @xmath546 and consisting of three line segments ; see figure [ fig : curve ] . we show that @xmath76 is the desired curve . ( up to a negligible error near the end - points ) with a curve consisting of a vertical part @xmath553 $ ] and two horizontal parts @xmath554 $ ] , @xmath555 $ ] . the desired estimate on the vertical part comes from the almost self - similarity of the cantor set @xmath424 with dimension strictly less than @xmath151 , whereas for the horizontal parts we have to make a bit more careful estimate.,scaledwidth=35.0% ] in fact for the vertical part @xmath556 $ ] , as @xmath551 is the middle point of @xmath557 with @xmath558 and @xmath508 , we have @xmath559 } \dist(z,\,e)^{\frac 1 { 1-q}}\ , ds(z)\lesssim |z_1-z_2| ^{\frac { q-2 } { q-1}},\ ] ] with the constant depending only on @xmath389 and @xmath400 . hence it suffices for us to consider the horizontal ones . first of all @xmath560 } \dist(z , e)^\frac{1}{1-q } \ , ds(z ) & \lesssim \sum_{i = n}^\infty 2^{i - n } \int_0^{p_i}t^\frac{1}{1-q}\ dt \lesssim \sum_{i = n}^\infty 2^{i - n}p_i^{\frac{q-2}{q-1}}\\ & \lesssim |z_1-z_2|^\frac{q-2}{q-1}\sum_{i = n}^\infty 2^{i - n}\left(\frac{p_i}{p_n}\right)^{\frac{q-2}{q-1}}.\end{aligned}\ ] ] therefore we are left with estimating the last sum in the above expression . this sum is bounded from above independently of @xmath105 , since @xmath561 where we have used the assumption @xmath562 to have convergence of the last sum via the fact that @xmath563 the estimate for @xmath564 $ ] is similar . hence we have shown the claim , and then the second part of the theorem follows . let @xmath565\cup ( -1,\,2)^2 \times ( 0.5,\,1):=((-1,\,2)^2\setminus e ) \times ( 0,\,0.5 ] \cup ( -1,\,2)^2 \times ( 0.5,\,1),\ ] ] where @xmath566 is compact and removable for @xmath382 for all @xmath385 but not for @xmath378 . such a set @xmath384 exists by theorem [ remove ] ( scale and translate if necessary ) . note that removability is a local question . namely @xmath384 is removable for @xmath378 if and only if for each @xmath402 there is @xmath578 such that @xmath579 see e.g. @xcite . hence if @xmath580 can be approximated by @xmath581 in the @xmath378-norm with @xmath582 , then by fubini s theorem and the fact that @xmath384 is removable for @xmath583 , for almost every @xmath572 we get a sequence , denoted by @xmath584 , approaching some @xmath585 in @xmath586 . note that @xmath585 coincides with @xmath10 on @xmath85 . this then contradicts the unremovability of @xmath384 since we chose @xmath10 arbitrarily ; notice that @xmath384 has @xmath370-lebesgue measure zero . we finally show that @xmath1 is homeomorphic to a ball via a locally bi - lipschitz map . towards this , for @xmath587 define @xmath588 ) ) \ ] ] for @xmath589 then @xmath590 is locally bi - lipschitz , and @xmath590 is a homeomorphism as it fixes the first two coordinates and is a homeomorphism with respect to the third one . moreover , @xmath591 is a lipschitz domain as the bottom of @xmath1 is mapped to a square in the @xmath592-plane and @xmath590 bi - lipschitz on the rest of the boundary of @xmath91 hence there is another ( locally ) bi - lipschitz homeomorphism @xmath593 mapping @xmath591 onto the unit ball . letting @xmath594 we conclude that @xmath1 is locally bi - lipschitz homeomorphic to a ball . l. ambrosio , n. fusco , d. pallara , _ functions of bounded variation and free discontinuity problems_. oxford mathematical monographs . the clarendon press , oxford university press , new york , 2000 . z. m. balogh , s. m. buckley , _ geometric characterizations of gromov hyperbolicity_. invent . * 153 * ( 2003 ) , no . 2 , 261301 .

we prove that for a bounded domain @xmath0 which is gromov hyperbolic with respect to the quasihyperbolic metric , especially when @xmath1 is a finitely connected planar domain , the sobolev space @xmath2 is dense in @xmath3 for any @xmath4 . moreover if @xmath1 is also jordan or quasiconvex , then @xmath5 is dense in @xmath3 for @xmath4 .

the existence of ultra high - speed stars in the halo of the milky way was recognized by hills ( 1998 ) as an inevitable consequence of the presence of a supermassive black hole ( smbh ) in a region of high stellar density such as the galactic center . hills estimated that the disruption of tightly bound binaries in the vicinity of the black hole may propel stars to speeds exceeding @xmath6 km / s , well beyond the escape velocity from the milky way . the discovery of a population of `` hypervelocity '' stars ( hvs , for short , brown et al . 2005 ) in radial velocity surveys of faint b - type stars in the galactic halo ( brown et al . 2006 , 2007a , b ) gave strong support to hills proposal and led to substantial theoretical interest in the topic . the latest compilation by brown , geller & kenyon ( 2008 ) lists 16 hvs , all receding at speeds exceeding @xmath7 km / s and reaching up to @xmath8 km / s in the galactic rest frame . hvs are widely thought to owe their extreme velocities to hills mechanism , since many of the stars appear to be short - lived , massive main sequence stars such as those often found near the galactic center . other mechanisms also capable of accelerating stars to high speeds , such as binary disruption in stellar clusters , have usually been disfavored on the grounds that the maximum ejection velocity is unlikely to exceed the escape velocity at the surface of a star , and thus would only be able to accelerate stars up to no more than a few hundred km / s . the mediation of the much deeper potential well of a supermassive black hole thus appears needed to explain the extreme velocities observed for the hvs population . although some hvs have very likely been ejected by the galactic smbh , there are a number of issues that remain unresolved in this scenario . for example , the velocity distribution of hvs and , in particular , the lack of _ very _ high - velocity stars ( i.e. , with speeds exceeding @xmath6 km / s ) , is not easily understood ( see , e.g. , sesana , haardt & madau 2007 ) . nor is the existence of at least one hvs moving at a speed greater than @xmath9 km / s in the galactic rest frame ( hd 271791 ; heber et al 2008 ) whose proper motion apparently rules out a galactic center origin . although small number statistics may explain away these concerns , we examine in this letter further peculiarities in the spatial distribution and kinematics of hvs and argue that these suggest , at least for some hvs , an origin different from the smbh - ejection mechanism . we begin by examining the distribution of travel times and the angular clustering of hvs compiled from the literature , and use numerical simulations to explore an alternative mechanism able to populate the galactic halo with high - speed stars . we end by considering several observational tests that may , in the future , be used to discriminate between these competing scenarios . we use primarily data from the brown et al ( 2006 , 2007a , b , hereafter bgkk , for short ) mmt and whipple surveys of b - type halo stars selected from the sloan digital sky survey . our analysis uses galactic rest - frame radial velocities , positions in the sky , and distances , as given by the authors . we note that distance estimates are only listed for stars with velocities exceeding @xmath10 km / s . these estimates typically assume that the stars are in the main sequence , although it is difficult to rule out that some ( or many ) are evolved stars in the blue horizontal branch ( bhb ) . this is one of the main systematic uncertainties in our analysis , but it is a shortcoming shared with all other attempts to study the kinematics of hvs stars with current datasets . the left panel in figure [ figs : vr_r ] shows the galactic rest frame radial velocity distribution of all stars in the bgkk survey , whereas the right panel shows the radial velocity of all stars with published distance estimates . these are all high - speed stars ( @xmath11 km / s ) out in the tail of the radial velocity distribution ( filled symbols ) . open symbols correspond to the @xmath12 new stars recently reported by brown , geller & kenyon ( 2008 , hereafter bgk08 ) and are added here only for completeness , since data for the underlying survey from which these stars have been identified are not yet publicly available . ( `` dotted '' symbols identify stars in the circle shown in fig . [ figs : brown3 ] , as discussed below . ) for illustration , we have added in figure [ figs : vr_r ] the position in the @xmath13-@xmath14 plane of several milky way satellite companions , as labeled . for each star with published distance estimate and radial velocity we compute a travel time from the galactic center , assuming purely radial orbits in the simple galactic potential model of bromley et al ( 2006 ) . this is given by a spherically symmetric density profile , @xmath15 , where @xmath16pc@xmath17 and @xmath18 pc ( kenyon et al 2008 ) . the thick solid curves in fig . [ figs : vr_r ] represent the loci of stars with constant travel time ( labeled in myr ) , whereas the dotted lines show the escape velocity from the milky way , assuming it is embedded within a cold dark matter halo with virial velocity ) and define the virial velocity as the circular velocity within a radius , @xmath19 , enclosing a mean overdensity of @xmath1 times the critical value for closure . the virial velocity defines implicitly the halo mass : for @xmath20 km / s , the mass within @xmath21 kpc / h , is @xmath22/h . we assume h@xmath23 throughout the paper . ] @xmath20 km / s and @xmath24 km / s , respectively . the hvs data in figure [ figs : vr_r ] is colored according to travel time , binned in equally spaced logarithmic intervals ( see also the histogram in the inset of fig . [ figs : brown3 ] ) . figure [ figs : vr_r ] illustrates a few interesting points . one is that very few stars are actually unbound if the virial velocity of the milky way is of the order of @xmath25 km / s , the circular speed at the solar circle . this is actually what is required by recent models of galaxy formation in order to match simultaneously the zero - point of the tully - fisher relation and the normalization of the galaxy luminosity function ( see , e.g. , croton et al 2006 for a full discussion ) . in other words , for @xmath20 km / s , the hvs speeds are unusually high , but not necessarily extreme , and only one hvs would be clearly unbound . this is important , since it suggests that more prosaic dynamical effects that do not rely on smbh ejection may be responsible for the unusual speeds of the hvs population . the second point to note is that the distribution of hvs travel times is not uniform . this is illustrated in the histogram shown in the inset of fig . [ figs : brown3 ] : 19 of the 30 hvs in the northern hemisphere seem to share a common travel time ( see bin centered at @xmath26 km / s ) . the peak in the histogram is higher than expected from a model where ejections occur uniformly in time . the model prediction is shown by the solid curve in the inset histogram , after taking into account the volume surveyed , the finite lifetime of late b - main sequence stars , and assuming the distribution of smbh - ejection velocities computed by bromley et al ( 2006 ) . the significance of the peak seems high : selecting at random 30 stars from such model yields a peak with 19 ( or more ) stars in fewer than 1 in 1000 trials . the small number of objects involved precludes a more conclusive assessment of this possibility , but we note that this is not necessarily inconsistent with smbh ejection . in this scenario , a `` preferred travel time '' may just reflect a burst of star formation that boosted the population of binaries in the galactic center region a few hundred myr ago . we discuss a different interpretation below . [ figs : brown3 ] shows an aitoff projection in galactic coordinates of all stars in the bgkk survey . high - speed stars ( including those reported recently by bgk08 ) are shown in color , coded according to their travel times , as shown in the bottom - left inset . the main point to note in this figure is that the hvs population is not uniformly distributed in the sky , and that its angular distribution appears inconsistent with a random sample of stars in the survey . one way of quantifying this is to focus on the region around the constellation of leo , which seems to contain the majority of hvs . we choose for this a 26@xmath27-radius circle centered at the position of leo i , itself a high - speed ( possibly unbound ) distant satellite companion of the milky way ( e.g. sales et al . 2007a , and mateo , olszewski & walker 2008 ) . a total of @xmath28 stars in the bgkk survey fall within the circle , or @xmath29 of the total of @xmath30 halo stars in the northern section of the bgkk survey . by contrast , @xmath31 of all stars with @xmath32 km / s fall within the vicinity of leo i , a proportion that rises to @xmath33 for stars with @xmath34 km / s . we may estimate the significance of this result by repeating the same calculation after randomly reassigning the measured radial velocities amongst the stars in the survey . a proportion at least as high as @xmath31 of @xmath32 km / s stars falls within the circle in fewer than @xmath35 in @xmath36 such random trials in @xmath37 for @xmath32 km / s stars , and decreases even further if the center of the circle is moved from leo i in the direction of leo iv . these estimates assume that the stars in the bgk08 and bgkk parent surveys have similar distributions in the sky . ] . for @xmath34 km / s stars , the corresponding number is fewer than @xmath35 in @xmath38 trials . also , since the majority of hvs are at galactic longitude @xmath39 ( see fig . [ figs : brown3 ] ) it is unlikely that the hvs anisotropy is the result of the slightly larger effective volume surveyed in the direction of the anti - galactic center . this supports the idea that the enhanced clustering of hvs in the direction of leo is real . the observed angular clustering is not easily explained in the smbh - ejection scenario , which predicts an hvs population distributed approximately isotropically across the sky . the anisotropic distribution and the preferred travel time discussed in the preceding subsections suggest that at least part of the hvs population may have a different origin from the smbh - ejection mechanism . we explore here an alternative scenario where these two peculiarities are reproduced naturally . our proposal envisions high - velocity stars in the halo as a result of the tidal disruption of a dwarf galaxy in the galactic potential . this is illustrated in fig . [ figs : vallstr ] , where we show the position of `` star '' particles in a cosmological simulation of the formation of a galaxy in the @xmath40cdm cosmogony . the simulation is one in the series presented by steinmetz and navarro ( 2002 ) and analyzed in detail in abadi , navarro & steinmetz ( 2006 ) , where we refer the interested reader to for further technical details . [ figs : vallstr ] shows the position of all stars in the @xmath13-@xmath14 plane , after rescaling the system to a virial velocity of @xmath20 km / s . this scaling allows us to compare the results directly with observations of the milky way . as in fig . [ figs : vr_r ] , the dotted lines indicate the escape velocity of the system , and the solid curves indicate loci of constant travel time . the data in fig . [ figs : vallstr ] are shown at a snapshot chosen a couple of hundred myr after the final merger / disruption of a dwarf satellite in the potential of the main galaxy . the stars of the disrupted satellite are shown in red in order to distinguish them from the rest of the stars in the simulated galaxy , shown in black . note that a long stream of stars formerly belonging to the satellite follow closely a line of constant travel time . these are stars that were stripped from the dwarf during its last pericentric passage , just before its main body merged with the central galaxy . the travel - time coherence in the stream ( or `` tidal tail '' ) results from the fact that all those stars were stripped from the dwarf at the same time . stars along the stream have different energies , but it is clear that the disruption event has been able to push some stars into very high - velocity orbits , some even exceeding the local escape speed from the system . > from the perspective of an observer near the center of the galaxy ( such as an observer at the sun in the milky way ) the stream would be seen projected onto a well - defined direction in the sky . this is shown in fig . [ figs : aitoff ] , which shows a projection on the sky of all stars in the halo of the simulated galaxy ( i.e. , excluding stars with distances less than @xmath41 kpc from the center ) . the points in color correspond to the tidal debris from the disrupting dwarf . in particular , blue dots highlight those with radial velocity exceeding @xmath42 km / s . note that , from the sun s perspective , the `` tidal tail '' of high velocity stars stripped from the dwarf falls within a well - defined region spanning a few tens of degrees in the sky . [ figs : vallstr ] and [ figs : aitoff ] thus illustrate that tidal disruption of a dwarf galaxy is able to push stars to speeds high enough to escape the galaxy , and provides a mechanism to populate the halo with high - velocity stars with kinematic and angular clustering peculiarities resembling those of the hvs population discussed above . this suggests that at least some stars of the hvs population may have originated in the recent disruption of a dwarf galaxy . how can this rather unorthodox proposal be validated / ruled out ? fortunately , the most natural predictions of this scenario differ significantly from those motivated by the smbh - ejection mechanism . in particular , a broader search for high - speed stars should confirm the enhancement of hvs in the constellation of leo . high - speed stars should also exist amongst all spectral types , and should contain a significant number of evolved stars and stars of low metallicity , which should make up the bulk of the stellar content of the dwarf . with enough statistics , a radial - velocity angular gradient amongst high - speed stars might also be observed , such as that shown in fig . [ figs : aitoff ] . the tail should also contain returning stars with negative velocity that are relatively nearby ( see , e.g. , fig . [ figs : vallstr ] ) so it would be worth checking existing surveys of nearby stars for unusual velocity patterns in the same general direction in the sky . we note that our argumentation rests on the assumption that the halo of the milky way is relatively massive ( i.e. , @xmath43 km / s ) . this is suggested by semianalytic galaxy formation models and is consistent with timing argument mass estimates for the milky way ( li & white 2007 ) . there is evidence , however , that the virial velocity of the galactic halo might be considerably smaller ( @xmath44 km / s ) , as indicated , for example , by recent analysis of milky way satellite data ( sales et al 2007b ) , of rave data for solar neighborhood stars ( smith et al 2007 ) , and of segue data on halo stars ( xue et al 2008 ) . this issue remains unresolved , but should a low mass for the milky way halo be confirmed , it would mean that a large fraction of hvs would be truly unbound and would argue against the tidal debris interpretation proposed here . although examining further a tidal debris origin for hvs appears worthwhile , it should be recognized that this proposal is not without shortcomings . one of the main difficulties is to explain why a tidal stream emanating from a dwarf should contain short - lived , massive main - sequence stars , unless the accreting dwarf was gas - rich and underwent a burst of star formation , perhaps triggered by pericentric passages prior to its final disruption . gas - rich dwarfs tend to be rather massive , and careful modeling is needed in order to examine whether the timescales really work out and to explain why the remains of the putative dwarf have escaped detection so far . systematic radial - velocity surveys of large numbers of faint stars across vast regions of the sky , such as those being planned by the gaia satellite , will undoubtedly be able to settle these questions in the foreseeable future . mga acknowledges laura v. sales for a careful reading and for comments on an early draft of this paper . we thank the anonymous referee for a constructive report and acknowledge useful discussions with warren brown during the kitp conference `` back to the galaxy ii '' . this research was supported in part by the national science foundation under grant no . phy05 - 51164 .

halo stars with unusually high radial velocity ( `` hypervelocity '' stars , or hvs ) are thought to be stars unbound to the milky way that originate from the gravitational interaction of stellar systems with the supermassive black hole at the galactic center . we examine the latest hvs compilation and find peculiarities that are unexpected in this black hole - ejection scenario . for example , a large fraction of hvs cluster around the constellation of leo and share a common travel time of @xmath0-@xmath1 myr . furthermore , their velocities are not really extreme if , as suggested by recent galaxy formation models , the milky way is embedded within a @xmath2 dark halo with virial velocity of @xmath3 km / s . in this case , the escape velocity at @xmath4 kpc would be @xmath5 km / s and very few hvs would be truly unbound . we use numerical simulations to show that disrupting dwarf galaxies may contribute halo stars with velocities up to and sometimes exceeding the nominal escape speed of the system . these stars are arranged in a thinly - collimated outgoing `` tidal tail '' stripped from the dwarf during its latest pericentric passage . we speculate that some hvs may therefore be tidal debris from a dwarf recently disrupted near the center of the galaxy . in this interpretation , the angular clustering of hvs results because from our perspective the tail is seen nearly `` end on '' , whereas the common travel time simply reflects the fact that these stars were stripped simultaneously from the dwarf during a single pericentric passage . this proposal is eminently falsifiable , since it makes a number of predictions that are distinct from the black - hole ejection mechanism and that should be testable with improved hvs datasets .

ob stars ( which as a group include stars of spectral type between o3 and b2 ) are the most massive main sequence objects . they are generally found in young clusters and associations , because their lifespans are too short to carry them far from their birthplaces . consequently , they are often subjected to considerable interstellar absorption . wolf - rayet ( wr ) stars are believed to be the evolved counterparts of at least some ob stars , and mostly correspond to a phase where the massive star has lost all its external hydrogen envelope through stellar winds ( conti 1976 ) . wolf - rayet stars generate intense stellar winds reaching rates of @xmath1 m@xmath2 yr@xmath3 and terminal speeds @xmath4 km s@xmath3 . the wolf - rayet photosphere arises not from the hydrostatic surface but in the wind itself which , besides he lines , is dominated either by ions of nitrogen ( wn stars ) or carbon and oxygen ( wc stars ) . hence the spectrum of a wr star is dominated by very broad lines of he and heavier elements . current estimates indicate that at least @xmath5 of wr stars are in multiple systems ( moffat 1995 ) . the presence of a companion can be suspected from a composite spectrum showing ob absorption lines or from the apparent `` dilution '' of the wr emission lines by extra continuum emission ( smith _ et al . confirmation of multiplicity is usually based on radial velocity studies . indirect confirmation can also be made by the detection of colliding wind effects ( e.g. tuthill _ et al . _ 1999 , williams 1999 ) . a small number of ob companions in very long period ( @xmath6 yr ) systems have also been found from speckle interferometry or direct imaging ( hartkopf _ et al . _ 1993 , williams _ _ 1997 , niemela _ et al . these systems are particularly interesting because the absolute magnitude of the wr component can be directly inferred from that of its companion , assuming the two stars are at the same distance . in the ir and radio regimes , wr stars are dominated by thermal , free - free emission from the dense , expanding envelope . however , @xmath7% of wr stars have been observed to have a radio spectral energy distribution consistent with non - thermal emission associated with synchrotron radiation and high - energy phenomena . eichler & usov ( 1993 ) have demonstrated how non - thermal radio emission could arise from the collision between the outflows from two early - type stars in a binary system . as it turns out , binary systems are over - represented in the sample of non - thermal emitters , which prompted van der hucht ( 1992 ) to suggest that all non - thermal wr emitters were actually in binary systems , with the wind collision being responsible for the non - thermal emission . the colliding wind model has been strikingly confirmed by radio observations of two non - thermal wr emitters ( williams _ et al . _ 1997 , dougherty _ 1996 ) , confirmed to be in binary systems with ob companions ( niemela _ et al _ 1998 ) . the non - thermal emission is found to be associated with a distinct region whose photocenter is located _ on or close to the line joining the wr star and its ob companion _ , the wr star itself being associated with a thermal source . however , not all wr+ob binaries are found to be non - thermal emitters . dougherty & williams ( 2000 ) have noted that wr+ob systems form two distinct groups , with most thermal emitters being in short period systems , while the non - thermal emitters are all in long period systems , with large component separation . since the non - thermal emission arises locally , at the wind - wind collision front , one might expect to observe non - thermal radio emission only from widely separated wr+ob binaries , whose large collision fronts are expected to be located well outside the extended radio photosphere of the wr wind . in short period binaries , the collision front occurs much deeper in the wind of the wr star , in which case non - thermal radio emission is not expected to be observed as the collision front lies inside the wr radio photosphere . more recently , the conjecture that all non - thermal emitters are colliding wind binaries has been put in doubt by wallace _ et al . _ ( 2000 ) , who have failed to identify companions for 8 wr stars with non - thermal radio emission , down to a mean projected separation of @xmath820 au . since this scale is within the distance of the radio photosphere , either non - thermal radio emission in these stars arises farther out in the wind itself ( possibly generated by intra - wind shocks ) , or non - thermal radio emission does arise in colliding winds but the radio photosphere is much smaller than predicted . if the latter hypothesis is correct though , all the apparently single non - thermal radio emitters should have companions on close orbits , and should have been identified as spectroscopic binaries . the single - star explanation remains to be confirmed . the colliding wind model , however , is successful in explaining non - thermal emission in long - period wr+ob systems . but are all long - period systems non - thermal emitters ? so far , there is one exception to the rule : the long period wr+ob system wr86 . however , while wr86 is not strictly defined as a non - thermal emitter , its spectral index is consistent with a `` composite '' thermal / non - thermal source , i.e. it is consistent with weak non - thermal emission ( dougherty & williams 2000 ) . this paper presents spatially resolved stis spectra of the components in the visual wr+ob systems wr86 , wr146 , and wr147 . the observations and data reduction are discussed in section 2 . spectral classification of the components is presented in section 3 . this allows us to estimate both the absolute luminosity of the components , and the mass - loss rates of the ob stars . results are discussed in the light of the colliding wind model in section 4 . we briefly summarize our findings in section 5 . long - slit spectroscopy of the stars wr86 , wr146 , and wr147 have been obtained with the stis spectrograph on board the _ hubble space telescope_. in each case , observations were carried out with the slit length oriented as close as possible to the apparent position angle of the binaries . ( 1998 ) have estimated the position angles to be pa=109@xmath99@xmath10 , 21@xmath94@xmath10 , and 350@xmath92@xmath10 for wr86 , wr146 , and wr147 , respectively . the exact orientation of the stis slit depends on the orientation of hst at the time of the observation ; hence the above pas were used as scheduling constraints . observations were finally carried out with pa=106.1@xmath10,18.8@xmath10 , and 360.2@xmath10 , for wr86 , wr146 , and wr147 , respectively . in each case , the orientation was such that the brightest component in the v band appeared on the larger ccd column number . the width of the 52@xmath110.5 arcsec slit is on the order of , or larger than , the separation between the components ; hence the small difference ( @xmath12 ) between the orientation angle at the time of observation and the pas of the systems has negligible effects on the throughput . the two stars in the wr147 system are clearly resolved by stis ; we used standard iraf aperture extraction to obtain their spectra . on the other hand , the relatively broad wings in the stis point spread function ( psf ) resulted in the wr86 and wr146 spectra being resolved , but with significant blending ( figure 1 ) . to complicate matters , the shape of the psf was found to be dependent on the wavelength , to a level that significantly affects the spectral extraction of barely resolved sources like those from wr86 and wr146 ( but is however of little consequence for clearly resolved sources such as in wr147 ) . we performed a multidimensional fit for each column on the ccd , to extract a blended double profile @xmath13 with the general shape : . we used this form for its simplicity and stability under the multi - dimensionnal fit . we did attempt to use other simple mathematical forms to describe the psf , e.g. moffat ( 1969 ) functions , only to obtain equivalent results in the spectral extraction . ] @xmath14 where @xmath15 and @xmath16 are the spatial locations of the stars along a column , and @xmath17 and @xmath18 are the amplitudes of the profiles for each star . the shape of the stellar profiles is governed by the dispersion parameter @xmath19 and the `` peakyness '' parameter @xmath20 ( which sets the relative strength of the psf wings ) . because the mean dispersion @xmath19 of the psf is found to be close to the pixel size , we re - sampled the profile over individual pixels @xmath21 from the continuous function @xmath13 using : @xmath22 where @xmath21 are integers representing the ccd lines . we then used the assumption that the separation between the stars should be constant and independent of the wavelength . to determine the separation between the stars , we first performed a fit of @xmath23 in the 6-dimensional parameter space @xmath24 and calculated the separation @xmath25 between the stars from the mean values of @xmath26 obtained from each ccd column . we then fixed @xmath27 and performed a second ( more robust ) fit in the 5-dimensional space @xmath28 , for each ccd column . using the derived values for @xmath25 and the stis pixel size of 0.0507 arcsec pixel@xmath3 , we estimate the separation along the slit between the components of wr86 , wr146 and wr147 to be 0.239@xmath290.006@xmath30 , 0.161@xmath290.005@xmath30 , and 0.624@xmath290.015@xmath30 , respectively . the pa of the slit was always within 13@xmath10 of the line joining the two stars , hence these values should be reliable estimates of the component separations . our values for wr146 and wr147 are consistent with the separation calculated from the wfpc2 image by niemela _ ( 1998 ) . for wr86 , our derived separation is outside of 1@xmath19 of the 0.286@xmath290.039@xmath30 range derived by niemela _ _ , but well within the 0.230@xmath290.013@xmath30 range cited in the hipparcos catalog . results of the multidimensional fits confirm the existence of a systematic variation in the psf pattern with wavelength ( figure 2 ) . the pattern also differs between wr86 and wr146 , despite the same spectral coverage , which suggests that the shape of the psf also depends on where the source falls along the slit ( the wr86 and wr146 systems have been recorded on slightly different ccd columns ) . this raises the possibility that the psf may be slightly different for each component in any one system , which may result in some inaccuracies in the spectral extraction . the two stellar components in both wr86 and wr146 were apparently separated reasonably well across most of the spectral range . the only exception occurs at wavelengths close to the very bright wr emission line civ @xmath315808 ( at which point the wr star is significantly brighter than the ob star ) , where a small contamination of the wr flux onto the ob companion spectrum apparently occurred . this is clearly due to an imperfect psf model . in particular , the wings of the psf showed evidence for a weak diffraction pattern , which our model does not reproduce . this results in some contamination of one spectrum by some light from the other component . however , this contamination is apparent only for the broad civ emission line , because of the large difference in the brightness of the wr and ob components at that wavelength , and is negligible elsewhere ( below instrumental noise levels ) . fortunately , this contaminated civ emission from the wr component can be unambiguously identified in the ob star spectra , whose spectrum does not normally exhibit such a broad emission feature at that wavelength . because the spectra are generally well separated , and because contamination effects are relatively small , we did not attempt to refine the psf model further , which would have required the use of extra parameters and would have made the multi - dimensional fit much more difficult to perform . the resolved spectra for all three stars are shown in figure 3 . it turns out that the brightest component in the v band in each pair is the wolf - rayet star , even though the continuum emission from the ob star is actually larger in wr86 and wr146 ( the wr stars in these systems are brighter in v because of their strong emission lines ) . contamination of the ob spectra by the very bright civ @xmath315812 line is very obvious in wr146 ( see figure 5 ) . examination of the civ @xmath315812 contamination profile in the spectrum of the o component shows that contamination becomes apparent as the monochromatic intensity from the wr star reaches @xmath32 times that of the o companion . since the relative wr intensity is below that level over the remainder of the spectrum , we conclude that contamination effects must be negligible over the remainder of the spectral range . hence , any feature observed elsewhere in the spectrum of the o star is assumed to be intrinsic . in wr86 , the components are further apart ; hence the extracted spectra are less susceptible to contamination effects . our extracted spectra of wr86 show evidence for a weak contamination of the civ@xmath315812 line , and also possibly from ciii @xmath315696 ( see figure 7 ) . both wr lines have monochromatic intensity reaching @xmath83 times that of the b star continuum . because these lines are the brightest features in the wr component of wr86 , and because they yield only weak contamination effects , we conclude that no other wr features contaminates the b spectrum significantly , and any other feature observed in the b star spectrum must be intrinsic . in each system , the slopes of the continua from the wr stars and the ob companions as well as the strength of the interstellar absorption lines are all consistent with equal amounts of reddening . this supports the idea that the components in each system are approximately at the same distance , and most likely to be physically related . in both wr146 and wr147 , we confirm that the wr ( o ) component is to the south ( north ) , consistent with the colliding wind interpretation of their radio maps ( see niemela _ 1998 ) . for wr86 , the wr component is to the north - west and the b component to the south - east . we did not find any trace of background , or `` nebular '' emission in the longslit spectral images , within the instrumental limits . attempts have been made to extract spectra at different locations along the slit , but only the wings of the psf from the stellar components and other instrumental features were detected . this lack of detection is significant for wr146 and wr147 , which are known to be colliding wind binaries . if there is any diffuse emission in the optical associated with the colliding wind region , we estimate that it must be weaker than 5 10@xmath33 ergs @xmath34 s@xmath3 @xmath3 arcsec@xmath3 . line identifications for each of the wr and ob star components are listed in tables 1 - 6 . the resolution of the stis spectra was @xmath35 , which is the accuracy in the central wavelength measurements . estimated errors on equivalent width measurements @xmath36 are listed individually . the error on @xmath36 depends on the strength of the line , and on whether it was resolved or in a blend . we do not give the mean central wavelengths of the lines in the wr stars , because the very broad profiles make the central wavelengths very unreliable for line identification . most of the bright wr lines are actually blends of several different lines ; the line identification and rest wavelength given in the tables is for the transition which most probably contributes to the largest part of the line emission . this is the v=9th magnitude system hd156327 , located at @xmath3717 18 23.06 @xmath38 - 34 24 30.6 ( j2000 ) . it was initially listed as a wolf - rayet binary with spectral type wc7+b0 v ( roberts 1962 ; smith 1968 ) , based on the presence of h and hei absorption lines in the blue . it was included in the sixth catalog of galactic wolf - rayet stars ( van der hucht _ et al . _ 1981 ) under the designation wr86 , and given a wc7+abs spectral type , implying that absorption lines could be intrinsic to the wr star , thus questioning its double star status . however , hd156327 was known to be a close visual double with separation @xmath39 ( jeffers _ et al . the fact that massey _ et al . _ ( 1981 ) failed to measure any radial velocity variation ruled out the idea of a _ close _ ob companion , strongly suggesting that the ob spectrum is associated with the visual companion ( see moffat _ et al . _ 1986 ) . in any case , the star continued to be referred to as a wc7+abs throughout the 1980s . the double star status was confirmed with speckle observations by hartkopf _ ( 1993 ) , who resolved the star into two components with a @xmath40 separation . the components were also clearly resolved by the wfpc2 camera on board hst ( niemela _ et al . _ 1998 ) . based on scanned image tube spectra of the pair , niemela _ et al . _ suggested the companion to be a b0 star ( detection of oii , siiii , and siiv ) with a luminosity class between i and iii . the system is now listed in the seventh catalog of wolf - rayet stars ( van der hucht 2001 ) as `` wc7 ( + b0 iii - i ) '' . our stis spectra confirm that the wr component is of subtype wc7 ( figure 4 , with line equivalent widths listed in table 1 ) . the ratio of the equivalent widths of the civ @xmath315801 and ciii @xmath315696 lines is log w@xmath41(civ 5801)/log w@xmath41(ciii 5696 ) @xmath42 0.16 , which is consistent with subtype wc7 in the quantitative classification system of crowther _ ( 1998 ) . for the o component ( figure 5 ; table 2 ) , our spectra confirm the b0 type , based on a comparison of the blue spectrum with the atlas of walborn & kirkpatrick ( 1990 ) . we attempt to better constrain the luminosity class based on the strength of the h@xmath43 absorption line , for which a calibration with the absolute magnitude has been derived by millward & walker ( 1985 ) . we measure @xmath44 the uncertainty being largely attributable to the blend with oii @xmath314349 . following the millward & walker calibration , this corresponds to an absolute magnitude @xmath45 . according to the b - star absolute magnitude calibration of schmidt - kaler ( 1982 ) , this makes the star a b0 giant of class iii . since the wfpc2 photometry shows the two stars to have the same @xmath46 ( within the @xmath47 observational errors ) , then the wr component is also estimated to have @xmath45 . this value is largely consistant with the range of values quoted by van der hucht ( 2001 ) for wc7 stars . luminosity classes of early b stars can also be estimated from the ratio of siiii @xmath315740 and hei @xmath315876 . comparison with the yellow - red atlas of walborn ( 1980 ) shows the spectrum to be generally consistent with luminosity class iii . while we can definitely rule out a class i for this object , it is not possible to rule out spectral class ii on the basis of the siiii @xmath315740 / hei @xmath315876 ratio . however , because the @xmath0 line shows no trace of overlying wind emission ( which occurs in most early b stars with luminosity class i - ii ) , we classify this system as wc7 + b0 iii . this star , located at @xmath3720 35 47.09 @xmath38 + 41 22 44.7 ( j2000 ) , was initially classified as wc6 by roberts ( 1962 ) , and as wc5 by smith ( 1968 ) . it was listed as a wc4 in the sixth catalog of galactic wolf - rayet stars ( van der hucht _ et al . _ 1981 ) . improved measurements of the line ratios led smith _ et al . _ ( 1990 ) to reclassify the wr star as wc6 . dougherty _ et al . _ ( 1996 ) observed the star at 1.6 ghz and 2.5 ghz with the merlin array , and resolved the source into two components , a thermal source and a non - thermal source , separated by @xmath48 milliarcseconds . they attributed the thermal source to the wr star , and the non - thermal source to a colliding - wind region between the wr star and an ob companion . ( 1996 ) also found weak hydrogen absorption lines ( @xmath49 , @xmath50 ) in an unresolved blue spectrum of wr146 , which they attributed to the unresolved companion . optical wfpc2 images from hst clearly resolved the object into two components separated by @xmath51 arcsecs , and with very similar colors ( niemela _ et al . an overlap of the optical images and radio maps showed the non - thermal source to be located between the optical components , confirming the colliding - wind binary hypothesis . assuming that the relative location of the non - thermal source matches the head of the bow shock , it is possible to calculate the ratio of the wind momentum fluxes . for wr146 , niemela _ et al . _ obtained a ratio @xmath52 . given the large mass - loss rate expected from the wc6 star , this requires the companion to have a relatively large mass - loss rate also . based on the momentum ratio and on the relative colors of the components , niemela _ et al . _ suggested the companion to be o6-o5 v - iii . more recently , dougherty _ et al . _ ( 2000 ) obtained a @xmath53spectrum of wr146 at the 4-m william herschel telescope ( wht ) . though blended with emission lines from the wr star , the relatively narrow absorption lines from the o companion were clearly identified . their @xmath314541 heii / @xmath314471 hei equivalent width ratio ( the principal diagnostic of the o - type sequence ) indicated a spectral type o8 . though several features were also suggestive of a high luminosity class , they did not assign a luminosity class due to a lack of the main luminosity diagnostic lines in their spectrum . the system is now listed as wc6+o8 in the seventh catalog of wolf - rayet stars ( van der hucht , 2001 ) . our stis spectra of the wr component ( figure 6 ; table 3 ) confirms the wc6 classification . we measure a line equivalent width ratio log w@xmath41(civ 5801)/log w@xmath41(ciii 5696 ) @xmath42 1.03 , consistent with subtype wc6 in the quantitative classification system of crowther _ et al . _ the lines in this star are especially broad , indicating a large wind terminal velocity ( @xmath54 km s@xmath3 as measured by eenens & williams 1994 ) . for the ob component ( figure 7 ; table 4 ) we measure a line ratio he ii @xmath314541 / he i @xmath314471 = 0.39 which is consistent with a spectral type o9 in the system of conti & alschuler ( 1971 ) and conti ( 1973 ) . the hei @xmath314471 line does look significantly broader than heii , which may be due to noise , but could also result from blending of the hei @xmath314471 with some other unidentified line . on close examination , the hei @xmath314471 profile in the b0 iii component of wr86 does look asymmetric , with the blue side of the line being unusually extended . if this is due to some unidentified is absorption feature ( we do indeed see a shallow absorption trough at the same wavelength in the wr spectrum which could also be the signature of this is feature , see figure 6 ) , then the ew of this line is most likely to be overestimated in wr146 . we consider the hei @xmath314388 line , and note that it is clearly weaker than heii @xmath314540 ; the spectral atlas of walborn & fitzpatrick ( 1990 ) shows that this generally does not occur in o9 type stars , where both lines have about the same ew . it is however consistent with spectral type o8 , and we thus also adopt this classification for the o star in wr146 . the main luminosity diagnostic for o8-o9 stars is the increase in the strength of the si iv @xmath554089 , 4116 at higher luminosities , and the change in niii @xmath554634 , 4640 and he ii @xmath314686 from absorption to emission ( walborn & fitzpatrick , 1990 ) . while the wht spectrum of dougherty _ et al . _ included both si iv lines , they were found to be blended with the @xmath49 line . unfortunately , neither the stis nor the wht spectrum covers the he ii @xmath314686 region . however , we note the presence of a weak emission feature centered on @xmath315696 which we attribute to ciii line emission . both the ciii @xmath315696 and h@xmath56 lines have been shown by walborn ( 1980 ) to behave like niii @xmath55 4634 , 4640 and he ii @xmath314686 , respectively . the h@xmath56 profile in the o star component shows a narrow but relatively shallow absorption trough flanked by relatively broad wings . the h@xmath56 line in the o star is coincident with the heii @xmath57 complex in the wolf - rayet star , which raises the possibility of contamination into the o spectrum , which would explain the broad wings . however , we argue that the broad h@xmath56 emission profile is intrinsic to the o star : ( 1 ) the intensity of the heii @xmath57 complex relative to the o star continuum is comparable to that of the broad shoulder in the civ @xmath58 complex ( between @xmath59 and @xmath60 ) , and we see no obvious contamination of the o star spectrum by the latter , ( 2 ) the h@xmath56 profile in the b0 component in wr86 does not show any broad wings , while it was more prone to being contaminated by the more intense heii @xmath57 complex from its wr component , ( 3 ) the centroid of the h@xmath56 absorption trough is 6.4larger than the measured lab value , which suggests that the central trough is being distorted as the line is filled up with h@xmath56 in emission , and ( 4 ) the overall shape of the h@xmath56 profile is very similar to that of other oif stars ( e.g. cygnus ob#7 and hd210839 see figure 6 in herrero _ _ 2000 ) , including both the central narrow absorption and the broad emission wings . the behavior of ciii @xmath315696 and h@xmath56 in the o component of wr146 clearly rules out any luminosity class fainter than ii . because the h@xmath56 line is not found to be strongly in emission , as in o8 i stars , a spectral type o8 ii is the most reasonable classification . a secondary indicator for the luminosity class is the ratio he i @xmath314388 / he i @xmath314471 , which correlates with the mass - loss rate . our spectrum yields he i @xmath314388 / he i @xmath314471 = 0.2 , which suggests that the star is of . the weakness of h@xmath43 ( @xmath61 ) is also consistent with large intrinsic luminosity ; the millward & walker ( 1985 ) relationship suggests @xmath62 which , for o8 , is consistent with luminosity class i - ii ( schmidt - kaler 1982 ) . from these lines of evidence , we classify this system as wc6 + o8 i - iif . this star , located at @xmath3720 36 43.65 @xmath38 + 40 21 07.3 ( j2000 ) , was resolved into a double radio source with a thermal lobe and a non - thermal lobe by churchwell _ williams ( 1996 ) hypothesized that the non - thermal lobe was the result of a colliding - wind interaction with an unseen companion . the companion was first identified as a faint component in an ir image of wr147 ( williams _ et al . _ 1997 ) , and confirmed in the optical by hst wfpc2 observations ( niemela _ et al . _ 1998 ) . based on the relative k magnitude between the wn8 star and the ir companion ( @xmath63=3.04@xmath290.09 ) , and assuming the wn8 star to have an absolute magnitude m@xmath64 , williams _ _ claimed the ob companion to be consistent with a spectral type b0.5 v. in the optical , the magnitude difference between the two components is observed to be @xmath65=2.16@xmath290.09 , which is too small for a b0.5 v companion , and niemela _ et al . _ ( 1998 ) proposed that the star is of earlier type . our stis data for the wr component in wr147 ( fig . 8 ; table 5 ) shows a spectrum typical of a wn8 star , with a relatively strong p cygni profile in hei @xmath315876 . also present are the weak heii @xmath556311 , 6406 , and 6527 lines all with p cygni profiles . weak nii @xmath555680 , 5686 lines are also detected , consistent with a late - type wn star . our wr147 spectrum in the blue regime is of very poor quality and is not shown in this paper . because of the considerable interstellar reddening , the wr147 system is several magnitudes fainter in the blue , and our hst stis exposures were not programmed appropriately for this relatively faint target . hence it is not possible to obtain a precise , reliable spectral type for the o component since classification of ob stars is largely based of the ratio of hei and heii lines in the blue . however , our spectrum shows a very weak hei @xmath315876 line , while hei @xmath316678 is too weak to even be detected ( fig.9 ; table 6 ) . since hei is very strong in late o stars and early b stars ( type b2 being where hei reaches a maximum ; see walborn & fitzpatrick 1990 ) , this suggests that this star is either earlier than o8 , or mid / late b. because he becomes doubly ionized in early o stars , we can also exclude a spectral type earlier than o5 . furthermore , the presence of high ionisation civ lines unambiguously rules out any spectral type b or later . these lines of evidence suggests that our star can only be in the range o5-o7 . an o5-o7 spectral type is consistent with the relatively strong civ @xmath555801,5812 lines , which are strongest in this spectral range ( walborn 1980 ) . because we lack a clear line ratio , and because there is considerable scatter among o stars in the equivalent widths of single species , it is however not possible to constrain the spectral type further . due to the lack of other objective classification criteria , the o5-o7 assignment must be regarded as tentative . the h@xmath56 region shows a very shallow absorption trough , flanked by broad emission wings , very similar to the profile observed in the o8 component in wr146 . the broad emission wings can not be explained by contamination from the wn8 star , since the two components are well resolved by stis and any ( very weak ) component arising in the very broad wing of the stis psf from the wn8 star should be subtracted out with the background in the aperture extraction procedure . furthermore , the centroid of the observed absorption trough is 8.9over the h@xmath56 laboratory wavelength ; this suggests that the line profile is significantly distorted as it is filled up with h@xmath56 emission arising in the wind . finally , the overall shape of the h@xmath56 profile is very similar to that of other o if stars ( e.g. cygnus ob#7 and hd210839 , herrero _ et al . _ 2000 ) . this @xmath0 emission suggests a substantial mass loss rate , which would imply that the star is a supergiant in the of category . however , ciii @xmath315696 is not clearly in emission ; this is more consistent with stars in the ( f ) category , which show a filling in of heii @xmath314686 but no strong emission lines . it is therefore not possible at this point to clearly distinguish between spectral class ia , iab , ib , or ii , but luminosity classes iii - v can be excluded . we tentatively classify this star as o5 - 7 i - ii(f ) . absolute magnitudes @xmath66 ( in the narrowband photometric system defined by westerlund 1966 ) have been estimated for galactic wr stars on the basis of cluster and association membership ( van der hucht 2001 , hereafter vdh01 ) . the narrowband @xmath67 is used because it avoids the brightest optical emission lines of wr stars . the distances of clusters and associations derived by lndstrom & stenholm ( 1984 ) are used . the mean absolute magnitudes @xmath68 for wc7 , wc6 , and wn8 stars are @xmath69 -4.5 , -3.5 , and -5.5 , respectively , with a standard deviation @xmath70 magnitude . the absolute magnitudes @xmath46 of the ob stars can be estimated , based on their spectral classification , using the relationships defined by schmidt - kaler ( 1982 , hereafter sk82 ) . while the @xmath67 and @xmath71 bands are not exactly the same , a comparison between @xmath71 and @xmath67 for a number of wr and ob stars shows that @xmath72 ( westerlund 1966 , see also turner 1982 ) . while the value of @xmath73 for any given wr star will be dependent on the strength of the optical lines , we can assume that @xmath74 to within less than half a magnitude . we thus compare the difference in absolute magnitudes as determined from the spectral types with the measured difference in @xmath71 derived using the wfpc2 images of these systems by niemela _ ( 1988 ) . for the wr86 system , the wc7 star is expected be in the range @xmath75 , and sk82 quotes a value of @xmath76 for a b0 iii star . the two estimates are thus consistent with each other , since the observed difference in @xmath71 magnitude for this system is @xmath77 . for the wr146 system , the wc6 star is expected to be in the range @xmath78 , while sk82 quotes a mean value of @xmath79 for o8 i - ii stars . the observed magnitude difference is @xmath80 ; there is thus a discrepancy of at least 2 magnitudes between the observed magnitude difference and that inferred from the spectral types . the two stars are definitely companions , as evidenced by the confirmation of a colliding wind region between the two components . the discrepancy thus can not be attributed to a chance alignment of two stars at different distances . if we assume that the o8 star is actually a main sequence object ( unlikely because of the @xmath0 line in emission ) , we get @xmath81 from sk82 , which still yields a difference of at least one magnitude . this means either that the wr star is too bright for its spectral type or , alternately , that the o star is too faint . an intriguing comparison can be made between wr146 and wr86 : both systems have components of approximately the same magnitude , but the wc5 star in wr146 is expected to be intrinsically fainter than the wc7 star in wr86 , while the o8 i - ii star in wr146 is expected to be intrinsically brighter than the b0 iii star in wr86 . for the wr147 system , the wn8 star is expected to be in the range @xmath82 . from sk82 , we get a range of values @xmath83 for o5 - 7 stars with luminosity classes i - iii . the observed magnitude difference is @xmath84 . again , we find that the wr component is too bright for its spectral type by at least 1.5 magnitudes or , alternately , that the o star is too faint by at least 1.5 magnitude . to be consistent with the expected magnitude of a wn8 star , the o star would have to be fainter than @xmath85 which , in the sk82 tables , is only consistent with stars of type b0 or later . a spectral type of b is clearly ruled out by our stis spectra . recently , absolute magnitudes of ob stars were re - evaluated , based on data by hipparcos ( wegner 2000 , hereafter w00 ) . the values of @xmath46 for giant and supergiant ob stars derived by w00 turn out to be fainter than the sk82 values by about 2 magnitudes . under the w00 system , the absolute magnitudes of wr146 and wr147 , as estimated from the spectral types would be consistent with the observed magnitudes . although the agreement is suggestive , one has to be concerned about the possible effects of the revised @xmath46 from the w00 system on the distance to galactic clusters and associations , and hence on the absolute magnitudes derived by vdh01 . one also has to consider the relatively large scatter in the absolute magnitudes of individual ob stars as derived from hipparcos parallaxes . for o7-b0 stars , the scatter can be as large as @xmath86 magnitude . there may also be systematic effects on parallax estimates of the ob star hipparcos sample , especially since most of the ob stars are beyond 200pc . hence , the values derived from w00 must be used with caution . some of the discrepancy may arise because of the uncertainty in the determination of the spectral - type of the ob components . in particular , the main criterion used in the determination of the luminosity class was the apparent filling in of the @xmath0 line , which suggests high mass - loss rates in wr146 and wr147 . however , both wr146 and wr147 are systems with colliding winds . if @xmath0 emission arises in the shock front , this could fill - up the @xmath0 absorption line in the spectra of the ob companions , mimicking the effect of intense mass - loss . if @xmath0 emission arises near the head of the bow shock front , which is relatively close to the o star in both wr146 and wr147 , then we would not be able to resolve them from the o star in the stis data presented here , and this might explain the apparent filling in of the emission lines . on the other hand , if @xmath0 emission was to arise downwind from the head of the bow shock , at some relatively large distance from both the wr and o components , then any extra @xmath0 emission would appear as a diffuse component on the stis data , which would not be strong enough ( as we have shown from the lack of any detected diffuse emission , see section 2 ) to account for the apparent filling in of the emission lines . the wr146 and wr147 systems exhibit strong non - thermal radio components . it has been found that the non - thermal emission occurs _ between _ the two stellar components , somewhat closer to the ob star ( dougherty _ et al . _ 1996 , williams _ 1997 ) . in the framework of the colliding wind model developed by eichler & usov ( 1993 ) , the shock front forms at a distance @xmath87 ( @xmath88 ) from the ob ( wr ) component . given @xmath89 the distance between the two stars , then : @xmath90 where @xmath91 is the wind momentum ratio . the ratio @xmath92 is dimensionless and independent of the projection of the system on the plane of the sky , and thus can measured directly from imaging . for wr146 and wr147 , @xmath93 0.25 and 0.14 , respectively , as measured directly from the combined radio maps and hst wfpc2 images ( niemela _ et al . these yield estimated values of @xmath94 and @xmath95 for wr146 and wr147 , respectively . from the spectral types , we estimate that the o components in wr146 and wr147 have mass - loss rates @xmath96 m@xmath2 yr@xmath3 ( see herrero _ et al . _ 2000 ) . if we assume a typical terminal velocity @xmath972000 km s@xmath3 for the o stars ( see prinja , barlow & howarth 1990 ) , we can estimate the mass - loss rate of the wr components and check for consistency . for wr146 , taking @xmath982900 km s@xmath3 as the terminal velocity of the wc6 star , this yields a mass - loss rate @xmath996.9 10@xmath100 m@xmath2 yr@xmath3 . this value is consistent with the mass loss rate value of @xmath101 derived from the radio emission of the wr star by dougherty _ ( 1996 ) . for wr147 , however , taking @xmath981100 km s@xmath3 as the terminal velocity of the wn8 star ( from eenens & williams 1994 ) , one gets @xmath1026.5 10@xmath103 m@xmath2 yr@xmath3 , which is an order of magnitude larger than the mass - loss rate value of @xmath104 estimated from the radio emission by williams _ et al . _ ( 1997 ) . for wr147 , williams _ ( 1997 ) have deduced a distance of @xmath105pc by comparing infrared photometry of wr147 and wr105 , the latter being another wn8 star suspected to be in the sgr ob1 association , whose distance is known . it is from this estimated distance and from vla measurements of the radio flux that williams _ et al . _ ( 1997 ) have estimated the mass - loss rate in the wr component of wr147 to be 4.2@xmath290.2 10@xmath100 m@xmath2 yr@xmath3 . the disagreement with our estimated value can be resolved in two ways : ( 1 ) we have overestimated @xmath106 by at least an order of magnitude , in which case the ob component in wr147 is unlikely to be an o supergiant , or ( 2 ) wr147 is more distant than estimated by williams _ ( 1997 ) , i.e. at least 1 kpc away . if the latter interpretation is true , then there exist significant differences between the spectral energy distributions of wr105 and wr147 . the former interpretation is more likely to be true , which means that we have overestimated the mass - loss rate of the o component because of an inaccurate assessment of the spectral type and luminosity class . the key element here is the observation of the filling in of the @xmath0 profile , which does suggest a high mass - loss rate and luminosity . clearly , a resolved spectrum of the ob component in wr147 spanning the whole optical range is required to resolve this issue . such an observation will have to be performed with the hubble space telescope , or with similar high spatial resolution from the ground . while wr86 does show evidence for some non - thermal emission , it is listed by dougherty & williams ( 2000 ) as having a `` composite '' spectral energy distribution ( as compared to a `` non - thermal '' spectral energy distribution for wr146 and wr147 , and a `` thermal '' energy distribution for suspected single wr stars ) . we believe that the fact that this star has only a weak component of non - thermal emission can be explained by the smaller mass - loss rate from the b component . while late of stars typically have derived mass - loss rates @xmath107 m@xmath2 yr@xmath3 , a star of spectral type b0 iii like the companion to wr86 is expected to have a mass - loss rate @xmath108 m@xmath2 yr@xmath3 , or about two orders of magnitude smaller than of stars ( e.g. runacres & blomme 1996 ) . assuming @xmath109 for the b0 iii star to be of the same order of magnitude as for the supergiant o stars in wr146 and wr147 , this means that the wind momentum from the b0 iii star must be two orders of magnitude smaller and @xmath87 , the distance from the ob star to the wind collision front , is expected to be one order of magnitude smaller for the wr86 system than it is for wr146 and wr147 . this means that the wr86 shock front is formed much closer to the ob component and most likely wraps around the ob component with a much smaller opening angle . hence , the total volume where non - thermal emission occurs should be significantly smaller , which accounts for the weaker non - thermal emission . this hypothesis can be tested by imaging the wr86 system in the ir / radio at very high spatial resolution , and locating the source of the weak non - thermal emission . we predict that the non - thermal emission occurs very close ( a few milliarcseconds ) to the ob component , where @xmath110 . it has been suggested by dougherty et al . ( 2000 ) that some apparent discrepancies in the luminosities of the components in wr146 might be resolved if the o8 companion was itself a wr+o binary . likewise , setia gunawan _ et al . _ ( 2000 ) have interpreted the observed 3.38 yr period in the 1.4 ghz radio emission as evidence for a third component in the system orbiting the o8 star . our resolved spectra of wr146 shows no clear evidence for a third component . it is true that the civ emission feature in the o8 star spectrum , which we marked as contaminated light from the wc6 star , does look significantly different from the actual civ profile in the wc6 spectrum , and thus one may argue that the so - called `` contaminated light '' could actually be the signature of another wr star orbiting the o8 component . we note however that ( 1 ) the fact that the psf is strongly dependent on the wavelength most likely introduces a dependency on wavelength for the amount of contaminated light which can distort the civ profile on the o8 star , and ( 2 ) the extracted spectra being the result of a multidimensional fit of the whole double psf profile , we do not necessarily expect the contamination to add up in a strictly linear fashion ( i.e. effects may be non - linear ) , which means that a disproportionately strong contamination could occur at the point where the civ line is brightest , hence distorting the civ profile in the o8 star spectrum . we therefore conclude that there is no evidence for another wr star orbiting the o8 star component on a tighter orbit . we can safely rule out the presence of any unresolved wr star , except for one that would be significantly fainter ( at least 1 magnitude ) than the resolved wc6 component . unless the wc6 is unusually bright for its spectral type , this leaves only a relatively faint wr star of spectral type wc3-wc4 or wn2-wn3 ( see van der hucht 2001 ) as a possible ( but unlikely ) candidate . if the o8 star is an unresolved double , it is more likely to be an ob+ob system . we briefly summarize our findings as follows : 1 . we have obtained resolved spectra of the components in the close visual binary systems wr86 , wr146 , and wr147 . wr86 is classified as wc7+b0 iii , with the wr component to the northwest and the b component to the southeast . wr146 is classified as wc6 + o8 i - iif , with the wr component to the south and the o component to the north . wr147 is classified as wn8 + o5 - 7 i - ii(f ) , with the wr component to the south and the o component to the north . the relative location of the wr and o components in the wr146 and wr147 systems is consistent with the colliding wind interpretation of their radio maps . absolute magnitudes @xmath46 of the ob stars have been derived based on the spectral type - magnitude relationship of schmidt - kaler ( 1982 ) , and compared with the estimated absolute magnitudes @xmath68 of the wolf - rayet stars ( from van der hucht 2001 ) . while the values are consistent for the wr86 system , we find a significant discrepancy in the wr146 and wr147 systems . for wr146 , it looks like the wc6 star is at least 2 magnitudes brighter than expected ( or the o8 i - iif star is at least 2 magnitudes fainter than expected ) . for wr147 the wn8 star appears to be at least 1.5 magnitudes brighter than expected ( or the o5 - 7 i - ii(f ) star is fainter than expected ) . 3 . from the spectral types , we estimate that the o components in wr146 and wr147 have mass - loss rates @xmath111 m@xmath2 yr@xmath3 . these values can be compared to the estimated values for the wr component mass - loss rate @xmath112 , which are linked to @xmath106 through the configuration of the colliding - wind systems . while the estimated value of @xmath106 for wr146 is consistent with @xmath112 , our value of @xmath106 for wr147 is an order of magnitude too large . this most likely indicates that our spectral classification is inaccurate , although it could also mean that current estimates of the distance to wr147 are too low . a more accurate spectral classification for wr147 is required to resolve the discrepancy , which will require new resolved spectroscopic observations of the ob component in wr147 . 4 . from the spectral type , we estimate the b component in wr86 to have @xmath113 m@xmath2 yr@xmath3 . the relatively smaller mass - loss rate in the ob component in wr86 must result in the colliding wind region being much smaller in volume and located much closer to the b star . hence the amount of non - thermal emission arising in the shock cone is expected to be much smaller . the reduced mass - loss rate from the b star and smaller volume of the resulting shock cone explains why wr86 is found to be a weak non - thermal emitter , while wr146 and wr147 are known strong non - thermal emitters . 5 . in none of the systems did we observe any trace of diffuse emission down to the instrumental limit . if there is any diffuse emission in the optical associated with the colliding wind interface , it must be weaker than 5 10@xmath33 ergs @xmath34 s@xmath3 @xmath3 arcsec@xmath3 . overall , we feel that the classification of ob stars , especially the determination of luminosity classes , is a difficult and non - trivial task . the main reason for this is the lack of availability of a uniform sequence of digital spectra spanning the whole spectral range from blue to red . existing atlases , while useful , are sometimes fragmentary , and most are based on photographic spectra . publication of a comprehensive atlas of digital spectra for ob stars based on ccd observations and covering the whole optical range from @xmath1144000 - 7000 would be very beneficial to this field . turner , d. g. 1982 , in : wolf - rayet stars : observations , physics , evolution ; proceedings of the symposium , cozumel , mexico , september 18 - 22 , 1981 . ( a82 - 48127 24 - 90 ) dordrecht , d. reidel publishing co. , p. 57 - 60 williams , p. m. , radio emission from the stars and the sun . asp conference series , volume 93 ; proceedings of a conference held at the university of barcelona ; barcelona ; spain ; 3 - 7 july 1995 , edited by a. r. taylor and j. m. paredes , p.15 williams , p. m. , 1999 , proceedings of the 193rd symposium of the international astronomical union held in puerto vallarta , mexico , 3 - 7 november 1998 " . edited by k. a. van der hucht , g. koenigsberger , and p. r. j. eenens . san francisco , calif . : astronomical society of the pacific , 1999 . , p.267 heii & 4338.7 & -25.0@xmath295.0 civ & 4441.5 & -31.9@xmath290.5 heii & 4541.6 & -15.0@xmath295.0 oiii / ov & 5592.2/5597.9 & -31.8@xmath292.0 ciii & 5695.9 & -205.@xmath2915 . civ & 5801.3/5812.0 & -295.@xmath2915 . hei & 5875.6 & -50.0@xmath295.0 heii & 6406.4 & -13.0@xmath293.0 heii & 6560.0 & -85.0@xmath2910 . ciii & 6744.4 & -120.@xmath2910 . h@xmath43 & 4340.5 & 4341.7 & 2.30@xmath290.20 oii & 4349.4 & 4349.0 & 0.70@xmath290.20 hei & 4387.9 & 4389.2 & 0.55@xmath290.05 oii & 4414.9 & 4417.3 & 0.35@xmath290.05 oii & 4448.3 & 4449.3 & 0.20@xmath290.05 hei & 4471.5 & 4472.4 & 1.00@xmath290.10 mgiii & 4479.0 & 4482.1 & 0.15@xmath290.05 nii & 4530.4 & 4530.7 & 0.15@xmath290.05 siiii & 4553.9 & 4554.2 & 0.35@xmath290.05 siiii & 4567.8 & 4569.1 & 0.30@xmath290.05 siiii & 5739.7 & 5741.6 & 0.30@xmath290.05 hei & 5875.6 & 5877.2 & 0.70@xmath290.05 h@xmath56 & 6561.9 & 6564.8 & 2.00@xmath290.05 hei / heii & 6678.1/6683.2 & 6679.8 & 0.80@xmath290.05 oiii / ov & 5592.2/5597.9 & -22.0@xmath293.0 ciii & 5695.9 & -55.0@xmath298.0 civ & 5801.3/5812.0 & -600.@xmath2935 . hei & 5875.6 & -140.@xmath2935 . heii & 6406.4 & -15.0@xmath293.0 heii & 6560.0 & -165.@xmath2910 . ciii & 6744.4 & -200.@xmath2915 . h@xmath43 & 4340.5 & 4339.6 & 1.50@xmath290.10 hei & 4387.9 & 4388.7 & 0.20@xmath290.05 hei & 4471.5 & 4470.0 & 0.80@xmath290.05 heii & 4541.5 & 4540.3 & 0.40@xmath290.05 ciii & 5695.9 & 5695.1 & -0.65@xmath290.05 civ & 5811.9 & 5811.2 & 0.22@xmath290.05 hei & 5875.6 & 5874.8 & 1.0@xmath290.10 h@xmath56 & 6561.9 & 6568.3 & -4.85@xmath290.25 hei / heii & 6678.1/6683.2 & 6678.3 & 0.15@xmath290.05 nii & 5679.6 & -6.0@xmath291.0 niv & 5736.9 & -5.8@xmath291.0 civ & 5801.3/5812.0 & -0.8@xmath290.2 hei & 5875.6 & -35.0@xmath291.0 heii & 6310.8 & -0.6@xmath290.2 niv & 6380.7 & -0.8@xmath290.2 heii & 6406.4 & -0.9@xmath290.2 niii & 6467.0/6478.7 & -3.2@xmath290.3 heii & 6527.1 & -1.0@xmath290.1 heii & 6560.0 & -45.0@xmath291.5 hei / heii & 6678.1/6683.2 & -30.0@xmath291.5

we present spatially resolved spectra of the visual wr+ob massive binaries wr86 , wr146 , and wr147 , obtained with the _ space telescope imaging spectrograph _ on board the _ hubble space telescope_. the systems are classified as follows : wr86 = wc7 + b0 iii , wr146 = wc6 + o8 i - iif , wr147 = wn8 + o5 - 7 i - ii(f ) . both wr146 and wr147 are known to have strong non - thermal radio emission arising in a wind - wind collision shock zone between the wr and ob components . we find that the spectra of their o companions show @xmath0 profiles in emission , indicative of large mass - loss rates , and consistent with the colliding - wind model . our spectra indicate that the b component in wr86 has a low mass - loss rate , which possibly explains the fact that wr86 , despite being a long period wr+ob binary , was not found to be a strong non - thermal radio emitter . because of the small mass - loss rate of the b star component in wr86 , the wind collision region must be closer to the b star and smaller in effective area , hence generating smaller amounts of non - thermal radio emission . absolute magnitudes for all the stars are estimated based on the spectral types of the components ( based on the tables by schmidt - kaler for ob stars , and van der hucht for wr stars ) , and compared with actual , observed magnitude differences . while the derived luminosities for the wc7 and b0 iii stars in wr86 are consistent with the observed magnitude difference , we find a discrepancy of at least 1.5 magnitudes between the observed luminosities of the components in each of wr146 and wr147 and the absolute magnitudes expected from their spectral types . in both cases , it looks as though either the wr components are about 2 magnitudes too bright for their spectral types , or that the o components are about 2 magnitudes too faint . we discuss possible explanations for this apparent discrepancy .

throughout the world , devastating earthquakes constantly occur with little or no advance warning . brune ( 1979 ) proposed that earthquakes may be inherently unpredictable since large earthquakes start as smaller earthquakes , which in turn start as smaller earthquakes , and so on . in his model , most of the fault is in a state of stress below that required to initiate slip , but it can be triggered and caused to slip by nearby earthquakes or propagating ruptures . any precursory phenomena will only occur when stresses are close to the yield stress . however , since even small earthquakes are initiated by still smaller earthquakes , in the limit , the region of rupture initiation where precursory phenomena might be expected is vanishingly small . even if every small earthquake could be predicted , one still faces the impossible task of deciding which of the thousands of small events will lead to a runaway cascade of rupture composing a large event . nevertheless , the discussion about the possibility of earthquake forecast continues to be open , and a wide spectrum of new spaceborne technologies for the earthquake study and forecast appeared during the last decades . the main advantage of spaceborne technologies is the ability to cover big territories and areas with difficult access . the list of these technologies is very large . as an example it is possible to mention the measurements of different ionospheric precursors of earthquakes including changes in electromagnetic elf radiation ( serebryakova et al . , 1992 , gokhberg et al . , 1995 ) , and ionospheric electron temperature ( sharma et al . , 2006 ) , and density ( trigunait et al . , 2004 ) ( see pulinets at al . , ( 2003 ) for a review ) . many efforts have been concentrated in the study of the ground deformation using the satellite radar interferometry , that makes it possible to determine the location and amount of coseismic surface displacements ( see for example satybala , 2006 ; schmidt and bergmann , 2006 , lasserre et al . , 2005 , funning et al . , the ir satellite thermal imaging data were used to study pre - earthquake thermal anomalies ( ouzounov and freund , 2004 ) . the anomalies in the surface latent heat flux data were also detected a few days prior to coastal earthquakes ( cervone et al . , 2005 , singh and ouzounov , 2003 ; dey at al . , 2004 ) . during last years , significant progress has been reached in the understanding how the complex set of phenomena , related to the earthquake gestation is reflected , at least partially , in the geological lineaments . in particular , cotilla - rodriguez and cordoba - barba ( 2004 ) studied the morphotectonic structure of the iberian peninsula and showed that the main seismic activity is concentrated on the first- and second rank lineaments , and some of important epicenters are located near the lineament intersections . stich et al . , ( 2001 ) found from the analysis of 721 earthquakes with magnitude between 1.5 and 5.0 , that the epicenters draw well - defined lineaments and show two dominant strike directions n120 - 130e and n60 - 70e , which are coincident with the known fault system of the area . distances within multiplets ( typically several tens of meters ) are smaller than the fracture radii of these events . carver et al . ( 2003 ) have used the srtm and landsat-7 digital data and paleoseismic techniques to identify active faults and evaluate seismic hazards on the northeast coast of kodiak island , alaska . arellano et al . ( 2004ab , 2005 ) studied the changes in the lineament structure caused by a 5.2 richter scale magnitude earthquake occurred january 27 , 2004 in southern peru . during last years this region is studied intensively using the ground based seismic network ( comte et al . , 2003 ; david et al . , 2004 ; legrand , 2005 ) as well as gps and sar interferometry data ( campos et al . , 2005 ) . the aster / terra high resolution multispectral images 128 and 48 days before and 73 days after the earthquake were used . it was shown that the lineament system is very dynamical , and significant numbers of lineaments appeared between four and one month before the earthquake . they also studied the changes in stripe density fields . these fields represent the density of stripes , calculated for each direction as a convolution between the corresponding circular masks and the image . the stripe density field residuals showed the reorientation of stripes , which agrees with the dilatancy models of earthquakes . these features disappear in the image obtained two months after the earthquake . analysis of the similar reference area , situated at 200 km from the epicenter , showed that in the absence of earthquakes both lineaments and stripe density fields remain unchanged . similar results were obtained later by bondur and zverev ( 2005 ) due to analysis of modis ( terra ) images of earthquake in california . singh v.p and r.p . singh ( 2005 ) used the lineament analysis to study changes in stress pattern around the epicenter of mw=7.6 bhuj earthquake . this earthquake occurred 26 january 2001 in india . indian remote sensing ( irs-1d ) liss data were used . the lineaments were extracted using high pass filter ( sobel filter in all directions ) . the results obtained also confirm that the lineaments retrieved from the images 22 days before the earthquake differ from the lineaments obtained 3 days after the earthquake . it was assumed that they are related to fractures and faults and their orientation and density give an idea about the fracture pattern of rocks . the results also show the high level of correlation between the continued horizontal maximum compressive stress deduced from the lineament and the earthquake focal mechanism . studies of lineament dynamics can also contribute to better understanding of the nature of earthquakes . to date significant number of theories has been developed to explain how an earthquake occur . one of the oldest is the elastic rebound theory , proposed by harry reid after the california 1906 earthquake ( reid , 1910 ) . it is based on the assumption that the rocks under stress deform elastically , analogous to a rubber band . strain builds up until either the rock break creating a new fault or movement occurs on an existing fault . as stored strain is released during an earthquake , the deformed rocks `` rebound '' to their undeformed shapes . the magnitude of the earthquake reflects how much strain was released . the seismic gap hypothesis states that strong earthquakes are unlikely in regions where weak earthquakes are common and the longer the quiescent period between earthquakes , the stronger the earthquake will be when it finally occur ( see kagan and jackson , 1995 , and references therein ) . the complication is that the boundaries between crustal plates are often fractured into a vast network of minor faults that intersect the major fault lines . when an earthquake relieves the stress in any of these faults , it may pile additional stress on another fault in the network . this contradicts the seismic gap theory because a series of small earthquakes in an area can then increase the probability that a large earthquake will follow . the theory of dilatancy states that an earthquake develops similarly to the rupture of a solid body ( whitcomb et al . , 1973 ; scholz et al . , 1973 ; griggs et al . , 1975 ) this approach has a physical basis in laboratory studies of rock samples , which showed that when rocks are compressed until they fracture , a dilatancy often occurs for a short time interval immediately before failure ( scholz , 1968 ) . mjachkin et al . ( 1975ab ) modified the dilatancy approach and formulated the theory of unstable avalanche crack formation . the model is based on the two phenomena : interaction between the stress fields of the cracks , and the localization of the process of the crack formation . the number and size of cracks increases gradually under the action of tensions below a critical value . when the density of cracks reaches some critical value , the rock breaks very quickly . this process develops due to merging of cracks as a result of interaction between their stress fields . however , the larger cracks have more probability to interact , and it supposes that a small number of large cracks is gradually formed , and their merging leads to the macro - destruction . during the earthquake gestation , a gradual increase in number and size of cracks occur in the whole volume of rock under compression . when the crack density reaches a critical value , the barriers between cracks are destroyed , and the velocity of deformation increases . finally , an unstable deformation develops and localizes in a narrow zone of future macro - rupture , the cracks orient along the future macro rupture , and a macro - crack is formed , producing an earthquake . however , this model was modified recently by introducing a concept of self - organized criticality , proposed by bak et al . ( 1988 ) for description of the behavior of complex systems . applied to earthquakes , this approach describes an interaction between the ruptures of different rank and the collective effects of rupture formation before a strong earthquake ( for example varnes , 1989 ; keilis - borok , 1990 ; sammis and sornette , 2002 ) . a wide area around the future epicenter reaches a metastable state , and the system turns to be very sensitive to small external actions . the concept of soc does not contradict the concept of dilatancy . however , it assumes that a significantly greater region is involved during the last stages of earthquake preparation than the dilatancy theories imply . unfortunately , the main processes leading to an earthquake develop deep inside the crust , and there is no way to realize direct measurements of any quantity . the unique possibility we have is to search for traces of these processes disseminated over the earth s surface . in this context , the lineament analysis could convert in the future in one of power tools for earthquake study , complementing other ground - based and satellite studies . nevertheless , despite promising results obtained , many important questions continue to be present . it is necessary to understand , whether the lineament system is always affected by earthquake ? how early before an earthquake is this alteration manifested ? how is it related to the earthquake magnitude and depth ? how different is it in case of different kinds of plate borders ? this study represent a first step in the search of some answers . for this study we used the the images from the advanced spaceborne thermal emission and reflection radiometer ( aster ) onboard the terra satellite . the satellite was launched to a circular solar - synchronous orbit with altitude of 705 km . the radiometer is composed by three instruments : visible and near infrared radiometer ( vnir ) with 15 m resolution ( bands 1 - 3 ) , short wave infrared radiometer ( swir ) with 30 m resolution ( bands 4 - 9 ) and thermal infrared radiometer tir with 90 m resolution ( bands 11 - 14 ) which measure the reflected and emitted radiation of the earths surface covering the range 0.56 to 11.3 @xmath1 m ( abrams , 2000 ) . the images were processed using the lineament extraction and stripes statistic analysis ( lessa ) software package ( zlatopolsky , 1992 , 1997 ) , which provides a statistical description of the position and orientation of short linear structures through detection of small linear features ( stripes ) and calculation of descriptors that characterize the spatial distribution of stripes . the program also makes it possible to extract the lineaments - straight lines crossing a significant part of the image . to make this extraction , a set of very long and very narrow ( a few pixels ) windows ( bands ) , crossing the entire image in different directions , was used . in each band the density of stripes , the direction of which is coincident with the direction of the band , is calculated . when the density of stripes overcomes a pre - established threshold , the chain of stripes along the band is considered as a lineament . the value of threshold depends on the brightness of the image , relief , etc . and is established empirically . previous studies showed that lineaments , extracted from the image by applying the lessa program , are strongly related to the main lineaments , obtained from the geomorphological studies ( zlatopolsky , 1992 , 1997 ) . the details about the application of lessa package for earthquake studies is given in ( arellano et al . , 2005 ) . during this study we analysed 5 earthquakes , occurred in the in the pacific coast of the south america and one earthquake occured in himalaya , china . table 1 resumes main characteristics of these earthquakes , indicating the date , country , geographic coordinates , magnitude , and depth of the earthquake . also the the aster images available for each earthquake are indicated , for example -126 means that the image 126 days before earthquake was used . the last column indicates that in all south american earthquakes number and orientation of lineaments suffered changes before the earthquake . in case of china earthquake , we can not give a defnite answer , because unfortunately the key images tens day before the earthquake were covered by clouds in appoximately 50% , that made the lineament analysis difficult ( last two lines , two areas cvering the hipocenter and close to hipocenter ) . neverthless , more sophysticated technique based on analysis of stipe density fields was able to detect the alterations in these fields related to the earthquake . the methodology of this analysis is given in ( arellano et al . currently we are preparring a manuscript dedicated especially to the analysis of this event . to illustrat the results obtained we give as an example a detailed analysis of 7.8 mw earthquake , which took place june 13 , 2005 in northern chile close to arica ( see figure 1 ) . the hipocenter was situated at 115 km deep in the crust . the coordinates were @xmath2 lat , @xmath3 long . in the top , a series of four band 3 aster ( vnir ) images around the hipocenter area are shown . it is possible to see , that the presence of clouds was low . the second line contains the images showing the systems of lineaments , obtainend from the images above using the lessa programm with a threhhold 120 ( zlatopolsky , 1992 , 1997 ) . it is posible to see clear time evolution of lineaments , experimenting strong increase in the number of lineaments 5 days before the earthquake . the third and fourth lines quanify this effect by calculaing the rose - diagramms and the radon transforms . reorientation of lineaments can be taken as an indirect evidence in favour to the theory of dilatncy . nevertheless , it is necessary to make more detailed studies to make definitive conclusions . .main characteristics of earthquakes analyzed [ cols="^,^,^,^,^,^,^,^",options="header " , ] in this study we used the multispectral satellite images from aster / terra satellite for detection and analysis of lineaments in the areas around strong earthquakes with magnitude more than 5 mw . a lineament is a straight or a somewhat curved feature in an image , which can be detected by a special processing of images , based on directional filtering and/or hough transform . it was established that the systems of lineaments are very dynamical . by analyzing 5 events of strong earthquakes , it was found that a significant number of lineaments appeared approximately one month before an earthquake , and one month after the earthquake the lineament configuration returned to its initial state . these features were not observed in the test areas , situated hundreds kilometers away from the earthquake epicenters . the main question is how the lineaments extracted from images of 15 - 30 m ( aster ) in resolution are able to reflect the accumulation of stress deep in the crust given that the ground deformations associated with these phenomena are about a few centimeters ? the nature of lineaments is related to the presence of faults and dislocations in the crust , situated at different depth . if a dislocation is situated close to the surface , the fault appears as a clear singular lineament . in the case of a deep located fault , we observe the presence of extended jointing zones , easily detectable in satellite images even up to 200 m resolution . nevertheless , how well lineaments can be detected strongly depends on a number of factors . in particular , it depends on the current level of stress in the crust . generally , an enlarged presence of lineaments indicates that in these regions the crust is more permeable , allowing the elevation of fluids and gases to the surface . accumulation of stress deep in the crust modifies all afore mentioned processes and leads to the variation in the density and orientationn of lineaments , previous to a strong earthquake . we acknowledge hiroji tsu ( geological survey of japan csj ) aster team leader , anne kahle ( jet propulsion laboratory jpl ) us aster team leader and the land processes distributed active archive center for providing the aster level 2 images . we acknowledge a. zlatopolsky for providing the lineament extraction and stripes statistic analysis ( lessa ) software package and helpful suggestions . we thank very much milton rojas gamarra for his assistance in image procesing . this work has been supported by dicyt / usach grant . + + * references * abrams , m. , the advanced spaceborne thermal emission and reflection radiometer ( aster ) : data products for the high spatial resolution imager on nasa s terra platform , international journal of remote sensing , 21(5 ) , 847 - 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over the last decades strong efforts have been made to apply new spaceborne technologies to the study and possible forecast of strong earthquakes . in this study we use aster / terra multispectral satellite images for detection and analysis of changes in the system of lineaments previous to a strong earthquake . a lineament is a straight or a somewhat curved feature in an image , which it is possible to detect by a special processing of images based on directional filtering and or hough transform . `` the lineament extraction and stripes statistic analysis '' ( lessa ) software package , developed by zlatopolsky ( 1992 , 1997 ) . we assume that the lineaments allow to detect , at least partially , the presence ruptures in the earths crust , and therefore enable one to follow the changes in the system of faults and fractures associated with strong earthquakes . we analysed 6 earthquakes occurred in the pacific coast of the south america and one earthquake in tibet , xizang , china with the richter scale magnitude @xmath0 mw . they were located in the regions with small seasonal variations and limited vegetation to facilitate the tracking of features associated with the seismic activity only . it was found that the number and orientation of lineaments changed significantly about one month before an earthquake approximately , after that the system gradually returns its initial state . this effect increases with the earthquake magnitude , and it is much more easily detectable in case of convergent plate boundaries ( for example , nazca and south american plates ) . the results obtained open a possibility to develop a methodology able to evaluate the seismic risk in the regions with similar geological conditions .

the radiative decay of mesons has been traditionally advocated as a good tool to investigate the nature of the controversial mesons . in particular the decay into @xmath3 has had a special attention @xcite . the scalar mesons have been those most thoroughly studied given the on going debate on whether they are @xmath4 states , tetraquarks , or meson - meson molecules as a particular case of the more general one , dynamically generated states from the meson - meson interaction in coupled channels , see @xcite for a recent review . in this work we want to call the attention to two particular mesons , the @xmath0 and @xmath1 states , because in a recent paper @xcite the two mesons were found as dynamically generated states from the @xmath5 interaction in the hidden gauge approach for the vector mesons @xcite . the attraction was stronger in the spin s=2 channel than in the scalar one , but in both channels there was enough attraction to generate bound states . in other channels the interaction was either very weak or repulsive , such that these two states stand as particular cases which can be viewed as largely being @xmath5 molecules in that framework . it is interesting to see how this idea immediately leads one to evaluate rather accurately the partial decay width of the two states into @xmath3 and this is the purpose of this paper . although ref . @xcite contains the first theoretical evaluation of the @xmath2 bound system , it is interesting to recall that , based on phenomenological properties ( the large @xmath6 versus @xmath7 ) , the @xmath2 molecular nature of the @xmath0 was also suggested in @xcite . the @xmath1 is , however , widely believed to be part of a p - wave nonet of @xmath4 states @xcite . the results of @xcite support the suggestion of @xcite for the @xmath0 as a @xmath2 molecule , but surprisingly also show that for spin s=2 the @xmath2 interaction is attractive and about three times larger than in the case of s=0 , thanks to which a stronger bound @xmath2 state appears for s=2 , which was identified in @xcite as the @xmath1 resonance . the experimental situation is rich in the case of the @xmath1 , which has a very pronounced peak in @xmath3 scattering going to pions . compatible results are found in different laboratories and using different methods , crystal ball @xcite , mark ii @xcite , jade @xcite , topaz @xcite , md-1 @xcite , cello @xcite and venus @xcite . the pdg quotes the result @xmath8 kev @xcite . recent results are also presented by the belle collaboration in @xcite and in @xcite , where the preferred solution gives @xmath9 kev . the situation of the @xmath10 decay of the @xmath0 is rather unclear . the latest edition of the pdg @xcite does not quote any value , superseding old results which were ambiguous . the belle collaboration has the most recent results in this direction @xcite . this work quotes a central value for the mass of the @xmath0 of 1470 mev , but with very large uncertainties , of the order of 255 mev , mostly of systematic origin . it also quotes a value for the radiative decay of this resonance , again with a very large uncertainty , @xmath11=(@xmath12 ) ev . the same work quotes more accurate values deduced by the crystal ball collaboration @xcite , @xmath11=(@xmath13 ) ev , with , however , much less statistics than belle . on the theoretical side , an evaluation of the radiative decay into @xmath14 of the @xmath0 has been done in @xcite , where using a model where the scalars are a mixture of @xmath4 and @xmath15 the authors find a small value between @xmath16 kev . much bigger values , of the order of @xmath17 kev are obtained in @xcite assuming the state to be basically a @xmath4 of non strange nature , although actually the value quoted is used as input to determine parameters of the theory . ratios of radiative widths between scalar states are also quoted in @xcite under the assumption that they are @xmath4 states mixing with glueballs . results for the @xmath3 decay of the @xmath1 state are also obtained in @xcite , where assuming that the resonance is a @xmath4 state , a satisfactory description of this decay rate together with that of the @xmath18 is obtained at the expense of fitting two free parameters . the novel picture of @xcite puts the @xmath0 and @xmath1 resonances on the same footing , allowing one to calculate the @xmath3 radiative width within the same formalism . this is the aim of the present paper . the evaluation presented here turns out to be rather simple technically , once the formalism for the generation of the two resonances is developed in @xcite . we will find that the widths obtained are rather precise , with respect to uncertainties from the parameters of the model , and agree well with the well known experimental results for the case of the @xmath1 , while the one for the @xmath0 follows the actual experimental trend that it is indeed about one order of magnitude smaller than that for the @xmath1 state . in @xcite the driving term for the @xmath19 interaction was obtained from the hidden gauge lagrangian @xmath20 where the symbol @xmath21 @xmath22 stands for the @xmath23 trace and @xmath24 is given by @xmath25~~,\ ] ] with @xmath26 , and @xmath27 mev is the pion decay constant . the @xmath23 matrix of @xmath28 is given by @xmath29 the interaction of @xmath30 of eq . ( [ eq:3 ] ) gives rise to a contact term @xmath31 and a three vector vertex given by @xmath32 with this information the driving term for the @xmath19 interaction is given by the diagrams of fig . [ fig:1 ] . driving terms of the @xmath19 interaction . the diagrams to the right sums the contribution of the first two diagrams.,width=264 ] this driving term , @xmath33 , is used as kernel in the bethe salpeter equation depicted in fig . [ fig:2 ] , diagrams summed up in the bethe salpeter equation.,width=283 ] which gives the solution @xmath34 with @xmath35 the loop function for the meson propagators conveniently regularized @xcite . the interaction is studied for @xmath36 and the projections over spin and isospin are performed . two states are obtained , visible in neat peaks of @xmath37 , which is depicted in fig . [ fig:3 ] . they correspond to the @xmath1 and @xmath0 , the later one appearing around @xmath38 mev in our approach , close to the preliminary results of the belle collaboration @xcite . the model of @xcite contains @xmath19 as basic components to form the scalar and tensor states . however , intermediate @xmath39 states , through the box and crossed box diagrams , were also considered . in addition , intermediate @xmath40 states , driven by pion exchange through anomalous @xmath41 couplings , were also taken into account . it was found there that the real parts of the @xmath39 and @xmath40 intermediate states mechanisms were individually small compared to the dominant tree level @xmath42 mechanisms and in addition there were cancellations between the @xmath39 and @xmath40 contributions , rendering the tree level @xmath19 terms largely dominant . the calculations of ref . @xcite were done using the on - shell approach of @xcite based on the n / d method , using a cut off in the three momentum in the loops , which was shown in @xcite to be equivalent to the use of dimensional regularization . this prescription then preserves the underlying symmetries and gauge invariance ( see a more detailed discussion in @xcite , page 5 ) . the approach of ref . @xcite uses a full relativistic treatment of the loop functions , which guarantees exact unitarity and analiticity of the amplitudes . nonrelativistic approximations are done in the evaluation of the @xmath43 potential , neglecting the three momentum of the vector mesons versus their mass . while this approximation is quite good for the @xmath0 state , for the case of the more bound @xmath1 resonance certainly it induces a larger correction , still under control as discussed in ref . @xcite ( see pag 4 of this reference ) , particularly because a small fine tuning of the parameters is allowed in the approach to fit one resonance mass , which allows one to cope with small correcions stemming from different sources . figure [ fig:3 ] shows results including also the box diagram accounting for @xmath39 decay , which plays a minor role in the binding of the two states but enlarges the width of the states due to the large phase space available for decay into two pions . the @xmath44 parameter in fig . [ fig:3 ] appears to account for @xmath45 off shell and is varied within reasonable values @xcite . the amplitude of fig . [ fig:2 ] can be parameterized as a breit - wigner amplitude , and using the spin projection operators of @xcite we find @xmath46 where @xmath47 are the polarization vectors of the @xmath48 for each @xmath49 of the four @xmath48 mesons involved ( 1 , 2 for the initial states and 3 , 4 for the final states ) . as shown in @xcite , because of the small three momenta of the @xmath48 mesons involved , only the spatial components of the @xmath48 polarization vectors are needed . these amplitudes correspond to a pole term as depicted in fig . [ fig:4 ] . a ) resonance pole representation of the amplitude of @xcite . b ) diagram depicting the coupling of the resonance to @xmath19.,width=283 ] since we are interested in the coupling of the resonance to the @xmath19 system , this is given by @xmath50 \label{eq:9}\\ s=0&&\nonumber\\ & & g_s\frac{1}{\sqrt 3}\epsilon^{(3)}_i\epsilon^{(4)}_i~~. \label{eq:10}\end{aligned}\ ] ] in both cases we are only interested in the isospin @xmath36 component , given by @xmath51 factor to account for identical particles in the sum over intermediate states ) and the phase convention @xmath52 . the use of this normalization will also account for the factor @xmath53 of symmetry that one has when dealing with identical particle in the final state . we only need the @xmath54 component of the amplitude @xmath55 . the @xmath54 component is given by @xmath56 times the @xmath36 components of eqs . ( [ eq:9 ] ) , ( [ eq:10 ] ) . here we present the formalism for the @xmath57 decay of the two resonances . since the resonances are formed from @xmath19 components , the two photons are radiated from these components . this is taken into account by loops involving the @xmath48 mesons , in a similar way as done in @xcite for the radiative decay into @xmath58 of the axial vector mesons generated dynamically from the interaction of vectors and pseudoscalars within the same hidden gauge formalism . taking into account that in the hidden gauge formalism the photons do not couple directly to the vector but indirectly through their conversion into @xmath48 , @xmath59 , @xmath60 , the picture we want for the @xmath57 decay of the resonances is given in fig . [ fig:5 ] . feynman diagrams to evaluate the radiative decay width of @xmath0 and @xmath1.,width=283 ] the fact that the photon couples to vectors by direct conversion into another vector , allows one to factorize the diagrams of fig . [ fig:5 ] into a strong part , @xmath61 , depicted in fig . [ fig:6 ] , followed by the photon coupling to either @xmath62 . strong part of feynman diagrams to evaluate the radiative decay width of @xmath0 and @xmath1.,width=283 ] note that whether we have the strong interaction terms , or the electromagnetic ones of fig . [ fig:5 ] , the loop contains @xmath63 alone since both , the @xmath64 contact term of the @xmath65 three leg vertex are zero . coming back to the diagrams of fig . [ fig:6 ] , we see that by definition of the potential or kernel of the interaction , see fig . [ fig:1 ] , the sum of the first two diagrams can be cast as the diagram of fig . [ fig:6]c , which is given by @xmath66 where @xmath67 stands for @xmath68 or @xmath69 , @xmath70 are the corresponding spin operators of eqs . ( [ eq:9 ] ) , ( [ eq:10 ] ) and @xmath71 stands for the loop function defined in eq . ( [ eq:6 ] ) evaluated at @xmath72 . however , according to eq . ( [ eq:6 ] ) , we are now at the pole of the amplitude , where @xmath73 , and , thus , we obtain @xmath74 which is the same coupling as in eqs . ( [ eq:9 ] ) , ( [ eq:10 ] ) including the isospin factor for @xmath54 . in other words , the addition of an extra bubble ( loop ) to the series of diagrams of fig . [ fig:2 ] leads to the same series at the pole of the resonance . this means that the coupling of two photons to the resonance is given by the diagram of fig . [ fig:7 ] . namely , that in the present case , and due to the peculiar couplings of the hidden gauge formalism , the coupling of @xmath57 to the dynamically generated @xmath19 resonances is given by the tree level diagram of fig . [ fig:7 ] alone . this makes the evaluation obviously very simple , and taking into account the coupling of the photon to the @xmath62 @xcite @xmath75 we find at the end the two amplitudes @xmath76 } \label{eq:15}\\ s=0&&\nonumber\\ t_{r\rightarrow\gamma\gamma}&=&-\displaystyle{\frac{1}{3}\frac{e^2}{2}\frac{g_s}{g^2}\epsilon_i(\gamma_1)\epsilon_i(\gamma_2)}~~. \label{eq:16}\end{aligned}\ ] ] feynman diagram equivalent to those of fig . [ fig:5 ] at the resonance pole energy.,width=170 ] we would like to make here some consideration concerning gauge invariance of the model . this problem was dealt with in detail in @xcite in the radiative decay of axial vector mesons to a pseudoscalar and a photon . in that case the low lying axial vectors were obtained dynamically from the interaction of a pseudoscalar and a vector within the same hidden gauge formalism used here . gauge invariance of the model was proved there by showing first how it works at tree level and then in the case of loops . we follow here the same strategy . first we show the gauge invariance of the tree level set of diagrams of fig . [ fig_new ] for the case @xmath77 ( it is sufficient to make the test for one photon since for two photons it follows a fortiori ) . feynman diagrams leading to a gauge invariance set in @xmath77.,width=302 ] the @xmath78 conversion proceeds via the term of eq . ( [ eq:14 ] ) and , up to a constant , replaces @xmath79 by @xmath80 . the test of gauge invariance proceeds finding a cancellation of terms upon the substitution of @xmath80 by @xmath81 , the photon momentum . we thus proceed by substituting @xmath79 by @xmath81 in the strong amplitude , @xmath82 . for the case of diagram a ) of fig . [ fig_new ] we find @xmath83 upon replacing the sum of polarizations in @xmath84 leading to @xmath85 we find that the second term in eq . ( [ eq:16.2 ] ) leads to a vanishing contribution of eq . ( [ eq:16.1 ] ) . the contribution of the diagram b ) can be obtained from the one of diagram a ) upon exchange @xmath86 , @xmath87 . the sum of polarizations for the intermediate @xmath48 meson leads to eq . ( [ eq:16.2 ] ) with @xmath88 and the contribution of the second term of the propagator vanishes equally . thus , only the @xmath89 part of the propagator contributes and leads to @xmath90 this last term provides a contribution equal , but with opposite sign , to the one of diagram c ) , the contact term which comes from the lagrangian @xcite for @xmath91 @xmath92 upon substitution of one @xmath93 by @xmath81 . this shows that the set of diagrams of fig . [ fig_new ] fulfills the gauge invariance requirement . the test of gauge invariance for the case of the loops contained in the dynamically generated states proceeds like in the case of the axial vector mesons by separating the intermediate propagator into its on shell and off shell parts @xmath94 which allows one to take into account the on shell cancellation found before . the rest of the terms vanish on shell and can be made to cancel a propagator . the cancellation of terms requires now some new diagrams like the one of fig . [ fig_newnew ] . terms encountered in the gauge invariant set of diagrams for @xmath77 in the case of loops.,width=132 ] yet , the interesting thing to observe is that in all terms needed , the photon always comes from a @xmath62 , the peculiar feature of vector meson dominance inherent in the hidden gauge formalism . it means that the @xmath82 interaction contains all these terms removing the @xmath95 coupling . terms like those in fig . [ fig_newnew ] , with two @xmath48 mesons propagating necessarily off shell in the loops appear in the renormalization procedure of @xcite and are effectively incorporated into the scheme through renormalized couplings and subtraction constants . as a consequence of this , the procedure followed here , coupling a @xmath95 to any final @xmath62 in the strong amplitude is the right thing to do , consistent with gauge invariance . we work in the coulomb gauge for the photons ( @xmath96 ) and to sum over the final ( transverse ) polarizations we use @xmath97 the final partial decay width , summing over final and averaging over initial state polarizations , are given by @xmath98 the numerical values require just the knowledge of the couplings @xmath99 and @xmath100 . using eqs . ( [ eq:7 ] ) and ( [ eq:8 ] ) and the results of fig . [ fig:3 ] ( where the spin projectors are excluded in @xmath37 ) we find @xmath101 in tables [ tab:1 ] , [ tab:2 ] we show the results of the couplings for different values of the @xmath44 used in the form factors . . resonance parameters and coupling constants obtained by fitting the results shown in fig . [ fig:3 ] for @xmath102 state with @xmath103 mev . [ tab:1 ] [ cols="^,^,^,^,^",options="header " , ] what we can see is that , independent of the value of @xmath44 , and hence the total width , the value of the couplings @xmath104 is rather stable with the results @xmath105 with uncertainties of the order of 10@xmath106 . with these values , the numerical results for the @xmath57 radiative widths are @xmath107 with estimated errors of 10@xmath106 . the results for the @xmath1 are in perfect agreement with the experimental data quoted in the introduction . in order to compare the results obtained for the @xmath0 with experiment we need also the branching ratio @xmath108 provided by the theory for this resonance . this number can be obtained from @xcite since the total width of the @xmath0 comes about @xmath109 from @xmath2 and @xmath110 from @xmath111 , out of which @xmath109 corresponds to @xmath112 decay . hence , we should compare our results of @xmath113 ev with those of the crystal ball collaboration @xcite of @xmath114 ev . the agreement is very good , but one is left to think why more accurate results are claimed in the crystal ball work than in @xcite in spite of having much less statistics . the estimated 10@xmath106 quoted errors are from the uncertainties in the model parameters . this certainly does not account for the systematic uncertainties related to how accurately the model can be a substitute for the underlying qcd dynamics of the problem . this is obviously difficult to quantize , like in other hadronic models , but should be kept in mind . admitting that the qcd dynamics is richer than the one provided by the hidden gauge mechanism used in the present approach , the hopes are that the model resulting from the present framework can be a good approximation to the real dynamics of the interaction of vector mesons in a certain energy regime where we move . how good this approximation is can only be found by testing the model with experimental data . the study done in this work on the radiative decay has passed this test . other tests would be most welcome to gradually find support for the idea of these two resonances as being , largely , dynamically generated states from the @xmath19 interaction , or @xmath19 bound states in the present case . certainly , precise measurement of the decay rate for the @xmath0 state , together with simultaneous results for both resonances in other models would be most helpful to further advance in our knowledge of the nature of these resonances . we have followed recent developments in which the @xmath1 and @xmath0 resonances appear as dynamically generated from the interaction of @xmath48 mesons using the hidden gauge formalism for vector mesons . we extended the formalism to account for the radiative decay of the resonances into @xmath3 . the extension has been done following the standard method to deal with dynamically generated resonances , in which the photons are coupled to the components of the resonance , in this case @xmath2 . this is technically implemented by means of loop functions which involve the photon couplings to the components of the resonance . in the present case , the peculiarity of the hidden gauge approach , in which the photons couple directly to one @xmath62 , allows a factorization of the strong part of the interaction and the final result is converted into a tree level contribution , hence rid of any ambiguity due to possible divergences of the loops . the results obtained for the radiative width of the @xmath1 are in perfect agreement with experimental data . so are those for the @xmath0 when they are compared with the experimental results of the crystal ball collaboration , or those of the more recent experiment by belle within its large errors . yet , the large systematic errors quoted in the work from belle , that has much better statistics , should raise some caution on these experimental numbers . with the ultimate goal of learning about the nature of the two resonances discussed , and having in mind the picture as dynamically generated states emerging from the @xmath2 interaction in the local hidden gauge approach , the test passed here in the radiative decay is a first step in the search of support for this idea , and further tests should be most welcome . to further strengthen this idea it would be most useful to have good results for the radiative decay width of the @xmath0 state , as well as results from other theoretical models for both resonances which could tell us how stringent is the test of this radiative decay to discriminate among different models . the work presented here should stimulate research along these lines . this work is partly supported by dgicyt contract number fis2006 - 03438 . this research is part of the eu integrated infrastructure initiative hadron physics project under contract number rii3-ct-2004 - 506078 . the work of h. n. is supported by japan society for the promotion of science ( no . 18 - 8661 ) , and that of j. y. is supported by japan society for the promotion of science ( no . 19 - 2831 ) . the work of s. h. is partially supported by the grant for scientific research ( no . c-20540273 ) from japan society for the promotion of science . m. r. pennington , nucl . suppl . * 181 - 182 * , 251 ( 2008 ) [ arxiv:0806.0328 [ hep - ph ] ] . r. molina , d. nicmorus and e. oset , phys . d * 78 * , 114018 ( 2008 ) [ arxiv:0809.2233 [ hep - ph ] ] . m. bando , t. kugo , s. uehara , k. yamawaki and t. yanagida , phys . lett . * 54 * , 1215 ( 1985 ) . m. bando , t. kugo and k. yamawaki , phys . rept . * 164 * , 217 ( 1988 ) . e. klempt and a. zaitsev , phys . rept . * 454 * , 1 ( 2007 ) [ arxiv:0708.4016 [ hep - ph ] ] . v. crede and c. a. meyer , prog . phys . * 63 * , 74 ( 2009 ) [ arxiv:0812.0600 [ hep - 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using recent results obtained within the hidden gauge formalism for vector mesons , in which the @xmath0 and @xmath1 resonances are dynamically generated resonances from the @xmath2 interaction , we evaluate the radiative decay of these resonances into @xmath3 . we obtain results for the width in good agreement with the experimental data for the @xmath1 state and a width about a factor two smaller for the @xmath0 resonance , which is also in agreement with the data of the crystal ball collaboration and with the more recent ones from the belle collaboration , which however have a very large uncertainty .

we separate the hamiltonian @xmath137 into a block - diagonal part @xmath138 that only acts within the ground and the excited state manifold , respectively , and an off - diagonal part @xmath59 that connects these two manifolds , @xmath139 to implement the schrieffer - wolff ( sw ) transformation @xcite , we construct a unitary transformation @xmath140 with some anti - hermitian @xmath56 to obtain a new hamiltonian @xmath141 , @xmath142 that contains no matrix elements that connect the ground and the excited states up to a desired order in @xmath59 . if we choose the anti - hermitian operator @xmath56 in such a way that @xmath143 = v\ ] ] holds , the leading order in @xmath59 cancels . if we keep the lowest order in @xmath59 , the hamiltonian @xmath141 is approximately given by @xmath144.\ ] ] the block - diagonal part @xmath138 of @xmath137 is given by @xmath145 and the interaction terms are @xmath146 from the condition in eq . ( [ eq : condition ] ) , we find @xmath147 and the effective hamiltonian for the decoupled ground state manifold becomes @xmath148 . \label{eq : effective_hamiltonian_ground_state}\end{aligned}\ ] ] here , we restrict our consideration to the @xmath63 subspace , and define the detuning @xmath64 of the laser frequency from the @xmath65 orbital transition . we omit all constant terms and neglect small energy shifts proportional to @xmath66 and @xmath67 . we start from a hamiltonian @xmath149 that describes two nv centers ( @xmath94 ) coupled to a common cavity mode and each driven by a laser of frequency @xmath41 , @xmath150 where we consider @xmath151 scattering on both nv centers and assume detunings @xmath95 and @xmath96 such that the cavity is excited only virtually . the effective coupling strength @xmath152 is given by @xmath153 where @xmath154 is the coupling strength of nv center @xmath107 to the cavity and @xmath155 is the rabi frequency of the @xmath107th laser field . to derive an effective interaction between the two nuclear spin qubits , we apply a second sw transformation to eliminate the cavity mode , i.e. to decouple the subspaces containing zero and one cavity photon , by choosing @xmath156 we obtain an effective hamiltonian through the unitary transformation @xmath157 where we also keep terms up to the lowest order in the off - diagonal matrix elements . the hamiltonian @xmath158 contains terms that only act on a single nuclear spin @xmath107 , @xmath159 and an interaction part @xmath160 that couples the two nuclear spin qubits , @xmath102 the last term in eq . ( [ eq : effective_h ] ) is zero in the considered subspace that contains no photons . 43ifxundefined [ 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/physreva.80.050302 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.93.130501 [ * * , ( ) ] http://dx.doi.org/10.1038/nature10900 [ * * , ( ) ] link:\doibase 10.1103/physrevb.81.035205 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.113.020506 [ * * , ( ) ] @noop link:\doibase 10.1103/physrevlett.102.057403 [ * * , ( ) ] link:\doibase 10.1126/science.1189075 [ * * , ( ) ] http://dx.doi.org/10.1038/nature10401 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.110.060502 [ * * , ( ) ] http://dx.doi.org/10.1038/nature12919 [ * * , ( ) ] link:\doibase 10.1126/science.1139831 [ * * , ( ) ] link:\doibase 10.1126/science.1176496 [ * * , ( ) ] http://dx.doi.org/10.1038/nphys2444 [ * * , ( ) ] http://dx.doi.org/10.1038/nphys2026 [ * * , ( ) ] link:\doibase 10.1126/science.1157233 [ * * , ( ) ] http://dx.doi.org/10.1038/nphys2545 [ * * , ( ) ] http://dx.doi.org/10.1038/ncomms4371 [ * * , ( ) ] http://dx.doi.org/10.1038/nnano.2014.2 [ * * , ( ) ] link:\doibase 10.1103/physreva.72.052330 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.96.070504 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.107.150503 [ * * , ( ) ] link:\doibase 10.1103/physrevx.4.031022 [ * * , ( ) ] @noop _ _ ( , , ) @noop ( ) http://stacks.iop.org/1367-2630/13/i=2/a=025019 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/j.physrep.2013.02.001 [ * * , ( ) ] link:\doibase 10.1146/annurev - conmatphys-030212 - 184238 [ * * , ( ) ] link:\doibase 10.1103/physrevb.80.241204 [ * * , ( ) ] @noop ( ) link:\doibase 10.1103/physrevb.77.155206 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.101.117601 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.79.075203 [ * * , ( ) ] link:\doibase 10.1103/physrev.149.491 [ * * , ( ) ] in link:\doibase 10.1007/978 - 3 - 540 - 36616 - 4_12 [ _ _ ] , , vol . ( , , ) pp . link:\doibase 10.1103/physrevlett.102.195506 [ * * , ( ) ] http://stacks.iop.org/1367-2630/13/i=2/a=025025 [ * * , ( ) ] link:\doibase 10.1126/science.1255541 [ * * , ( ) ] link:\doibase 10.1103/physreva.89.052317 [ * * , ( ) ] link:\doibase 10.1021/nl061342r [ * * , ( ) ] http://dx.doi.org/10.1038/ncomms6718 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.100.073001 [ * * , ( ) ]

defect centers in diamond are exceptional solid - state quantum systems that can have exceedingly long electron and nuclear spin coherence times . so far , single - qubit gates for the nitrogen nuclear spin , a two - qubit gate with a nitrogen - vacancy ( nv ) center electron spin , and entanglement between nearby nitrogen nuclear spins have been demonstrated . here , we develop a scheme to implement a universal two - qubit gate between two distant nitrogen nuclear spins . virtual excitation of an nv center that is embedded in an optical cavity can scatter a laser photon into the cavity mode ; we show that this process depends on the nuclear spin state of the nitrogen atom . if two nv centers are simultaneously coupled to a common cavity mode and individually excited , virtual cavity photon exchange can mediate an effective interaction between the nuclear spin qubits , conditioned on the spin state of both nuclei , which implements a universal controlled-@xmath0 gate . we predict operation times below 100 nanoseconds , which is several orders of magnitude faster than the decoherence time of nuclear spin qubits in diamond . _ introduction . _ substantial experimental progress has been made in demonstrating the viability of nuclear spins coupled to nitrogen - vacancy ( nv ) centers in diamond as qubits . compared with the nv electron spin , the nuclear spin offers for superior coherence properties , but so far , a scheme for the necessary two - qubit gates is lacking . candidate nuclear spins are the intrinsic nitrogen nuclear spin ( @xmath1n or @xmath2n ) @xcite or incidental proximal nuclear spins ( e.g. @xmath3c ) @xcite . decoherence times of @xmath4 at room temperature have been measured @xcite , and elementary single - qubit operations were implemented , including manipulation @xcite , initialization @xcite and high - fidelity single - shot readout @xcite . it was further demonstrated that the nitrogen nuclear spin can be a functioning part of a small quantum register @xcite , or can act as a quantum memory to store and later retrieve the nv electron spin state @xcite . nuclear spin entanglement has been studied both experimentally @xcite and theoretically @xcite . however , a deterministic long - distance coupling scheme that does not utilize prior electron entanglement has not yet been demonstrated . the coupling of nuclear spins is fundamentally required in the context of quantum information processing , e.g. to perform universal quantum computation @xcite . in this article , we develop and analyze a mechanism to optically generate a controlled quantum gate between two distant nitrogen nuclear spins ( fig . [ fig : nv_center ] ) . the coupling between the nuclear spins is achieved by exchanging virtual cavity photons among two nv centers . external laser photons incident on each nv center can be scattered into the cavity mode , or vice versa , by exciting electronic raman - type transitions between the nv ground and excited state . we find that in the appropriate parameter regime , the scattering process depends on the nitrogen nuclear - spin state and can be completely suppressed for a specific nuclear spin configuration by properly tuning the laser frequency . this nuclear - spin dependent scattering mediates an effective interaction between two nitrogen nuclear spins . for a specific interaction time , a universal controlled-@xmath5 ( @xmath6 ) gate is implemented , which is equivalent to cnot up to single qubit operations . a quantitative analysis of the proposed mechanism yields gate operation times below 100 nanoseconds , which is more than four orders of magnitude shorter than the decoherence time of several milliseconds for the nitrogen nuclear spin . while cavity - mediated coupling between nv _ electron _ spins relies on the zero field splitting @xcite , the coupling of nv nuclear spins has its physical origin in the hyperfine interaction . _ model . _ we start our analysis by describing a single nv center coupled to a single cavity mode and to an external laser field . the extension to two nv centers interacting with the same cavity mode , as required for the two - qubit gate , is straightforward and will be given later . to model the combined system of a single nv center , an optical cavity and the external laser , we use the time - dependent hamiltonian @xmath7 where @xmath8 describes the electron ( e ) and nuclear ( n ) spin systems coupled through hyperfine ( hf ) interactions , @xmath9 the coupling to the cavity , and @xmath10 the interaction with the laser field . in the presence of an external magnetic field @xmath11 along the defect symmetry axis ( @xmath12 axis ) , the electron spin ( @xmath13 ) and nuclear spin ( @xmath14 ) hamiltonians are given by ( @xmath15)@xcite @xmath16 here , @xmath17 mhz / g is the electron gyromagnetic ratio , @xmath18 ev is the energy gap between ground and excited state , and @xmath19 and @xmath20 with the zero - field spin splittings of the ground ( @xmath21 ghz ) and excited state ( @xmath22 ghz ) . the nuclear gyromagnetic ratio is denoted @xmath23 and q is the nuclear electric quadrupole coupling ( see tab . [ tab : values ] ) . to describe the orbital degree of freedom , we use pauli matrices @xmath24 @xmath25 , and choose the ground and excited states as @xmath26 eigenvectors with eigenvalues @xmath27 and @xmath28 , respectively . we only consider the lower orbital branch of the excited state doublet ( @xmath29 ) . this is justified by naturally occurring strain fields of 10 ghz and more @xcite , which split the excited state into two well - separated orbital branches . hyperfine interaction in the excited state is modeled by a diagonal hyperfine tensor , which has the same form as in the ground state @xcite . however , since the electron density at the nitrogen site is larger in the excited state @xcite , the hyperfine interaction is about 20 times stronger compared to the ground state according to measurements under ambient conditions @xcite . the difference @xmath30 between the hyperfine coupling in the ground and the excited state forms the basis of the nuclear - spin dependent light scattering effect which we predict . working at magnetic field strengths away from the ground and excited state level anticrossings , electron - nuclear spin flip - flop processes are energetically suppressed . therefore , we neglect the transverse part of the hyperfine tensor and only include the longitudinal coupling . denoting the hyperfine coupling strengths by @xmath31 and @xmath32 for the ground and excited state ( tab . [ tab : values ] ) , we arrive at @xmath33 where @xmath34 and @xmath35 . .[tab : values]relevant nuclear - spin parameters for the nv center . [ cols="<,^,^",options="header " , ] we consider the nv electronic orbital transition between the ground and excited state to be coupled to a single mode of the optical cavity , which , in the rotating - wave approximation , is described by @xmath36 , where @xmath37 is the cavity frequency , @xmath38 is cavity - photon annihilation ( creation ) operator , @xmath39 the coupling strength ( which be assumed real ) , and @xmath40 . the external laser is described by a classical field of frequency @xmath41 that excites electronic orbital transitions between states having the same spin projections @xmath42 and @xmath43 , @xmath44 . here , @xmath45 is the complex rabi frequency that depends on the phase of the laser field . the hamiltonian @xmath46 can be made time - independent by transforming into a rotating frame , @xmath47 with @xmath48 , and we obtain @xmath49 . the transformed part @xmath50 of the electronic hamiltonian is obtained by replacing @xmath51 with the detuning @xmath52 in @xmath53 . in the hamiltonian @xmath9 , the transformation causes a shift of the cavity frequency to @xmath54 , which is the detuning of the laser from the cavity mode . the laser hamiltonian @xmath10 becomes time - independent , @xmath55 . _ nuclear - spin dependent scattering . _ virtually exciting the nv center by the external laser field can finally lead to an excitation of the cavity mode through the coupling @xmath39 . we describe this process by using quasi - degenerate perturbation theory in terms of a schrieffer - wolff ( sw ) transformation @xcite to eliminate the intermediate virtual transition to the excited state , and obtain a model that effectively describes the scattering of a laser photon into the cavity mode , and vice versa , that particularly depends on the nitrogen nuclear spin projection @xmath43 . it is exactly this spin - dependent scattering that eventually enables a conditional two - qubit quantum gate . to implement the sw transformation , we construct an anti - hermitian operator @xmath56 such that @xmath57 = v$ ] ( see appendix ) , where the part @xmath58 only acts on the ground and excited state manifold , respectively , and @xmath59 describes transitions between these two hilbert subspaces . in the transformed hamiltonian @xmath60 , we keep the lowest order in the interaction @xmath59 and continue with the effective hamiltonian @xmath61/2 $ ] . the effective ground - state hamiltonian becomes ( see appendix ) @xmath62 . \nonumber\end{aligned}\ ] ] here , we restrict our consideration to the @xmath63 subspace , and define the detuning @xmath64 of the laser frequency from the @xmath65 orbital transition ( fig . [ fig : nv_center ] ) . we omit all constant terms and neglect small energy shifts proportional to @xmath66 and @xmath67 . on the basis of previous experimental work @xcite using the @xmath68n nuclear spin as a qubit , we choose the nuclear spin sublevels @xmath69 and @xmath70 as the computational basis . we can neglect the @xmath71 state because the transition frequency between these two levels is well separated from other transitions @xcite . from eq . ( [ eq : effective_hamiltonian_ground_state ] ) , one can see that the effective coupling of the nv center to the cavity via the virtual laser excitation depends on the spin state of nitrogen nucleus and can , e.g. , be completely suppressed for one of the two spin states . this is the case if the laser frequency is chosen such that e.g. @xmath72 , where only scattering from the @xmath73 state is possible . by using @xmath74 and @xmath75 , we find the qubit hamiltonian @xmath76 with an effective coupling strength @xmath77 scattering only from the @xmath78 state is possible for @xmath79 occurring with the same effective coupling strength @xmath80 [ eq . ( [ eq : effective_single_qubit_coupling ] ) ] ; however , we concentrate on @xmath73 scattering in the following . _ spin - spin interactions . _ to understand the scattering mechanism of a laser photon into the cavity mode qualitatively , we so far neglected spin - mixing terms in the lower branch of the excited state doublet @xcite . however , to make quantitative predictions of the effective scattering process , we take into account the fine structure of the excited state manifold . so far , electronic spin - spin interactions were only incorporated by the zero - field splittings @xmath81 and @xmath82 . in the limit of high strain considered here , the two branches of the excited - state orbital doublet split and anticrossings in the lower branch mix spin states with different quantum numbers @xmath42 the hamiltonian describing the spin mixing is @xcite @xmath83 where transitions between the excited state orbitals have been neglected due to the high strain , and the fine structure parameters are given by @xmath84 ghz and @xmath85 ghz @xcite . the effective coupling strength @xmath86 analogous to eq . ( [ eq : effective_single_qubit_coupling ] ) can be obtained by adding @xmath87 to the bare nv hamiltonian @xmath88 , and then performing the sw transformation . in doing so , we assume the cavity to be populated by at most one photon , and only if the nv center is in the ground state . in the excited state , we need to include all spin states @xmath89 . the effective ground state hamiltonian in the case of @xmath73 scattering has the same form as given in eq . ( [ eq : hamiltonian_mi1_scattering ] ) with a different coupling strength @xmath90 . the detuning - dependent part @xmath91 is plotted in fig . [ fig : comparison_geff ] . ratio @xmath92 of coupling strengths with ( @xmath86 ) and without ( @xmath80 ) spin - spin interaction in the excited state . magnetic field strengths @xmath93 are chosen such that @xmath43 is a good quantum number in the ground and excited state . ] _ controlled quantum gate . _ for the two - qubit gate , we consider two nv centers ( @xmath94 ) coupled to the same cavity mode and each individually driven by a laser of frequency @xmath41 [ fig . [ fig : nv_center ] ( c ) ] . in the following , we keep only the lowest order of the interaction parts , and consider @xmath73 scattering on both nv centers . furthermore , we assume detunings @xmath95 and @xmath96 such that the cavity is excited only virtually , which , in turn , leads to an effective interaction between the two nv centers . to describe this interaction , we apply a second sw transformation to @xmath97 to eliminate the cavity mode by choosing @xmath98 , which leads to an effective hamiltonian @xmath99 , where again only the lowest order contribution of the off - diagonal elements is kept ( see appendix ) . @xmath100 comprises single - qubit terms @xmath101 and a two - qubit interaction term , @xmath102 here , @xmath103 is the nuclear spin state of both nv centers 1 and 2 , and the effective two - qubit coupling strength @xmath104 is found to be @xmath105 where @xmath106 denotes the phase of the @xmath107th laser field , @xmath108 . quantitative predictions of @xmath104 are plotted in fig . [ fig : g12 ] ( a ) . ( a ) effective two - qubit coupling strength @xmath104 as a function of @xmath109 for different values of the laser rabi frequency @xmath110 ( see legend , valid for all figures ) and @xmath111 mhz for @xmath68n nuclear spins . ( b ) time @xmath112 to generate a cz gate between the two nuclear spins as a function of @xmath109 for @xmath111 mhz . ( c ) and ( d ) equivalent for @xmath113n nuclear spins . all calculations performed at @xmath114 g. ] since @xmath115 = 0 $ ] , the time evolution @xmath116 generated by the hamiltonian @xmath100 can be written as @xmath117 where @xmath118 is a single - qubit rotation of nuclear spin @xmath107 and @xmath119 describes a two - qubit operation generated by the interaction part @xmath120 . in eq . ( [ eq : time_evolution ] ) , the time evolution of the cavity field has been omitted , since the nuclear spin degree of freedom has been decoupled from the cavity field by the above transformation . in the following , we only concentrate on the two - qubit interaction part , and disregard single - qubit rotations since they can be undone afterwards , e.g. , by off - resonant excitation of the ground - state electronic spin transition , thereby implementing a phase gate on the n nuclear spin @xcite , or direct driving of the nuclear spin transitions @xcite . for an operation time of @xmath121 , a cz gate is implemented on the two nuclear spin qubits , @xmath122 from which cnot can be created using additional hadamard gates @xcite . in fig . [ fig : g12 ] ( b ) , values of @xmath112 are shown for different rabi frequencies @xmath45 . as the main result of our paper , we find fast operation times clearly below 100 ns . in our calculations , we assumed large detunings @xmath123 and @xmath124 to justify the effective model used . _ conclusions . _ nitrogen nuclear spins in diamond have proved to be highly promising candidates to physically realize qubits . we have presented a theoretical proposal for the implementation of a controlled optical cavity - mediated quantum gate between two nitrogen nuclear spin qubits intrinsic to nv centers in diamond . gate operation can be achieved within 100 nanoseconds or less , which is more than four orders of magnitude below the nuclear - spin decoherence time . assuming @xmath125 ns [ see fig . [ fig : g12 ] ( b ) ] , the cavity loss rate @xmath126 must not exceed values of @xmath127 mhz , which requires @xmath128 factors of @xmath129@xmath130 @xcite . silica microsphere cavities can reach such values @xcite and progress towards photonic crystal cavities in bulk diamond exceeding @xmath128-factors of @xmath131 has been recently achieved @xcite . in addition to the presented findings , an equivalent analysis for the @xmath113n nuclear spin with @xmath132 show that the proposed scheme also works for this isotope if the computational basis is chosen as @xmath133 and @xmath134 [ fig . [ fig : nv_center ] ( e ) ] . we find the same effective scattering rate @xmath80 [ eq . ( [ eq : effective_single_qubit_coupling ] ) ] for @xmath135 scattering for laser detunings @xmath136 . including spin - spin interactions , the effective two - qubit coupling strength @xmath104 and the gate time @xmath112 show qualitatively the same behavior as for the @xmath68n nuclear spin , and are depicted in figs . [ fig : g12 ] ( c ) and ( d ) . during the fast electronic excitation cycles , the nuclear spins are subject to a time - varying hyperfine interaction . however , using a spin - fluctuator model , it has been shown that nuclear spin state will be unaffected and coherence can be preserved @xcite . together with elementary and experimentally demonstrated single - qubit operations , the realization of a universal cz gate makes the nitrogen nuclear spin valuable for quantum computation in addition to its remarkable quality as a quantum memory @xcite . _ acknowledgements . _ we acknowledge funding from the dfg within sfb 767 and from the bmbf under the program q.com-hl .

most of the ongoing theoretical and observational effort in cosmology is driven by its three most famous mysteries : inflation , dark matter and dark energy . in comparison , the unexplained origin of large scale cosmic magnetic fields may seem like many other problems in astrophysics technically difficult but of lesser fundamental interest . this may , to some extent , be the case for magnetic fields in mature galaxies , where they could be generated through a dynamo mechanism @xcite . however , explaining their presence in young protogalaxies , clusters and , possibly , in the intergalactic space @xcite , is more challenging . evolution of magnetism in forming cosmic structures is highly non - linear which makes it difficult to conclusively rule out the existence of yet unknown astrophysical mechanisms that could generate magnetic fields on large scales and high redshifts . another possibility is that magnetic fields existed before astrophysical structures began to form @xcite . a primordial magnetic field ( pmf ) could be produced in the aftermath of cosmic phase transitions @xcite or in specially designed inflationary scenarios @xcite . measurements of cosmic microwave background ( cmb ) temperature and polarization could decisively prove their primordial origin if they contained magnetic signatures present at the time of last scattering . a discovery of pmf would have profound implications for our understanding of the early universe , with critical insights into fundamental problems such as the matter - antimatter asymmetry @xcite . a stochastic pmf influences cmb observables in several ways . magnetic stress - energy perturbs the metric which leads to cmb anisotropies , while the lorentz force deflects moving electrons and protons coupled to photons . recently , it has been suggested @xcite that small - scale fields can appreciably alter the recombination history and , consequently , the distance to last scattering , because of the enhanced small scale baryonic inhomogeneities . here we focus on another signature of pmf the faraday rotation ( fr ) of cmb polarization . fr produces a @xmath0 mode type polarization with a characteristic spectrum @xcite as well as non - trivial 4-point correlations of the cmb temperature and polarization . in @xcite , we examined detectability of pmf using different correlators and evaluated their relative merits . we fond that a planck - like experiment can detect scale - invariant pmf of ng strength using the fr diagnostic at @xmath1ghz , while realistic future experiments at the same frequency can detect @xmath2 . this is comparable or better than other cmb probes of pmf , and using multiple frequency channels can further improve on these prospects . at a given direction @xmath3 on the sky , cmb is characterized by its intensity and two additional stokes parameters , @xmath4 and @xmath5 , quantifying its linear polarization . while @xmath6 and @xmath7 are the quantities that experiments directly measure , their values depend on the choice of the coordinate axes . instead , it has become customary to interpret polarization maps by separating them into parity - even and parity - odd patterns , or the so - called @xmath8 and @xmath0 modes @xcite . existence of intensity fluctuations at last scattering implies generation of @xmath8 modes , which by now have been observed and found to be consistent with the spectrum of temperature anisotropies . on the other hand , @xmath0 modes would not be generated at last scattering unless there were gravitational waves or other sources of metric perturbations with parity - odd components such as cosmic defects @xcite or magnetic fields @xcite . weak lensing ( wl ) of cmb photons by the large scale structures along the line of sight distorts polarization patterns generated at last scattering and converts some of the @xmath8 mode into @xmath0 modes , which is expected to be measured with upcoming cmb experiments . a primordial magnetic field present at and just after last scattering will faraday - rotate the plane of polarization of the cmb photons . the rotation angle along @xmath3 is given by @xmath9 where @xmath10 is the differential optical depth , @xmath11 is the line of sight free electron density , @xmath12 is the thomson scattering cross - section , @xmath13 is the scale factor , @xmath14 is the observed wavelength of the radiation , @xmath15 is the `` comoving '' magnetic field , and @xmath16 is the comoving length element along the photon trajectory . statistically homogeneous , isotropic and gaussian distributed stochastic magnetic fields can be characterized by a two - point correlation function in fourier space b_i ( * k * ) b_j ( * k* ) = ( 2)^3 ^(3)(*k * + * k* ) [ ( _ ij - k_i k_j ) s(k ) [ bcorr ] where @xmath17 is the symmetric magnetic power spectrum , and where we omit the anti - symmetric contribution that quantifies the amount of magnetic helicity because only @xmath17 contributes to the cmb observables evaluated in this paper . the shape of @xmath17 depends on the mechanism responsible for production of pmf and generally can be taken to be a power law up to a certain dissipation scale : s(k ) k^2n-3 & + 0 & . [ eq : singlepb ] the dissipation scale , @xmath18 , should , in principle , be dependent on the amplitude and the shape of the magnetic fields spectrum . we assume that @xmath18 is determined by damping into alfven waves @xcite and can be related to @xmath19 as 1.4 h^1/2 ( 10 ^ -7 gauss b_eff ) , [ kibeff ] where @xmath19 is defined as the effective homogeneous field strength that would have the same total magnetic energy density . @xmath19 is related to the fraction of magnetic energy density to the total radiation density , @xmath20 , via @xcite b_eff= 3.25 10 ^ -6 gauss . [ beff - omega ] the generation of cmb polarization and the fr happen concurrently during the epoch of last scattering . however , as we have shown in @xcite , assuming an instantaneous last scattering , i.e. that @xmath8 modes were produce first and subsequently rotated by pmf , results in relatively minor inaccuracies . in this approximation , the relation between the spherical expansion coefficients of the @xmath8 , @xmath0 and @xmath21 fields can be written as b_lm= _ lm_l_1 m_1_lm e_l_1 m_1 m^lm_l_1m_1 , [ eq : blm ] where @xmath22 is defined in terms wigner @xmath23-@xmath24 symbols @xcite . we note that @xmath0 modes from wl can also be schematically written as ( [ eq : blm ] ) but with a different mixing matrix @xmath22 . importantly , the mixing matrix for wl has a parity opposite to that of fr so that the two rotations are orthogonal to each other , making it possible to reconstruct them separately . and @xmath25 ( solid red ) , and a causal spectrum with @xmath26 and @xmath27 , at @xmath28 ghz . the black short - dash line is the input e - mode spectrum , the black dash - dot line is the contribution from inflationary gravitational waves with @xmath29 , while the black long - dash line is the expected contribution from gravitational lensing by large scale structure.,scaledwidth=48.0% ] in fig . [ fig : cl ] we show the b - mode auto - correlation spectra due to fr by stochastic magnetic fields with two different spectra . one , with @xmath30 , corresponds to nearly scale - invariant pmf generated via an inflationary mechanism @xcite , while the other , with @xmath26 , corresponds to pmf produced causally in phase transitions @xcite , e.g. at the time of electroweak or qcd symmetry breaking . also shown is the @xmath8 mode auto - correlation spectrum which acts as a source for the fr @xmath0 modes , as well as @xmath0 modes from inflationary gravitational waves with @xmath29 , and the expected contribution from wl . the fr induced @xmath0 mode spectra have certain characteristic features . in the case of the nearly scale invariant magnetic spectrum , the spectrum is oscillatory . in fact , the shape of the b - mode spectrum mimics that of the e - mode , except for the lack of exponential damping on small scales . this is because @xmath8 modes are suppressed by the silk damping , while pmf can remain correlated on small scales . the damping of the fr induced b - mode power is due to averaging over many random rotations along the line of sight . this translates into a @xmath31 suppression of the angular spectrum , i.e. asymptotically we have @xmath32 at large @xmath33 . in the case of @xmath26 , expected for causally generated magnetic fields @xcite , the fr produced b - mode can dominate the signal at high @xmath33 . it is thus interesting to consider possibility of future @xmath0 mode experiments specially designed to look for cosmological signals at sub - arcmin scales . as a function of the magnetic spectral index @xmath34 for the three estimators , @xmath35 ( solid red ) , @xmath36 ( dashed green ) and @xmath37 ( dotted blue ) . the top panel is for e1 , the middle panel is for e2 and the lower panel is for e3.,scaledwidth=50.0% ] spatially dependent fr produces additional non - gaussian signatures in the cmb polarization . namely , a particular realization of the fr distortion field correlates the respective legendre coefficients @xmath38 and @xmath39 . in fact , as shown in @xcite , it is possible to reconstruct the distortion field @xmath40 from specially constructed linear combinations of products @xmath41 . the additional correlations induced by fr also manifest themselves as connected 4-point functions of the cmb , which , in turn , provide a measurement of the distortion spectrum @xmath42 @xcite . in principle , one can construct four estimators of the distortion spectrum , based on products of two cmb fields one of which contains polarization : @xmath43 and @xmath36 . of these four , the first two receive a large contribution to their variance from the usual scalar adiabatic gaussian perturbations which makes it harder to find the fr signal . in @xcite , we studied the last two , i.e we considered estimators based on 4-point correlations @xmath44 and @xmath45 , and compared them to the 2-point function @xmath46 . we asked which of the estimators has the highest signal to noise for several types of magnetic field spectra and for a range of experimental sensitivities . we have fully accounted for the contamination by weak lensing , which contributes to the variance , but whose contribution to the 4-point correlations is orthogonal to that of fr . to forecast the detectability of fr we looked at three experimental setups : a planck - like satellite @xcite ( e1 ) , a ground- or balloon - based experiment realistically achievable in the next decade ( e2 ) , and a future dedicated cmb polarization satellite ( e3 ) . the forecasts depend on the fraction of the sky covered by the experiment , @xmath47 , which is close to unity for e1 and e3 , and will be smaller for e2 . we present our forecasted bounds on the pmf fraction in fig . [ fig : ovsn ] subject to specifying @xmath47 , which only appears under a quartic root in the bounds on @xmath19 . as fig . [ fig : ovsn ] demonstrates , fr will be a very promising diagnostic of pmf . in particular , future generation of sub - orbital or space - based cmb polarization experiments will be able to detect scale - invariant magnetic fields as weak as @xmath48 g based on the measurement at @xmath1 ghz frequency . measurements at multiple frequencies can further significantly improve on these prospects . the relative strengths of the three estimators , demonstrated in fig . [ fig : ovsn ] can be understood as follows . generally , the @xmath36 and @xmath37 estimators have a larger number of independent modes contributing to the signal than the @xmath0-mode spectrum . thus , in principle , it is not surprising if they result in a higher signal to noise . however , whether that is the case depends on the experimental noise level , and the distribution of power in the given combination of cmb fields and in the magnetic field . for a scale - invariant pmf spectrum , the @xmath0-mode is essentially a copy of the @xmath8-mode , with most of the @xmath0-mode power being on scales where the @xmath8-modes are also most prominent . this results in a strong correlation between @xmath8 and @xmath0 for scale - invariant fields . in the case of the @xmath37 correlation , the underlying @xmath49 and @xmath8 ( @xmath0 is obtained by a scale - invariant rotation of @xmath8 ) fields peak on rather different scales . namely , @xmath49 peaks at @xmath50 while @xmath8 peaks at @xmath51 . in other words , the intrinsic correlation between @xmath49 and @xmath8 is already suboptimal , translating into a lesser correlation between @xmath49 and @xmath0 . thus , for experiments with sufficiently low noise levels , such as e1 , e2 and e3 considered in this paper , the @xmath36 estimator performs better than @xmath37 for scale - invariant fields . this would not necessarily remain true if polarization measurements had a significantly higher experimental noise . for causally generated pmf with steeply rising ( `` blue '' ) spectra , as in the @xmath26 case , the fr power is concentrated on very small scales , far away from the scales at which any of the unrotated cmb fields have significant power . this means that the @xmath0-modes in the observable range are obtained either by a rotation of @xmath8-modes far away from their peak power scale , or by a rotation of peak @xmath8-mode by a negligible angle . this means that @xmath8 and @xmath0 fields peak at very different scales , with their correlation being close to zero over the observable scales . in this case , we see that the @xmath0-mode spectrum , i.e. the @xmath35 correlation , has the highest signal to noise . when interpreting the forecasted bounds on the magnetic field energy fraction or the effective magnetic field strength in fig . [ fig : ovsn ] , several points must be kept in mind . first , the constraints are obtained after setting the dissipation scale to be given by eq . ( [ kibeff ] ) . for scale - invariant fields it makes no difference , as in this case most of the signal is on scales larger than the magnetic dissipation scale , and @xmath18 does not contribute to the normalization of the spectra when @xmath52 . in fact , for scale - invariant fields , the effective field @xmath19 defined via eq . ( [ beff - omega ] ) is the same as the commonly used @xmath53 , which is the field smoothed on a given scale @xmath54 . thus , our forecasts of the minimum detectable @xmath19 for scale - invariant fields can be directly compared to most other bounds in the literature . cmb is less sensitive to magnetic fields with blue spectra because most of the anisotropies are concentrated on very small scales . this is what fig . [ fig : ovsn ] is showing too . here we note that fig . [ fig : ovsn ] assumes that the spectrum will keep rising at the same steep rate ( @xmath26 ) all the way to the dissipation scale . on the other hand , simulations @xcite suggest that the spectrum becomes less steep , with @xmath55 over some intermediate range @xmath56 , implying a smaller net magnetic energy fraction @xmath57 . big bang nucleosynthesis constrains this fraction to be less than @xmath58% @xcite . note that the expected bound from planck ( e1 ) in fig . [ fig : ovsn ] for @xmath26 will be at least an order of magnitude stronger , while e2 and e3 will improve on the bbn bound by two orders of magnitude . here it is worth keeping in mind that our bounds are on the magnetic field contribution at the time of last scattering . while it is expected that the fields remain effectively frozen - in between the time of nucleosynthesis and last scattering , with a relatively slow time evolution of the dissipation scale , this is still an approximation . in any case , it is interesting to know how good the resolution of future @xmath0 mode experiments can be , since the fr contribution from causally generated pmf appears to dominate over other cosmological sources on small scales . in the case of scale - invariant fields , existing bounds on the magnetic field strength from wmap are at a level of a few ng @xcite . these bounds are based on the anisotropies induced by the metric fluctuations sourced by magnetic fields , and ignore the fr effect . in refs . @xcite the wmap bound using fr was obtained at the @xmath59 level . as one can see from fig . [ fig : ovsn ] , planck ( e1 ) can almost match today s bounds for scale invariant ( @xmath60 ) fields using the @xmath36 estimator at only one frequency , while future probes , such as e2 and e3 , can improve the bounds by an order of magnitude ! this suggests that the mode coupling estimators of fr can be a very powerful direct probe of scale - invariant pmf . finally , as mentioned already , the bounds are based on using a single frequency band , while using several bands can further improve the constraints . while our estimates look quite promising , they are still preliminary and ignore the potentially devastating foreground effects . prior to reaching our detectors , cmb must pass through the magnetic field of our own galaxy and will experience fr in which @xmath0 modes are produced . it remains to be shown to what extent the fr due to the galaxy can be subtracted from the cosmological fr signal . l. m. widrow , rev . phys . * 74 * , 775 ( 2002 ) [ astro - ph/0207240 ] . a. neronov and i. vovk , science * 328 * , 73 ( 2010 ) [ arxiv:1006.3504 [ astro-ph.he ] ] . d. grasso and h. r. rubinstein , phys . rept . * 348 * , 163 ( 2001 ) [ astro - ph/0009061 ] . t. vachaspati , phys . lett . b * 265 * , 258 ( 1991 ) . j. m. cornwall , phys . rev . d * 56 * , 6146 ( 1997 ) [ hep - th/9704022 ] . m. s. turner and l. m. widrow , phys . d * 37 * , 2743 ( 1988 ) . b. ratra , astrophys . j. * 391 * , l1 ( 1992 ) . t. vachaspati , phys . lett . * 87 * , 251302 ( 2001 ) [ astro - ph/0101261 ] . k. jedamzik and t. abel , arxiv:1108.2517 [ astro-ph.co ] . a. kosowsky , t. kahniashvili , g. lavrelashvili and b. ratra , phys . rev . d * 71 * , 043006 ( 2005 ) [ astro - ph/0409767 ] . l. pogosian , a. p. s. yadav , y. -f . ng and t. vachaspati , phys . d * 84 * , 043530 ( 2011 ) [ erratum - ibid . d * 84 * , 089903 ( 2011 ) ] [ arxiv:1106.1438 [ astro-ph.co ] ] . s. yadav , l. pogosian and t. vachaspati , arxiv:1207.3356 [ astro-ph.co ] . m. kamionkowski , a. kosowsky and a. stebbins , phys . lett . * 78 * , 2058 ( 1997 ) [ astro - ph/9609132 ] . u. seljak and m. zaldarriaga , phys . lett . * 78 * , 2054 ( 1997 ) [ astro - ph/9609169 ] . u. seljak , u. -l . pen and n. turok , phys . lett . * 79 * , 1615 ( 1997 ) [ astro - ph/9704231 ] . t. r. seshadri and k. subramanian , phys . lett . * 87 * , 101301 ( 2001 ) [ astro - ph/0012056 ] . m. kamionkowski , phys . lett . * 102 * , 111302 ( 2009 ) [ arxiv:0810.1286 [ astro - ph ] ] . s. yadav , r. biswas , 1 , m. su and m. zaldarriaga , phys . d * 79 * , 123009 ( 2009 ) [ arxiv:0902.4466 [ astro-ph.co ] ] . v. gluscevic , m. kamionkowski and a. cooray , phys . d * 80 * , 023510 ( 2009 ) [ arxiv:0905.1687 [ astro-ph.co ] ] . k. jedamzik , v. katalinic and a. v. olinto , phys . d * 57 * , 3264 ( 1998 ) [ astro - ph/9606080 ] . k. subramanian and j. d. barrow , phys . d * 58 * , 083502 ( 1998 ) [ astro - ph/9712083 ] . r. durrer and c. caprini , jcap * 0311 * , 010 ( 2003 ) [ astro - ph/0305059 ] . k. jedamzik and g. sigl , phys . d * 83 * , 103005 ( 2011 ) [ arxiv:1012.4794 [ astro-ph.co ] ] . 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cosmic microwave background ( cmb ) polarization b - modes induced by faraday rotation ( fr ) can provide a distinctive signature of primordial magnetic fields because of their characteristic frequency dependence and because they are only weakly damped on small scales . fr also leads to mode - coupling correlations between the e and b type polarization , and between the temperature and the b - mode . these additional correlations can further help distinguish magnetic fields from other sources of b - modes . we review the fr induced cmb signatures and present the constraints on primordial magnetism that can be expected from upcoming cmb experiments . our results suggest that fr of cmb will be a promising probe of primordial magnetic fields .

due to their layered structure high-@xmath4 superconductors such as @xmath2 have strongly anisotropic properties . the electronic conductivity perpendicular to the @xmath5planes is between two and four orders smaller than in the planes , and the effective interlayer superexchange about five orders weaker than the cu - o - cu in - plane superexchange . @xcite nevertheless , a finite electronic interlayer coupling is essential for 3d antiferromagnetic ( af ) order or 3d bulk superconductivity ( sc ) to occur . @xcite @xmath6 has been an ideal playground for experimental and theoretical studies of interlayer interactions . @xcite it is amenable to doping and offers examples where a small modification of the crystal structure can change the ground state . particularly interesting is the case of @xmath3with @xmath7 , where bulk sc is suppressed and replaced by a static order of charge and spin stripes . @xcite concomitant with stripe order a transition from lto to ltt is observed . @xcite there is growing evidence that the stripe ordered ltt phase causes an electronic decoupling of the @xmath5planes . @xcite the complexity of the involved electronic , magnetic , and structural interactions , however , poses a challenge for an unambiguous experimental analysis . therefore , the focus of the present work lies on lightly doped samples ( @xmath8 ) , where the influence of structure and magnetism on the electronic transport may be deciphered more easily . there is no sc or long range stripe order involved , and the af order is commensurate as long as one does not cool below the spin glass transition . @xcite early magneto - resistance and magnetization measurements on @xmath6 , and more recently on @xmath9 have shown that in the lto phase the electronic interlayer transport depends on how the af sublattices are stacked along the c - axis . @xcite here , similar magneto - resistance experiments on a @xmath10 single crystal are reported . this compound exhibits the same sequence of structural transitions as @xmath11 , thus providing the opportunity to analyze the electronic interlayer coupling in the lightly doped ltt phase . the paper is organized as follows . in the next section the experimental methods are described . the results are presented in sec . [ results ] . there are three parts with focus on the crystal structure , the magneto - transport , and complementary magnetization measurements . in sec . [ discussion ] it is shown how these properties are connected and enable an interpretation of the electronic interlayer transport in the ltt phase . at the end of this section implications for @xmath3 are pointed out . the @xmath10 single crystal with a diameter of @xmath12 5 mm was grown by the travelling - solvent floating - zone method in an atmosphere of flowing oxygen gas at a pressure of @xmath13 atm . as grown the crystal contains a considerable amount of excess oxygen , which was removed by annealing in ar at 900@xmath14c for 24 h. the electric resistance @xmath15 of bar shaped samples was measured with the four terminal method for currents @xmath16 and magnetic fields @xmath17 applied perpendicular and parallel to the @xmath5 planes . the leads were attached with silver epoxy , carefully cured to reduce the contact resistance . the x - ray diffraction experiments were performed at beamline x22c of the national synchrotron light source at a photon energy of 8.9 kev . scattering vectors @xmath18 are specified in units of @xmath19 , where @xmath20 , @xmath21 and @xmath1 are the lattice parameters of the orthorhombic unit cell . @xcite at room temperature @xmath22 @xmath23 , @xmath24 @xmath23 , and @xmath25 @xmath23 , while at 20 k in the ltt phase @xmath26 @xmath23 , and @xmath27 @xmath23 . the experiment was performed in reflection geometry on a polished surface which , due to twinning in the orthorhombic phase , is normal to either [ 1 , 0 , 0 ] or [ 0 , 1 , 0 ] . the static magnetization @xmath28 at constant temperatures and the static susceptibility@xmath29 at constant magnetic fields were measured with a squid ( superconducting quantum interference device ) magnetometer . all studied crystal pieces are twinned in the @xmath30-plane , i.e. , when measuring along the orthorhombic in - plane axes one averages over domains with the @xmath20 and @xmath21 axes interchanged . the degree of twinning was determined for each sample by bulk magnetization measurements and will be indicated wherever of relevance . data with dominant contribution of the @xmath20-axis ( @xmath21-axis ) will be indexed with @xmath31 ( @xmath32 ) . single crystal x - ray diffraction experiments were performed since the interpretation of the transport measurements requires a detailed knowledge of the structure . at high temperature @xcite @xmath10 transforms from the high - temperature tetragonal ( htt ) phase with space group @xmath33 to the lto phase with space group @xmath34 . this transition also occurs in @xmath2 . @xcite however , the eu - doped compound shows a second transition at @xmath35 from lto to ltt with space group @xmath36 . the nature of these transitions has been discussed in numerous studies . @xcite in first approximation all phases can be described by different pattern of tilted @xmath37 octahedra , parameterized by the tilt angle @xmath38 and the tilt direction @xmath39 , measured as the in - plane angle between the tilt axis and the [ 100 ] direction , see fig . [ fig2](d ) . in the htt phase @xmath40 . in the lto phase @xmath41 and @xmath42 , while in the ltt phase @xmath41 and @xmath43 . @xmath38 is on the order of several degree and approximately the same in the lto and ltt phase . thus , the major change at the lto@xmath44ltt transition is a @xmath45 rotation of the tilt axis . note that in the ltt phase @xmath39 changes sign from plane to plane , i.e. , the tilt axes in adjacent layers are orthogonal . there have been questions about whether lightly doped @xmath46 becomes truly tetragonal , or assumes the low - temperature less - orthorhombic ( ltlo ) phase with space - group @xmath47 , which is an intermediate phase between lto and ltt with @xmath48 . @xcite the following results will clarify this point . figure [ fig1 ] shows scans through the ( 4 , 0 , 0)/(0 , 4 , 0 ) reflections . above 125 k their is only one pair of reflections , i.e. , the sample is in the lto phase . upon cooling two additional peaks with reduced split appear , indicating a coexistence of the lto and the ltlo phase . below 120 k the transformation towards ltlo is completed . the orthorhombic strain quickly decreases and below 90 k the crystal is in the ltt phase . a summary of the temperature dependence of some structural properties is given in fig [ fig2 ] . panel ( a ) shows the lattice parameters @xmath20 and @xmath21 , panel ( b ) the sum of the integrated intensity of the ( 5 , 1 , 0)/(-1 , 5 , 0 ) super structure reflections which are allowed in the ltlo and ltt phases only . figure [ fig2](c ) shows the orthorhombic strain @xmath49 , and fig . [ fig2](d ) calculated values for @xmath50 $ ] , where @xmath51 and @xmath52 are the lattice parameters in the lto phase just above the structural transition . in the lto phase @xmath39 was set zero . the x - ray diffraction results clearly demonstrate that the low temperature transition in @xmath10 is a sequence of two transitions : a discontinuous lto@xmath53ltlo transition and a continuous ltlo@xmath53ltt transition . the temperature range of the ltlo phase is very sensitive to excess oxygen , and likely to shrink under more reducing annealing conditions . figure [ fig3](a ) shows the @xmath1-axis resistivity @xmath54 for different magnetic fields @xmath55 . the overall trend is an insulating behavior . however , the magnetic field dependence reveals some dramatic changes as a function of temperature . above the neel temperature of @xmath56 k the field dependence is very small . between @xmath57 and @xmath35 a strong decrease of @xmath58 with increasing @xmath17 is observed . finally , in the ltt phase the field dependence is again small . right at the transition one can see that @xmath59 decreases on cooling , while @xmath60 increases by an equal amount . this shows that the @xmath1-axis transport in the ltt phase is distinct from both the zero field and the high field regime in the lto phase . interestingly , the average @xmath61/2 $ ] shows no significant change at @xmath35 suggesting that primarily the magnetic scattering dependent transport is affected by the structural transformation . the nature of the changes @xmath58 undergoes at the structural transition for @xmath62 is even more obvious in the magneto - resistance curves in fig . [ fig4 ] . in the af ordered lto phase @xmath63 shows a sharp drop which grows with decreasing temperature and reaches @xmath64 at 130 k , fig . [ fig4](a ) . this is so to speak the normal behavior that is also observed in the af ordered lto phase of pure @xmath2 . @xcite it is well established , that the effect is connected to the _ spin - flip _ transition at @xmath65 which alters the spin structure along the @xmath1-axis . @xcite corresponding magnetization data for @xmath10 will be discussed in sec . the new observation is that , in the ltlo and ltt phases , this jump in @xmath63 quickly decreases , becomes hysteretic , and at @xmath66 k amounts to @xmath675% only , fig . [ fig4](b ) . a microscopic interpretation is given in sec . [ discussion ] . a much weaker field dependence of @xmath58 was observed for @xmath68 and @xmath69 . note that the crystal used for the @xmath58 measurements is largely detwinned , i.e. , for 80% of the sample @xmath70 . figure [ fig3](b ) compares @xmath54 for @xmath71 t and 7 t. figure [ fig5 ] compares @xmath63 for all three field directions at @xmath72 k in the lto phase and at @xmath73 k in the ltt phase . in the lto phase a negative magneto - resistance of several percent at 7 t is observed , which is slightly larger for @xmath74 than for @xmath75 , consistent with results for @xmath9 . @xcite in the ltt phase this weak magneto - resistance decreases by one order of magnitude . it is well known that in @xmath6 and in @xmath9 a _ spin - flop _ with concomitant features in the magneto - resistance occurs for @xmath76 and critical fields up to 20 t , depending on the temperature . @xcite in ref . it was suggested that the spin - flop field may decrease substantially in the ltt phase . based on the current data one can safely say that at least up to 7 t no spin - flop takes place in the ltt phase of @xmath10 . measurements of the in - plane resistivity @xmath77 are presented in fig . [ fig3](c ) . the crystal used here is only slightly detwinned , i.e. , for 55% of the sample @xmath78 . at zero field @xmath79 shows a minimum at 200 k and a sharp increase at the lto@xmath44ltlo transition . for @xmath68 and @xmath69 the magneto - resistance at 7 t is very small and barely visible in the @xmath80-dependent data . field loops @xmath81 at fixed temperature show a negative magneto - resistance of less than 1% at 7 t in the lto phase and a one order of magnitude smaller effect in the ltt phase ( not shown ) . for @xmath55 a significant decrease of @xmath79 is observed in the af ordered lto phase , reaching 8% at 130 k and 7 t , see fig . [ fig3](c ) . furthermore , the field loops @xmath81 show the same type of sharp drop at @xmath65 as for @xmath82 and @xmath55 , just much smaller ( not shown ) . in the ltt phase the magneto - resistance is again extremely small . intuitively it is not obvious why , in the lto phase , the in - plane resistivity should decrease at a transition that effects how the spin sublattices are staggered along the @xmath1-axis , but leaves the in - plane spin structure unchanged . the first explanation that comes to mind is that , because of the extreme anisotropy @xmath83 , a minor misalignment of the crystal or of the contacts caused an admixture of a @xmath1-axis component . since the crystal for @xmath77 was quite small we can not rule out this source of error . on the other hand , similar observations have been reported for the lto phase of @xmath2 . @xcite in recent theoretical studies the effect was ascribed to a less anisotropic localization length in the high field regime ( @xmath84 ) . @xcite it was suggested that this results in a more 3d like variable - range - hopping , making more out - of - plane states available for @xmath30-plane transport . assuming this is true , it is clear from the present data that this channel and , thus , @xmath77 become independent of @xmath55 in the ltt phase , because as the spin - flip induced magneto - resistance of @xmath58 disappears , so does the associated change of the @xmath1-axis localization length . the magnetization measurements were performed on a bulky @xmath85 g single crystal . note that similar measurements on a @xmath86 crystal and on @xmath46 polycrystals have been discussed in ref . . the present sample is our first lightly doped crystal and features very sharp transitions . the presentation of data will be limited to @xmath62 , since no significant effects have been observed for @xmath87 and fields up to 7 t , consistent with the absence of significant magnetic field effects in @xmath77 . figure [ fig6](a ) presents the static susceptibility @xmath88 for different @xmath62 . the van vleck susceptibility @xmath89 of the europium ions provides by far the largest contribution ( solid line ) . figure [ fig6](b ) shows the same data after subtraction of @xmath89 , which can now be compared to pure @xmath2 . for @xmath90 t a sharp nel peak at @xmath57 and a jump at @xmath35 are observed . for @xmath91 t and higher fields the susceptibility in the af ordered lto phase starts to increase significantly . the same behavior is observed in @xmath6 . @xcite in contrast , in the ltlo and ltt phases the susceptibility is elevated at any field and shows almost no field dependence . at @xmath92 t the susceptibility increases monotonous with decreasing @xmath80 . as is well documented , the nel peak is the fingerprint of a weak spin canting perpendicular to the @xmath5 planes , caused by dzyaloshinsky moriya ( dm ) superexchange . @xcite each plane carries a weak ferromagnetic moment ( wfm ) which orders antiparallel in adjacent layers for @xmath93 . when the external field @xmath55 exceeds the spin - flip field @xmath65 , needed to overcome the interlayer coupling @xmath94 , the spin lattice of every other layer rotates by @xmath95 , with the effect that the wfm of all planes become parallel to the field . as a result the susceptibility in the lto phase increases . note that this is the reason why for @xmath96 t the peak does not represent @xmath57 , but the temperature below which @xmath97 and wfm start to order antiparallel . the changes across the lto@xmath53ltlo@xmath53ltt transition are also apparent in the magnetization curves @xmath28 . the data in figs . [ fig6](c ) and [ fig6](d ) are after subtraction of the linear @xmath98 van vleck contribution . in the lto phase the spin - flip transition grows sharper and larger for @xmath93 . again , this is the normal behavior found in @xmath2 . @xcite in contrast , below the structural transition no spin - flip transition is observed . the @xmath28 curves are close to being linear in the studied field range , indicating a significant change of the magnetic coupling between the planes . close to @xmath35 the magnetization at maximum field in the lto and ltt phase differs only slightly . the susceptibility at 7 t in fig . [ fig6](b ) shows even better that there is no significant anomaly at the lto@xmath53ltlo@xmath53ltt transition . this implies that the wfm do not change their size across the transition , and at 7 tesla are ferromagnetically aligned in all three phases . there is a small number of interesting theoretical studies on this new magnetic state , motivated by experiments on @xmath99 . @xcite however , the static magnetization presented in fig . [ fig6 ] and in ref . seems to escape these earlier calculations , in particular with respect to the structure dependence of the @xmath28 curves and the saturation field and moment of the wfm in the ltt phase . @xcite figure [ fig7 ] compares the resistivity drop @xmath100 with the moment change @xmath101 at the spin - flip transition . in the lto phase the data are qualitatively the same as for @xmath2 , @xcite whereas in the ltlo and ltt phase one can see the dramatic drop of these quantities . note that @xmath101 reflects the af coupled part of the wfm only . the total wfm , which also consists of a non - spin - flip part ( in particular in the ltt phase ) , continues to grow on cooling ( cf . fig . 22 in ref . ) . [ dis ] in several theoretical studies , motivated by the experiments on @xmath6 and @xmath9 , it was pointed out that the electronic transport between the @xmath5 planes does not depend on the direction of the weak ferromagnetic moments , but on the relative orientation @xmath102 of the spin @xmath103 sublattices in neighbor planes . @xcite the apparent reason is that holes in an antiferromagnet prefer to hop between sublattices with same spin direction . as is shown schematically in fig . [ fig8 ] for the lto phase , this implies that interlayer hopping at low fields ( @xmath104 ) takes place predominantly along the @xmath21-axis , whereas above the spin - flip field ( @xmath105 ) it takes place predominantly along the @xmath20-axis . the negative @xmath1-axis magneto - resistance then requires that , microscopically ( not measured ) , the interlayer hopping resistance along @xmath20 in the high - field regime is smaller than along @xmath21 in the low - field regime ( @xmath106 ) . an intuitive explanation for this is that @xmath107 , although the details are known to be more complicated . @xcite in the ltt phase the situation is quite different ( fig . [ fig8 ] ) . the octahedral tilt axes have rotated by @xmath108 in adjacent layers . the magnetization measurements on @xmath46 presented here and in ref . , as well as neutron diffraction experiments on @xmath109 in ref . show that , due to dm superexchange , spins follow the alternating rotation of the tilt axes . this means that spins are canted out - of - plane , but now form a non - collinear spin structure . both the tetragonal symmetry ( @xmath110 ) and the non - collinear spin structure ( @xmath111 ) cause a frustration of the interlayer superexchange , resulting in the absence of a well - defined spin - flip in the @xmath28 curves , see fig . [ fig6](d ) . moreover , the two sketched ltt spin configurations with antiparallel ( left ) and parallel ( right ) alignment of the wfm are energetically nearly equivalent , and should both populate the ground state . @xcite what are the consequences for the @xmath1-axis magneto - transport in the ltt phase ? because @xmath110 and @xmath111 , both interlayer hopping directions are structurally and magnetically equivalent . moreover , the application of a high magnetic field @xmath112 has no effect on @xmath102 , although it shifts the magnetic ground state towards the one in the right panel with parallel wfm . hence , the ltt phase is expected to be `` spin - valve '' inactive , consistent with the dramatic decrease of the magneto - resistance in fig . [ fig3](a ) and fig . [ fig4](b ) . the remaining magneto - resistance of @xmath58 for @xmath55 and its field hysteresis at low temperatures , fig . [ fig4](b ) , still lack interpretation . it is unclear whether these features are intrinsic or due to structural imperfections of the ltt phase , resulting from a limited domain size and ltlo or lto like domain boundaries . @xcite nevertheless , these features seem to correspond with the hysteresis and remanent moment observed in the magnetization curves throughout the entire af ordered ltt phase of @xmath46 ( @xmath8 ) . @xcite the ltlo phase , represented by the middle panels in fig . [ fig8 ] , is expected to show some intermediate behavior . in the temperature range @xmath113 , where this phase assumes 100% volume fraction , it offers a unique opportunity to study the interlayer magneto - transport as a function of @xmath114 and @xmath115 . figure [ fig9](a ) shows the temperature dependence of @xmath114 and @xmath116 for @xmath117 , normalized to their values at @xmath35 . the correct way to compare these properties is after division by their values in the lto phase , extrapolated into the ltt phase ; see functions @xmath118 and @xmath119 . the result is shown in fig . [ fig9](b ) . several scenarios are possible . if @xmath116 ( blue circles ) depends primarily on the spin orientation @xmath102 , then it should be proportional to @xmath120 ( red triangles ) . however , it is more likely that @xmath121 also depends on the orthorhombic strain , which produces the anisotropy of the interlayer hopping along @xmath20 and @xmath21 in first place , so that one may expect @xmath121 to be in first approximation proportional to @xmath122 ( black squares ) . the similarity between the temperature dependencies of @xmath116 and @xmath123 clearly shows that these quantities are connected . within the experimental error of the independent x - ray diffraction and magneto - resistance measurements it is , however , not possible to decide on the exponent @xmath124 . to isolate the effects of @xmath114 and @xmath102 on @xmath116 , one could study the magneto - resistance in the lto phase under pressure . pressure is known to reduce the orthorhombic strain . what has been learned that may apply to the stripe ordered ltt phase of @xmath3 ? the stripe phase consists of spin stripes coupled antiphase across the charge stripes ( fig . 3 in ref . the stripe direction is parallel to the cu - o - cu bond but rotates by @xmath125 in adjacent planes , similar to the octahedral tilt axes . in zero field spins are parallel to the stripes , resulting in a non - collinear spin structure ( @xmath111 ) . it is easy to see that for this type of system a large normal state @xmath1-axis magneto - resistance may not be observed for any field direction . first the tetragonal symmetry offers no advantage for any interlayer hopping direction . second in terms of the simple hopping picture often applied to the lightly doped compounds , i.e. , hole and spin swap sites , interlayer hopping in the stripe phase always creates frustrated spin moments between antiphase spin stripes . the application of a high magnetic field @xmath55 was shown to have no effect on the magnetic order . @xcite even if spin stripes carry a wfm due to dm superexchange ( which is still unknown ) the net wfm of each plane cancels out because of the phase shift by @xmath126 across charge stripes . hence , no spin - flip transition can be induced , ruling out a similarly strong @xmath1-axis magneto - resistance as in the af ordered lto phase of the lightly doped compounds . application of a high magnetic field parallel to the @xmath5planes produces a collinear spin structure , i.e. , spins within the stripes rotate until they are approximately perpendicular to the field . @xcite however the stripes themselves do not rotate . thus , even if slight lattice distortions due to , e.g. , the charge stripes would lift the structural frustration , the topology of the interlayer superexchange for a collinear spin configuration compares to a checkerboard pattern with @xmath127 and @xmath128 . as discussed in refs . , this indeed perfect magnetic and electronic decoupling of the planes seems responsible for the frustration of the interlayer josephson coupling and the concomitant loss of 3d superconducting phase coherence . in summary , the magneto - transport of lightly hole doped @xmath10 has been explored and linked to structural and magnetic properties . it was shown that the low temperature structural transition from orthorhombic to tetragonal symmetry and from collinear to non - collinear spin structure eliminates the spin - valve type contribution to the interlayer magneto - resistance . after calculating out spin orientation dependent effects by averaging high and low field data , the interlayer transport appears largely unaffected by the structural transition . in contrast , the transition triggers a significant increase of the in - plane charge carrier localization . the author thanks j. m. tranquada for fruitful discussions , and p. reutler and g. dhalenne for support during the crystal growth experiment at the laboratoire de physico - chimie de letat solide in orsay . the work at brookhaven was supported by the office of science , u.s . department of energy under contract no . de - ac02 - 98ch10886 . the @xmath28 curves in the ltt phase are still poorly understood . there is no sharp spin - flip , as well as no spontaneous weak ferromagnetism ( @xmath132 t ) . although in the low field regime @xmath133 is significantly larger than in the lto phase , which indicates that the af interlayer coupling is much weaker , it still takes very high fields ( @xmath134 t ) to ferromagnetically align all wfm . let us describe the af order of the lto phase as @xmath135 , where @xmath136 and @xmath137 means wfm moment up and down . then , the zero field ground state of the ltt phase may possibly look like @xmath138 . there are random stacks with @xmath139 , @xmath140 and @xmath141 of different length . the size of these domains is likely to vary also in the @xmath30 plane , and will most likely be smaller than in the lto phase ( due to the first order nature of the lto@xmath53ltlo transition ) . this may cause a broad distribution of local critical fields and , thus , smear out the spin - flip to a degree that the @xmath28 curves become effectively linear in a wide field range .

the electronic interlayer transport of the lightly doped antiferromagnet @xmath0 has been studied by means of magneto - resistance measurements . the central problem addressed concerns the differences between the electronic interlayer coupling in the tetragonal low - temperature ( ltt ) phase and the orthorhombic low - temperature ( lto ) phase . the key observation is that the spin - flip induced drop in the @xmath1-axis magneto - resistance of the lto phase , which is characteristic for pure @xmath2 , dramatically decreases in the ltt phase . the results show that the transition from orthorhombic to tetragonal symmetry and from collinear to non - collinear antiferromagnetic spin structure eliminates the strain dependent anisotropic interlayer hopping as well as the concomitant spin - valve type transport channel . implications for the stripe ordered ltt phase of @xmath3 are briefly discussed .

this paper continues to develop the line of ideas in @xcite , which are all motivated by the mzard - parisi formula for the free energy in the diluted spin glass models originating in @xcite . this formula is closely related to the original parisi formula @xcite for the free energy in the sherrington - kirkpatrick model @xcite , but at the same time it is more complicated , because it involves a more complicated functional order parameter that encodes some very special structure of the distribution of all spins ( or all multi - overlaps ) rather than the distribution of the overlaps . an important progress was made by franz and leone in @xcite , who showed that the mzard - parisi formula gives an upper bound on the free energy ( see also @xcite ) . the technical details of their work are very different but , clearly , inspired by the analogous earlier result of guerra @xcite for the sherrington - kirkpatrick model , which lead the to the first proof of the parisi formula by talagrand in @xcite . another proof of the parisi formula was given later in @xcite , based on the ultrametricity property for the overlaps proved in @xcite using the ghirlanda - guerra identities @xcite ( the general idea that stability properties , such as the aizenman - contucci stochastic stability @xcite or the ghirlanda - guerra identities , could imply ultrametricity is due to arguin and aizenman , @xcite ) . the proof there combined the cavity method in the form of the aizenman - sims - starr representation @xcite with the description of the asymptotic structure of the overlap distribution that follows from ultrametricity and the ghirlanda - guerra identities @xcite . the mzard - parisi ansatz in the diluted models builds upon the ultrametric parisi ansatz in the sk model , so it is very convenient that ultrametricity for the overlaps can be obtained just as easily in the diluted models as in the sk model , simply because the ghirlanda - guerra identities can be proved in these models in exactly the same way , by using a small perturbation of the hamiltonian of the mixed @xmath1-spin type . however , as we mention above , the mzard - parisi ansatz describes the structure of the gibbs measure in these models in much more detail , as we shall see below . some progress toward explaining the features of this ansatz beyond ultrametricity was made in @xcite , where the so - called hierarchical exchangeability of the pure states and the corresponding aldous - hoover representation were proved . this representation looks very similar to what one expects in the mzard - parisi ansatz , but lacks some additional symmetry . one example where this additional symmetry can be proved rigorously was given in @xcite for the @xmath2-rsb asymptotic gibbs measures in the diluted @xmath0-spin model , where it was obtained as a consequence of the cavity equations for spin distributions developed rigorously in @xcite . the main contribution of this paper is to show how this result can be extended to all finite - rsb asymptotic gibbs measures for all diluted models . namely , we will show that one can slightly modify the hamiltonian in such a way that all finite - rsb asymptotic gibbs measures satisfy the mzard - parisi ansatz as a consequence of the ghirlanda - guerra identities and the cavity equations . before we can state our main results , we will need to introduce necessary notations and definitions , as well as review a number of previous results . let @xmath3 be an integer fixed throughout the paper . a random clause with @xmath0 variables will be a random function @xmath4 on @xmath5 symmetric in its coordinates . the main examples we have in mind are the following . * example 1 . * ( @xmath0-spin model ) given an inverse temperature parameter @xmath6 , the random function @xmath7 is given by @xmath8 where @xmath9 is a random variable , typically , standard gaussian or rademacher . * example 2 . * ( @xmath0-sat model ) given an inverse temperature parameter @xmath6 , the random function @xmath7 is given by @xmath10 where @xmath11 are i.i.d . bernoulli random variables with @xmath12 we will denote by @xmath13 independent copies of the function @xmath7 for various multi - indices @xmath14 . given a parameter @xmath15 , called connectivity parameter , the hamiltonian of a diluted model is defined by @xmath16 where @xmath17 is a poisson random variable with the mean @xmath18 , and the coordinate indices @xmath19 are independent for different pairs @xmath20 and are chosen uniformly from @xmath21 . the main goal for us would be to compute the limit of the free energy @xmath22 as @xmath23 goes to infinity . the formula for this limit originates in the work of mzard and parisi in @xcite . to state how the formula looks like , we need to recall several definitions that will be used throughout the paper . * ruelle probability cascades ( rpc , @xcite ) . * given @xmath24 , consider an infinitary rooted tree of depth @xmath25 with the vertex set @xmath26 where @xmath27 , @xmath28 is the root of the tree and each vertex @xmath29 for @xmath30 has children @xmath31 for all @xmath32 . each vertex @xmath33 is connected to the root @xmath28 by the path @xmath34 we will denote the set of vertices in this path by @xmath35 we will denote by @xmath36 the distance of @xmath33 from the root , i.e. @xmath1 when @xmath37 . we will write @xmath38 if @xmath39 and @xmath40 if , in addition , @xmath41 , in which case we will say that @xmath33 is a descendant of @xmath42 , and @xmath42 is an ancestor of @xmath33 . notice that @xmath43 if and only if @xmath44 the set of leaves @xmath45 of @xmath46 will sometimes be denoted by @xmath47 . for any @xmath48 , let @xmath49 be the number of common vertices ( not counting the root @xmath28 ) in the paths from the root to the vertices @xmath33 and @xmath42 . in other words , @xmath50 is the distance of the lowest common ancestor of @xmath33 and @xmath42 from the root . let us consider parameters @xmath51 that will appear later in the c.d.f . of the overlap in the case when it takes finitely many values ( see ( [ zetap ] ) below ) , which is the usual functional order parameter in the parisi ansatz . for each @xmath52 , let @xmath53 be a poisson process on @xmath54 with the mean measure @xmath55 with @xmath56 , and we assume that these processes are independent for all @xmath33 . let us arrange all the points in @xmath53 in the decreasing order , @xmath57 and enumerate them using the children @xmath58 of the vertex @xmath33 . given a vertex @xmath59 and the path @xmath60 in ( [ pathtoleaf ] ) , we define @xmath61 and for the leaf vertices @xmath62 we define @xmath63 these are the weights of the ruelle probability cascades . for other vertices @xmath64 we define @xmath65 this definition obviously implies that @xmath66 when @xmath67 . let us now rearrange the vertex labels so that the weights indexed by children will be decreasing . for each @xmath52 , let @xmath68 be a bijection such that the sequence @xmath69 is decreasing . using these local rearrangements we define a global bijection @xmath70 in a natural way , as follows . we let @xmath71 and then define @xmath72 recursively from the root to the leaves of the tree . finally , we define @xmath73 the weights ( [ vs ] ) of the rpc will be accompanied by random fields indexed by @xmath45 and generated along the tree @xmath46 as follows . * hierarchical random fields . * let @xmath74 be i.i.d . random variables uniform on @xmath75 $ ] . given a function @xmath76^r \to [ -1,1]$ ] , consider a random array indexed by @xmath77 , @xmath78 note that , especially , in subscripts or superscripts we will write @xmath79 instead of @xmath80 . we will also denote by @xmath81 and @xmath82 copies of the above arrays that will be independent for all different multi - indices @xmath83 the function @xmath84 above is the second , and more complex , functional order parameter that encodes the distribution of spins inside the pure states in the mzard - parisi ansatz , as we shall see below . this way of generating the array @xmath85 using a function @xmath76^r \to [ -1,1]$ ] is very redundant in a sense that there are many choices of the function @xmath84 that will produce the same array in distribution . however , if one prefers , there is a non - redundant ( unique ) way to encode an array of this type by a recursive tower of distributions on the set of distributions that is more common in physics literature . * extension of the definition of clause . * let us extend the definition of function @xmath7 on @xmath5 to @xmath86^k$ ] as follows . often we will need to average @xmath87 over @xmath88 ( or some subset of them ) independently of each other , with some weights . if we know that the average of @xmath89 is equal to @xmath90 $ ] then the corresponding measure is given by @xmath91 we would like to denote the average of @xmath92 again by @xmath92 , which results in the definition @xmath93 here is how this general definition would look like in the above two examples . in the first example of the @xmath0-spin model , using that @xmath94 , we can write @xmath95 and , clearly , after averaging , @xmath96 in the second example of the @xmath0-sat model , using that @xmath97 , we can write @xmath98 and after averaging , @xmath99 * the mzard - parisi formula . * let @xmath100 and @xmath101 be poisson random variables with the means @xmath102 and @xmath103 correspondingly and consider @xmath104 for @xmath105 and @xmath106 let @xmath107 denote the average over @xmath108 and consider the following functional @xmath109 that depends on @xmath25 , the parameters ( [ zetas ] ) and the choice of the functions @xmath84 in ( [ mpfop ] ) . then the mzard - parisi ansatz predicts that @xmath110 at least in the above two examples of the @xmath0-spin and @xmath0-sat models . we will see below that all the parameters have a natural interpretation in terms of the structure of the gibbs measure . * franz - leone upper bound . * as we mentioned in the introduction , it was proved in @xcite that @xmath111 for all @xmath23 , in the @xmath0-spin and @xmath0-sat models for even @xmath0 . their proof was rewritten in a slightly different language in @xcite to make it technically simpler , and it was observed by talagrand in @xcite that the proof actually works for all @xmath3 in the @xmath0-sat model . as a natural starting point for proving matching lower bound , a strengthened analogue of the aizenman - sims - starr representation @xcite for diluted models was obtained in @xcite in the language of the so called asymptotic gibbs measures . we will state this representation in theorem [ th1 ] below for a slightly modified hamiltonian , while also ensuring that the asymptotic gibbs measures satisfy the ghirlanda - guerra identities . to state this theorem , we need to recall a few more definitions . * asymptotic gibbs measures . * the gibbs ( probability ) measure corresponding to a hamiltonian @xmath112 on @xmath113 is defined by @xmath114 where the normalizing factor @xmath115 is called the partition function . to define the notion of the asymptotic gibbs measure , we will assume that the distribution of the process @xmath116 is invariant under the permutations of the coordinates of @xmath117 - this property is called symmetry between sites , and it clearly holds in all the models we consider . let @xmath118 be an i.i.d . sequence of replicas from the gibbs measure @xmath119 and let @xmath120 be the joint distribution of the array @xmath121 of all spins for all replicas under the average product gibbs measure @xmath122 , @xmath123 for any @xmath124 and any @xmath125 . we extend @xmath120 to a distribution on @xmath126 simply by setting @xmath127 for @xmath128 let @xmath129 denote the set of all possible limits of @xmath130 over subsequences with respect to the weak convergence of measures on the compact product space @xmath126 . because of the symmetry between sites , all measures in @xmath129 inherit from @xmath120 the invariance under the permutation of both spin and replica indices @xmath131 and @xmath132 by the aldous - hoover representation @xcite , @xcite for such distributions , for any @xmath133 , there exists a measurable function @xmath134 ^ 4\to\{-1,+1\}$ ] such that @xmath135 is the distribution of the array @xmath136 where the random variables @xmath137 are i.i.d . uniform on @xmath75 $ ] . the function @xmath138 is defined uniquely for a given @xmath139 up to measure - preserving transformations ( theorem 2.1 in @xcite ) , so we can identify the distribution @xmath135 of array @xmath140 with @xmath138 . since @xmath138 takes values in @xmath141 , the distribution @xmath135 can be encoded by the function @xmath142 where @xmath143 is the expectation in @xmath144 only . the last coordinate @xmath145 in ( [ sigma ] ) is independent for all pairs @xmath146 , so it plays the role of ` flipping a coin ' with the expected value @xmath147 . therefore , given the function ( [ fop ] ) , we can redefine @xmath138 by @xmath148 without affecting the distribution of the array @xmath140 . we can also view the function @xmath149 in ( [ fop ] ) in a more geometric way as a random measure on the space of functions , as follows . let @xmath150 and @xmath151 denote the lebesgue measure on @xmath75 $ ] and let us define a ( random ) probability measure @xmath152 on the space of functions of @xmath153 $ ] , @xmath154 equipped with the topology of @xmath155 , dv)$ ] . we will denote the scalar product in @xmath155 , dv)$ ] by @xmath156 and the corresponding @xmath157 norm by @xmath158 . the random measure @xmath159 in ( [ gibbsw ] ) is called an asymptotic gibbs measure . the whole process of generating spins can be broken into several steps : 1 . generate the gibbs measure @xmath160 using the uniform random variable @xmath161 ; 2 . consider i.i.d . sequence @xmath162 of replicas from @xmath159 , which are functions in @xmath163 ; 3 . plug in i.i.d . uniform random variables @xmath164 to obtain the array @xmath165 ; 4 . finally , use this array to generate spins as in ( [ sigmatos ] ) . for a different approach to this definition via exchangeable random measures see also @xcite . from now on , we will keep the dependence of the random measure @xmath159 on @xmath161 implicit , denote i.i.d . replicas from @xmath159 by @xmath166 , which are now functions on @xmath75 $ ] , and denote the sequence of spins ( [ sigmatos ] ) corresponding to the replica @xmath167 by @xmath168 because of the geometric nature of the asymptotic gibbs measures @xmath159 as measures on the subset of @xmath155,dv)$ ] , the distance and scalar product between replicas play a crucial role in the description of the structure of @xmath159 . we will denote the scalar product between replicas @xmath167 and @xmath169 by @xmath170 , which is more commonly called the overlap of @xmath167 and @xmath169 . let us notice that the overlap @xmath171 is a function of spin sequence ( [ spinsell ] ) generated by @xmath167 and @xmath169 since , by the strong law of large numbers , @xmath172 almost surely . * the ghirlanda - guerra identities . * given @xmath124 and replicas @xmath173 , we will denote the array of spins ( [ spinsell ] ) corresponding to these replicas by @xmath174 we will denote by @xmath175 the average over replicas @xmath167 with respect to @xmath176 . in the interpretation of the step ( ii ) above , this is the same as averaging over @xmath177 in the sequence @xmath178 . let us denote by @xmath179 the expectation with respect to random variables @xmath161 , @xmath180 and @xmath145 . we will say that the measure @xmath159 on @xmath163 satisfies the ghirlanda - guerra identities if for any @xmath181 any bounded measurable function @xmath182 of the spins @xmath183 in ( [ sn ] ) and any bounded measurable function @xmath184 of one overlap , @xmath185 another way to express the ghirlanda - guerra identities is to say that , conditionally on @xmath183 , the law of @xmath186 is given by the mixture @xmath187 where @xmath188 denotes the distribution of @xmath189 under the measure @xmath190 , @xmath191 the identities ( [ gg ] ) are usually proved for the function @xmath182 of the overlaps @xmath192 instead of @xmath183 , but exactly the same proof yields ( [ gg ] ) as well ( see e.g. section 3.2 in @xcite ) . it is well known that these identities arise from the gaussian integration by parts of a certain gaussian perturbation hamiltonian against the test function @xmath182 , and one is free to choose this function to depend on all spins and not only overlaps . * modification of the model hamiltonian . * next , we will describe a crucial new ingredient that will help us classify all finite - rsb asymptotic gibbs measures . let us consider a sequence @xmath193 of independent gaussian random variables satisfying @xmath194 where @xmath195 is a fixed small parameter , and consider the following random clauses of @xmath196 variables , @xmath197 we will denote by @xmath198 and @xmath199 independent copies over different multi - indices @xmath14 . we will define a perturbation hamiltonian by @xmath200 where @xmath201 are poisson random variables with the mean @xmath23 independent over @xmath202 and @xmath203 are chosen uniformly from @xmath21 independently for different indices @xmath14 . notice that , because of ( [ gsvar ] ) , this hamiltonian is well defined . we will now work with the new hamiltonian given by @xmath204 notice that the second term @xmath205 is of the same order as the model hamiltonian , but its size is controlled by the parameter @xmath195 . for example , if we consider the free energy @xmath206 corresponding to this modified hamiltonian , letting @xmath207 go to zero will give the free energy of the original model . finally , as in ( [ deftheta ] ) , let us extend the definition of @xmath208 by @xmath209 to @xmath86^d$ ] from @xmath210 * the cavity equations for the modified hamiltonian . * let us now recall the cavity equations for the distribution of spins proved in @xcite . these equations will be slightly modified here to take into account that the perturbation hamiltonian @xmath211 will now also contribute to the cavity fields . we will need to pick various sets of different spin coordinates in the array @xmath140 in ( [ sigma ] ) , and it is inconvenient to enumerate them using one index @xmath212 . instead , we will use multi - indices @xmath213 for @xmath124 and @xmath214 and consider @xmath215 where all the coordinates are uniform on @xmath75 $ ] and independent over different sets of indices . similarly , we will denote @xmath216 for convenience , below we will separate averaging over different replicas @xmath217 , so when we average over one replica we will drop the superscript @xmath217 and simply write @xmath218 now , take arbitrary integers @xmath219 such that @xmath220 the index @xmath221 will represent the number of replicas selected , @xmath222 will be the total number of spin coordinates and @xmath223 will be the number of cavity coordinates . for each replica index @xmath224 we consider an arbitrary subset of coordinates @xmath225 and split them into cavity and non - cavity coordinates , @xmath226 the following quantities represent the @xmath131th coordinate cavity field of the modified hamiltonian ( [ hammain ] ) in the thermodynamic limit , @xmath227 where @xmath228 and @xmath229 are poisson random variables with the mean @xmath196 and @xmath102 , independent of each other and independent over @xmath202 and @xmath212 . compared to @xcite , now we have additional terms in the second line in ( [ ai ] ) coming from the perturbation hamiltonian ( [ hnpertmain ] ) . next , let us denote @xmath230 where @xmath107 denotes the uniform average over @xmath231 . recall that @xmath175 denotes the average with respect to the asymptotic gibbs measure @xmath159 . define @xmath232 then we will say that an asymptotic gibbs measure @xmath159 satisfies the cavity equations if @xmath233 for all choice of @xmath234 and sets @xmath235 . * the aizenman - sims - starr type lower bound . * consider a random measure @xmath159 on @xmath163 in ( [ spaceh ] ) and let @xmath236 be generated by a replica @xmath149 from this measure as in ( [ sg ] ) . from now on we will denote by @xmath237 a poisson random variable with the mean @xmath238 and we will assume that different appearances of these in the same equation are independent of each other and all other random variables . this means that if we write @xmath239 and @xmath240 , we assume them to be independent even if @xmath241 happens to be equal to @xmath242 . consider @xmath243 for @xmath105 and @xmath244 again , compared to @xcite , we have additional terms in the second line in ( [ aiag ] ) and ( [ bag ] ) coming from the perturbation hamiltonian ( [ hnpertmain ] ) . consider the following functional @xmath245 the following is a slight modification of the ( lower bound part of the ) main result in @xcite in the setting of the diluted models . [ th1 ] the lower limit of the free energy in ( [ fnmod ] ) satisfies @xmath246 where the infimum is taken over random measures @xmath159 on @xmath163 that satisfy the ghirlanda - guerra identities ( [ gg ] ) and the cavity equations ( [ sc ] ) . we will call the measures @xmath159 that appear in this theorem asymptotic gibbs measures , because that is exactly how they arise in @xcite . the main difference from @xcite is that we also include the requirement that the measures @xmath159 satisfy the ghirlanda - guerra identities in addition to the cavity equations . this can be ensured in exactly the same way as in the sherrington - kirkpatrick model by way of another small perturbation of the hamiltonian ( see e.g. @xcite , where this was explained for the @xmath0-sat model ) . we are not going to prove theorem [ th1 ] in this paper , because it does not require any new ideas which are not already explained in @xcite , and the main reason we stated it here is to provide the motivation for our main result below . of course , the proof involves some technical modifications to take into account the presence of the new perturbation term ( [ hnpertmain ] ) , but these are not difficult . instead , we will focus on the main new idea and the main new contribution of the paper , which is describing the structure of measures @xmath159 that satisfy the ghirlanda - guerra identities and the cavity equations in the case when the overlap @xmath247 of any two points @xmath149 and @xmath248 in the support of @xmath159 takes finitely many , say , @xmath249 values , @xmath250 for any @xmath24 - the so called @xmath25-step replica symmetry breaking ( or @xmath25-rsb ) case . to state the main result , let us first recall several known consequences of the ghirlanda - guerra identities . * consequences of the ghirlanda - guerra identities . * by talagrand s positivity principle ( see @xcite ) , if the ghirlanda - guerra identities hold then the overlap can take only non - negative values , so the fact that the values in ( [ finiteoverlap ] ) are between @xmath251 and @xmath2 is not a constraint . another consequence of the ghirlanda - guerra identities ( theorem 2.15 in @xcite ) is that with probability one the random measure @xmath159 is concentrated on the sphere on radius @xmath252 , i.e. @xmath253 since we assume that the overlap takes finitely many values , @xmath159 is also purely atomic . finally ( see @xcite or theorem 2.14 in @xcite ) , with probability one the support of @xmath159 is ultrametric , @xmath254 by ultrametricity , for any @xmath255 , the relation defined by @xmath256 is an equivalence relation on the support of @xmath159 . we will call these @xmath257 equivalence clusters simply @xmath221-clusters . let us now enumerate all the @xmath255-clusters defined by ( [ qclusters ] ) according to gibbs weights as follows . let @xmath258 be the entire support of @xmath159 so that @xmath259 . next , the support is split into @xmath260-clusters @xmath261 , which are then enumerated in the decreasing order of their weights @xmath262 , @xmath263 we then continue recursively over @xmath30 and enumerate the @xmath264-subclusters @xmath265 of a cluster @xmath266 for @xmath267 in the non - increasing order of their weights @xmath268 , @xmath269 thus , all these clusters were enumerated @xmath270 by the vertices of the tree @xmath46 in ( [ atree ] ) . it is not a coincidence that we used the same notation as in ( [ vs2 ] ) . it is another well - known consequence of the ghirlanda - guerra identities that the distribution of these weights coincides with the reordering of the weights of the ruelle probability cascades as in ( [ vs2 ] ) with the parameters ( [ zetas ] ) given by @xmath271 for @xmath272 the @xmath273-clusters are the points of the support of @xmath159 - these are called pure states . they were enumerated by @xmath274 and , if we denote them by @xmath275 , @xmath276 recall that we generate the array @xmath277 ( or @xmath278 for general index @xmath14 ) by first sampling replicas @xmath167 from the measure @xmath159 ( which are functions on @xmath75 $ ] ) and then plugging in i.i.d . uniform random variables @xmath279 , i.e. @xmath280 . in the discrete setting ( [ gdiscrete ] ) , this is equivalent to sampling @xmath33 according to the weights @xmath281 and then plugging in @xmath279 into @xmath282 i.e. @xmath283 . therefore , in order to describe the distribution of the array @xmath284 it is sufficient to describe the joint distribution of the arrays @xmath285 in addition to the fact that @xmath286 corresponds to some reordering of weights of the ruelle probability cascades , it was proved in @xcite that if the measure @xmath159 satisfies the ghirlanda - guerra identities then ( see theorem @xmath2 and equation ( 36 ) and ( 37 ) in @xcite ) : 1 . the arrays @xmath287 and @xmath288 are independent ; 2 . there exists a function @xmath289^{2(r+1)}\to[-1,1]$ ] such that @xmath290 where , as above , @xmath291 and @xmath292 for @xmath293 are i.i.d . uniform random variables on @xmath75 $ ] . the mzard - parisi ansatz predicts that in the equation ( [ sigmaf2 ] ) one can replace the function @xmath84 by a function that does not depend on the coordinates @xmath294 , which would produce exactly the same fields as in ( [ mpfopagain ] ) . we will show that this essentially holds for finite - rsb asymptotic gibbs measures . * consequence of the cavity equations . * the main result of the paper is the following . [ th2 ] if a random measure @xmath159 on @xmath163 in ( [ spaceh ] ) satisfies the ghirlanda - guerra identities ( [ gg ] ) and the cavity equations ( [ sc ] ) and the overlap takes @xmath249 values in ( [ finiteoverlap ] ) then there exists a function @xmath289^{r+2}\to[-1,1]$ ] such that @xmath295 where @xmath296 and @xmath292 for @xmath293 are i.i.d . uniform random variables on @xmath75 $ ] . in other words , the cavity equations ( [ sc ] ) allow us to simplify ( [ sigmaf2 ] ) and to get rid of the dependence on the coordinates @xmath297 for @xmath298 . notice that , compared to the mzard - parisi ansatz , we still have the dependence on @xmath296 in ( [ sigmaf3 ] ) . however , from the point of view of computing the free energy this is not an issue at all , because the average in @xmath296 is on the outside of the logarithm in ( [ ppg ] ) and when we minimize over @xmath159 in ( [ ppgeq ] ) , we can replace the average over @xmath296 by the infimum . of course , the infimum over @xmath159 in ( [ ppgeq ] ) could involve measures that are not of finite - rsb type , and this is the main obstacle to finish the proof of the mzard - parisi formula , if this approach can be made to work . if one could replace the infimum in ( [ ppgeq ] ) over measures @xmath159 that satisfy the finite - rsb condition in ( [ finiteoverlap ] ) ( in addition to the cavity equations and the ghirlanda - guerra identities ) then , using theorem [ th2 ] and replacing the average over @xmath296 by the infimum , we get the lower bound that essentially matches the franz - leone upper bound , except that now we have additional terms in the second line in ( [ aiag ] ) and ( [ bag ] ) compared to ( [ aibef ] ) and ( [ bef ] ) coming from the perturbation hamiltonian ( [ hnpertmain ] ) . however , these terms are controlled by @xmath207 in ( [ gsvar ] ) and , letting it go to zero , one could remove the dependence of the lower bound on these terms and match the franz - leone upper bound . the main goal of this paper is to show that the function @xmath84 that generates the array @xmath299 in ( [ sigmaf2 ] ) , @xmath300 can be replaced by a function that does not depend on the coordinates @xmath297 for @xmath301 . we will show this by induction , removing one coordinate at a time from the leaf @xmath33 up to the root @xmath28 . our induction assumption will be the following : for @xmath302 , suppose that , instead of ( [ sigmaf2 ] ) , the array @xmath299 for @xmath303 , is generated by @xmath304 for some function @xmath84 that does not depend on the coordinates @xmath297 for @xmath305 . notice that this holds for @xmath306 , and we would like to show that one can replace @xmath84 by @xmath307 that also does not depend on @xmath308 for @xmath309 , without affecting the distribution of the array @xmath299 . often , we will work with a subtree of @xmath46 that ` grows out ' of a vertex at the distance @xmath1 or @xmath310 from the root , which means that all paths from the root to the vertices in that subtree pass through this vertex . in that case , for certainty , we will fix the vertex to be @xmath311 = ( 1,2,\ldots , p)$ ] or @xmath312 $ ] . we will denote by @xmath313}}}$ ] the expectation with respect to the random variables @xmath297 , @xmath314 indexed by the descendants of @xmath311 $ ] , i.e. @xmath311\prec \beta$ ] , and by @xmath313,i}}}$ ] the expectation with respect to @xmath314 for @xmath311\prec \beta$ ] . our goal will be to prove the following . [ sec6ith1 ] under the assumption ( [ indass ] ) , for any @xmath315 and any @xmath312\preceq \alpha_1,\ldots,\alpha_k \in { \mathbb{n}}^r$ ] , the expectation @xmath313,i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_1}\cdots { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_k}$ ] with respect to @xmath316\preceq \beta}$ ] does not depend on @xmath317}$ ] almost surely . here @xmath212 is arbitrary but fixed and @xmath318 need not be different , so the quantities @xmath313,i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_1}\cdots { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_k}$ ] represent all possible joint moments with respect to @xmath316\preceq \beta}$ ] of the random variables @xmath299 for @xmath312\preceq \alpha\in { \mathbb{n}}^r$ ] . it will take us the rest of the paper to prove this result , and right now we will only explain why it completes the induction step . the reason is identical to the situation of an exchangeable sequence @xmath319 ( say , bounded in absolute value by one ) such that all moments @xmath320 for @xmath315 with respect to @xmath321 do not depend on @xmath322 . in this case if we choose any function @xmath323 with this common set of moments then the sequences @xmath324 and @xmath325 have the same distribution , which can be seen by comparing their joint moments . for example , we can choose @xmath326 for any @xmath327 from the set of measure one on which all moments @xmath320 coincide with their average values @xmath328 . we can do the same in the setting of theorem [ sec6ith1 ] , which can be rephrased as follows : for almost all @xmath329}$ ] and @xmath317}$ ] , @xmath330,i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_1}\cdots { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_k } = { { \mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_1}\cdots { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_k}\ ] ] for all @xmath315 and @xmath312\preceq \alpha_1,\ldots,\alpha_k \in { \mathbb{n}}^r$ ] , where @xmath313}}}$ ] now also includes the average in @xmath317}.$ ] this means that we can find @xmath317}=\omega_{[p+1]}^*$ ] such that the equality of all these moments holds for almost all @xmath329}$ ] . if we now set @xmath331}^ * , ( \omega_\beta^i)_{\beta\in p(\alpha ) } \bigr)\ ] ] then by comparing the joint moments one can see that @xmath332 which completes the induction step . the proof of theorem [ sec6ith1 ] will proceed by a certain induction on the shape of the configuration @xmath318 , where by the shape of the configuration we essentially mean the matrix @xmath333 ( or its representation by a tree that consists of all paths @xmath334 ) . it is clear that the quantity @xmath330,i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_1}\cdots { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_k}\ ] ] depends on @xmath312\preceq \alpha_1,\ldots,\alpha_k \in { \mathbb{n}}^r$ ] only through the shape . the induction will be somewhat involved and we will explain exactly how it will work toward the end of the paper , once we have all the tools ready . however , we need to mention now that the induction will have an important _ monotonicity property _ : whenever we have proved the statement of theorem [ sec6ith1 ] for some @xmath318 , we have also proved it for any subset of these vertices . at this moment , we will suppose that @xmath312\preceq \alpha_1,\ldots,\alpha_k \in { \mathbb{n}}^r$ ] are such that the following holds : 1 . for any subset @xmath335 , @xmath313,i}}}\prod_{\ell\in s } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}$ ] does not depend on @xmath317}$ ] . then over the next sections we will obtain some implications of this assumption using the cavity equations . finally , in the last section we will show how to use these implications inductively to prove theorem [ sec6ith1 ] for any choice of @xmath318 . of course , the starting point of the induction will be the case of @xmath336 that we will obtain first . in fact , in this case the statement will be even stronger and will not assume that ( [ indass ] ) holds ( i.e. we only assume ( [ sigmaf2 ] ) ) . [ sec2ilem1 ] for any @xmath337 and any @xmath311\prec \alpha \in { \mathbb{n}}^r$ ] , the expectation @xmath313,i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha}$ ] with respect to @xmath338\prec \beta}$ ] does not depend on @xmath339\prec \beta}$ ] almost surely . * proof . * consider @xmath311\prec \alpha , \beta\in{\mathbb{n}}^r$ ] such that @xmath340 . by ( [ sigmaf2 ] ) , it is clear that the overlap of two pure states satisfies @xmath341 where @xmath342 denotes the expectation in random variables @xmath343 that depend on the spin index @xmath131 . by construction , we enumerated the pure states @xmath275 in ( [ gdiscrete ] ) so that @xmath344 and , since @xmath340 , we get that , almost surely , @xmath345,i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha } { { \mathbb{e}_{[p],i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\beta}\bigr).\ ] ] if we denote @xmath346}$ ] , @xmath347\prec \eta\preceq \alpha}$ ] , @xmath348\prec \eta\preceq \beta}$ ] and @xmath349}$ ] then @xmath330,i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha } = \varphi(v , v_1,u),\,\ , { { \mathbb{e}_{[p],i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\beta}=\varphi(v , v_2,u)\ ] ] for some function @xmath350 the random variables @xmath351 are independent , and the above equation can be written as @xmath352 for almost all @xmath353 . this means that for almost all @xmath354 , the above equality holds for almost all @xmath355 . let us fix any such @xmath354 and let @xmath356 be the image of the lebesgue measure on @xmath75^{r - p}$ ] by the map @xmath357^{p+1},du)$ ] . then the above equation means that , if we sample independently two points from @xmath356 , with probability one their scalar product in @xmath157 will be equal to @xmath255 . this can happen only if the measure @xmath356 is concentrated on one point in @xmath157 , which means that the function @xmath358 does not depend on @xmath359 . before we start using the cavity equations , we will explain a property of the ruelle probability cascades that will play the role of the main technical tool throughout the paper . the property described in this section will be used in two ways - directly , in order to obtain some consequences of the cavity equations , and indirectly , as a representation tool to make certain computations possible . this property is proved in theorem 4.4 in @xcite in a more general form , but here we will need only a special case as follows . let us consider a random variable @xmath360 taking values in some measurable space @xmath361 ( in our case , this will always be some nice space , such as @xmath75^n$ ] with the borel @xmath117-algebra ) and let @xmath362 be its independent copies indexed by the vertices of the tree @xmath59 excluding the root . recall the parameters @xmath363 in ( [ zetas ] ) . let us consider a measurable bounded function @xmath364 and , recursively over @xmath365 , define functions @xmath366 by @xmath367 where the expected value @xmath368 is with respect to @xmath360 . in particular , @xmath369 is a constant . let us define @xmath370 for @xmath371 and @xmath372 let us point out that , by the definition ( [ ch54xp ] ) , @xmath373 and , therefore , for each @xmath374 we can think of @xmath375 as a change of density that yields the following conditional distribution on @xmath376 given @xmath377 , @xmath378 for @xmath379 , @xmath380 is just a probability distribution on @xmath381 let us now generate the array @xmath382 for @xmath59 iteratively from the root to the leaves as follows . let @xmath383 for @xmath384 be i.i.d . random variables with the distribution @xmath380 . if we already constructed @xmath382 for @xmath385 then , given any @xmath267 , we generate @xmath386 independently for @xmath124 from the conditional distribution @xmath387 , and these are generated independently over different such @xmath388 notice that the distribution of the array @xmath389 depends on the distribution of @xmath360 , function @xmath390 and parameters @xmath363 . with this definition , the expectation @xmath391 for a finite subset @xmath392 can be written as follow . for @xmath393 , let @xmath394 slightly abusing notation , we could also write this simply as @xmath395 . given a finite subset @xmath396 , let @xmath397 then , the above definition of the array @xmath398 means that @xmath399 simply , to average over @xmath400 we need to use changes of density over all vertices in the paths from the root leading to the vertices @xmath401 . the meaning of the above construction will be explained by the following result . recall the ruelle probability @xmath402 cascades in ( [ vs ] ) and define new random weights on @xmath45 , @xmath403 by the change of density proportional to @xmath404 . we will say that a bijection @xmath405 of the vertices of the tree @xmath46 preserves the parent - child relationship if children @xmath406 of @xmath33 are mapped into children of @xmath407 , @xmath408 for some @xmath409 . another way to write this is to say that @xmath410 for all @xmath411 for example , the bijection @xmath412 defined in ( [ permute ] ) , ( [ vs2 ] ) , is of this type . theorem 4.4 in @xcite gives the following generalization of the bolthausen - sznitman invariance property for the poisson - dirichlet point process ( proposition a.2 in @xcite ) . there exists a random bijection @xmath413 of the vertices of the tree @xmath46 , which preserves the parent - child relationship , such that @xmath414 and these two arrays are independent of each other . this result will be more useful to us in a slightly different formulation in terms of the sequence @xmath415 in ( [ vs2 ] ) . namely , if we denote by @xmath416 then the following holds . [ th4label ] there exists a random bijection @xmath413 of the vertices of the tree @xmath46 , which preserves the parent - child relationship , such that @xmath417 and these two arrays are independent of each other . * proof . * we have to apply twice the following simple observation . suppose that we have a random array @xmath418 of positive weights that add up to one and array @xmath419 generated along the tree similarly to @xmath398 above - namely , @xmath420 for @xmath384 are i.i.d . random variables with some distribution @xmath380 and , if we already constructed @xmath421 for @xmath385 then , given any @xmath267 , we generate @xmath422 independently for @xmath124 from some conditional distribution @xmath387 , and these are generated independently over different such @xmath388 suppose that @xmath418 and @xmath419 are independent . consider any random permutation @xmath413 that preserves the parent - child relationship , which depends only on @xmath418 , i.e. it is a measurable function of this array . then the arrays @xmath423 are independent and @xmath424 this is obvious because , conditionally on @xmath425 , the array @xmath426 is generated exactly like @xmath427 along the tree , so its conditional distribution does not depend on @xmath425 . one example of such permutation @xmath425 is the permutation defined in ( [ vsall ] ) , ( [ permute ] ) , ( [ vs2 ] ) , that sorts the cluster weights indexed by @xmath52 defined by @xmath428 namely , for each @xmath52 , we let @xmath68 be a bijection such that the sequence @xmath429 is decreasing for @xmath124 ( we assume that all these cluster weights are different as is the case for the ruelle probability cascades ) , let @xmath71 and define @xmath430 recursively from the root to the leaves of the tree . let us denote @xmath431 notice that this sorting operation depends only on @xmath418 , so it does not affect the distribution of @xmath419 . now , let us show how ( [ th3eq ] ) implies ( [ th4eq ] ) . first of all , the permutation @xmath425 in the equation ( [ th4eq ] ) is just the sorting operation described above , @xmath432 let @xmath412 be the permutation in ( [ permute ] ) , ( [ vs2 ] ) and , trivially , @xmath433 since the sorting operation does not depend on how we index the array . on the other hand , by the definition ( [ tildevs ] ) and the fact that @xmath434 , @xmath435 also , since the permutation @xmath412 depends only on @xmath436 , by the above observation , the arrays @xmath437 and @xmath438 are independent and @xmath439 comparing with the definition ( [ tildevs ] ) , this gives that @xmath440 and all together we showed that @xmath441 since we already use the notation @xmath425 , let us denote the permutation @xmath425 in ( [ th3eq ] ) by @xmath442 . then ( [ th3eq ] ) implies @xmath443 finally , since the sorting permutation @xmath412 depends only on the array @xmath436 , by the above observation , the array @xmath444 is independent of @xmath286 and has the same distribution as @xmath445 . this finishes the proof . in this section , we will obtain some general consequences of the cavity equations ( [ sc ] ) that do not depend on any inductive assumptions . in the next section , we will push this further under the assumption ( m ) made in section [ sec2ilabel ] . first of all , let us rewrite the cavity equations ( [ sc ] ) taking into account the consequences of the ghirlanda - guerra identities in ( [ gdiscrete ] ) and ( [ sigmaf2 ] ) . let us define @xmath446 and let @xmath447 . we will keep the dependence of @xmath448 on @xmath223 implicit for simplicity of notation . then ( [ ulbar2 ] ) can be redefined by ( using equality in distribution ( [ sigmaf2 ] ) ) @xmath449 moreover , if we denote @xmath450 then the cavity equations ( [ sc ] ) take form @xmath451 we can also write this as @xmath452 we will now use this form of the cavity equations to obtain a different form directly for the pure states that does not involve averaging over the pure states . let us formulate the main result of this section . let @xmath453 be the @xmath117-algebra generated by the random variables that are not indexed by @xmath59 , namely , @xmath454 for various indices , excluding the random variables @xmath455 and @xmath456 that are indexed by @xmath59 . let @xmath457 be the set of indices @xmath14 that appear in various @xmath458 in ( [ aieps ] ) , i.e. @xmath14 of the type @xmath459 or @xmath460 . let @xmath461 and let @xmath462 notice that with this notation , conditionally on @xmath453 , the random variables @xmath463 and @xmath464 in ( [ aialpha ] ) , ( [ sec3xiialpha ] ) for @xmath274 can be written as @xmath465 for some function @xmath466 and @xmath467 ( that implicitly depend on the random variables in ( [ ff ] ) ) and @xmath468 in the setting of the previous section , let @xmath469 in ( [ xr ] ) and let @xmath470 be the array generated along the tree using the conditional probabilities ( [ ch51transitionprime ] ) . recall that this means the following . the definition in ( [ ch54xp ] ) can be written as @xmath471 where @xmath472 is the expectation in @xmath362 , and the definition in ( [ ch31wp ] ) can be written as @xmath473 then the array @xmath470 is generated along the tree from the root to the leaves according to the conditional probabilities in ( [ ch51transitionprime ] ) , namely , given @xmath474 we generated @xmath382 by the change of density @xmath475 . let us emphasize one more time that this entire construction is done conditionally on @xmath453 . also , notice that the coordinates @xmath476 in ( [ zeealpha ] ) were independent for different @xmath14 , but the corresponding coordinates @xmath477 of @xmath478 are no longer independent , because @xmath390 and the changes of density @xmath479 depend on all of them . as in ( [ zeealpha ] ) and ( [ xichi ] ) , let us denote @xmath480 we will prove the following . [ sec4th ] the equality in distribution holds ( not conditionally on @xmath453 ) , @xmath481 * proof . * as in ( [ sec2eq1 ] ) and ( [ rabsec2 ] ) , we can write @xmath482 where @xmath342 denotes the expectation in random variables @xmath314 in ( [ sialpha ] ) that depend on the spin index @xmath131 , and @xmath483 . in the cavity equations ( [ scnew ] ) , let us now make a special choice of the sets @xmath484 . for each pair @xmath485 of replica indices such that @xmath486 , take any integer @xmath487 and consider a set @xmath488 of cardinality @xmath489 . let all these sets be disjoint , which can be achieved by taking @xmath490 for each @xmath224 , let @xmath491 then a given spin index @xmath492 appears in exactly two sets , say , @xmath484 and @xmath493 , and the expectation of ( [ scnew ] ) in @xmath494 will produce a factor @xmath495 . for each pair @xmath485 , there will be exactly @xmath496 such factors , so averaging in ( [ scnew ] ) in the random variables @xmath494 for all @xmath492 will result in @xmath497 approximating by polynomials , we can replace @xmath498 by the indicator of the set @xmath499 for any choice of constraints @xmath500 taking values in @xmath501 . therefore , ( [ scagain ] ) implies @xmath502 using the property ( i ) above the equation ( [ sigmaf2 ] ) , which as we mentioned is the consequence of the ghirlanda - guerra identities , we can rewrite the left hand side as @xmath503 moreover , it is obvious from the definition of the array @xmath299 in ( [ sialpha ] ) that the second expectation depends on @xmath504 only through the overlap constraints @xmath505 , or @xmath506 . on the other hand , on the right hand side of ( [ scf ] ) both @xmath507 and @xmath464 depend on the same random variables through the function @xmath508 . if we compare ( [ tildevs ] ) and ( [ valpha ] ) and apply theorem [ th4label ] conditionally on @xmath453 , we see that that there exists a random bijection @xmath413 of the vertices of the tree @xmath46 which preserves the parent - child relationship and such that @xmath509 and these two arrays are independent of each other ( all these statement are conditionally on @xmath453 ) . if we denote by @xmath510 the conditional expectation given @xmath453 then this implies that @xmath511 since the distribution of @xmath512 does not depend on the condition and @xmath513 , taking the expectation gives @xmath514 this proves that @xmath515 again , the second expectation in the sum on the right depends on @xmath504 only through the overlap constraints @xmath505 and , since the choice of the constraints was arbitrary , we get @xmath516 for any @xmath517 . clearly , one can express any joint moment of the elements in these two arrays by choosing @xmath518 large enough and choosing @xmath519 and the sets @xmath520 properly , so the proof is complete . we will continue using the notation of the previous section , only in this section we will take @xmath521 in theorem [ sec4th ] . let us recall the assumption ( m ) made at the end of section [ sec2ilabel ] : we consider some @xmath312\preceq \alpha_1,\ldots,\alpha_k \in { \mathbb{n}}^r$ ] such that the following holds : 1 . for any subset @xmath335 , @xmath313,i}}}\prod_{\ell\in s } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}$ ] does not depend on @xmath317}$ ] . in this section , we will obtain a further consequence of the cavity equations using that @xmath313,i}}}\prod_{\ell\leq k } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}$ ] does not depend on @xmath317}$ ] , but a similar consequence will hold for any subset of these vertices . let us denote by @xmath313}}}$ ] the expectation with respect to the random variables @xmath522 indexed by the ancestors @xmath523 $ ] of @xmath311 $ ] . we will use the same notation @xmath313}}}$ ] to denote the expectation with respect to the random variables @xmath524 for @xmath523 $ ] conditionally on @xmath524 for @xmath525 $ ] and all other random variables that generate the @xmath117-algebra @xmath453 in ( [ ff ] ) . given any finite set @xmath526 , let us denote @xmath527 then the following holds for @xmath312\preceq \alpha_1,\ldots,\alpha_k \in { \mathbb{n}}^r$ ] . [ sec5lem1 ] if @xmath528 and @xmath313,i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{c}$ ] does not depend on @xmath317}$ ] then @xmath330}}}{{\tilde{\xi}}}_1^{{c } } { { \tilde{\xi}}}_2^{{c } } = { { \mathbb{e}_{[p]}}}{{\tilde{\xi}}}_1^{{c } } { { \mathbb{e}_{[p]}}}{{\tilde{\xi}}}_2^{{c } } \label{sec3lem1eq}\ ] ] almost surely . * proof . * first of all , @xmath330 } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c } } = { { \mathbb{e}_{[p]}}}\prod_{i\leq 2 } { { \mathbb{e}_{[p],i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{{c } } = \prod_{i\leq 2 } { { \mathbb{e}_{[p],i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{{c}}\ ] ] almost surely , since @xmath313,i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{c}$ ] does not depend on @xmath317}$ ] . similarly , @xmath330 } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{{c } } = { { \mathbb{e}_{[p]}}}{{\mathbb{e}_{[p],i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{{c } } = { { \mathbb{e}_{[p],i } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{{c}}\ ] ] almost surely and , therefore , @xmath529 } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c } } - { { \mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c } } { { \mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c } } \bigr)^2 \nonumber \\ = & \ { \mathbb{e}}\bigl({{\mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}}\bigr)^2 -2 { \mathbb{e}}\bigl({{\mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}}\bigr ) \bigl({{\mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c } } { { \mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}}\bigr ) + { \mathbb{e}}\bigl({{\mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c } } { { \mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}}\bigr)^2 . \label{sec3above}\end{aligned}\ ] ] let us now rewrite each of these terms using replicas . let @xmath530 and for @xmath531 let @xmath532 for arbitrary @xmath533\preceq \alpha^j_1,\ldots,\alpha^j_k \in { \mathbb{n}}^r$ ] such that @xmath534 for any @xmath535 in other words , @xmath536 are copies of @xmath537 that consists of the descendants of different children of @xmath311 $ ] . therefore , we can write @xmath330 } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c}_j } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}_j } = { { \mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c}_{j ' } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}_{j ' } } \,\,\mbox { and } \,\ , { { \mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{{c}_j } = { { \mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{{c}_{j'}}\ ] ] almost surely for any @xmath538 and @xmath539 } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}}\bigr)^2 = & \ { \mathbb{e } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c}_1 } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}_1 } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c}_2 } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}_2 } , \\ { \mathbb{e}}\bigl({{\mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}}\bigr ) \bigl({{\mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c } } { { \mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}}\bigr ) = & \ { \mathbb{e } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c}_1 } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}_1 } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c}_2 } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}_3 } , \\ { \mathbb{e}}\bigl({{\mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c } } { { \mathbb{e}_{[p ] } } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}}\bigr)^2 = & \ { \mathbb{e } } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c}_1 } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}_2 } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_1^{{c}_3 } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_2^{{c}_4}.\end{aligned}\ ] ] by theorem [ sec4th ] , this and ( [ sec3above ] ) imply that @xmath540 repeating the above computation backwards for @xmath541 instead of @xmath149 gives @xmath542}}}{{\tilde{\xi}}}_1^{{c } } { { \tilde{\xi}}}_2^{{c } } - { { \mathbb{e}_{[p]}}}{{\tilde{\xi}}}_1^{{c } } { { \mathbb{e}_{[p]}}}{{\tilde{\xi}}}_2^{{c } } \bigr)^2 = 0\ ] ] and this finishes the proof . by analogy with ( [ sec2walpha ] ) and ( [ sec2expectw ] ) , let us rewrite the expectation @xmath313}}}$ ] with respect to the random variables @xmath543 for @xmath544 $ ] in terms of the expectation with respect to the random variables @xmath545 for @xmath544 $ ] , writing explicitly the changes of density @xmath546 as in lemma [ sec5lem1 ] , let @xmath528 for some @xmath312\preceq \alpha_1,\ldots,\alpha_k \in { \mathbb{n}}^r$ ] , let @xmath547,{c } ) = \bigl\{\beta \ \bigr|\ [ p+1]\preceq \beta\preceq \alpha , \alpha\in{c } \bigr\}\ ] ] and define @xmath548,{c } } = \prod_{\alpha\in p([p],{c } ) } w_\alpha . \label{sec5wpc}\ ] ] with this notation , we can rewrite ( [ sec3lem1eq ] ) as @xmath330}}}\xi_1^{{c } } \xi_2^{{c } } w_{[p],{c } } = { { \mathbb{e}_{[p]}}}\xi_1^{{c}}w_{[p],{c}}\ , { { \mathbb{e}_{[p]}}}\xi_2^{{c}}w_{[p],{c } } \label{sec5eq1}\ ] ] almost surely . notice that in ( [ sec3lem1eq ] ) almost surely meant for almost all random variables in ( [ ff ] ) that generate the @xmath117-algebra @xmath453 and for almost all @xmath382 for @xmath549 $ ] that are generated conditionally on @xmath453 according to the changes of density in ( [ sec3walpha ] ) . however , even though in ( [ sec5eq1 ] ) we simply expressed the expectation with respect to @xmath382 for @xmath311\prec \alpha$ ] using the changes of density explicitly , after this averaging both sides depend on @xmath550 for @xmath549 $ ] , so almost surely now means for almost all random variables in ( [ ff ] ) that generate the @xmath117-algebra @xmath453 and for almost all @xmath550 for @xmath549 $ ] . the reason we can do this is very simple . notice that @xmath508 in ( [ aieps ] ) can be bounded by @xmath551 which , by the assumption ( [ gsvar ] ) , is almost surely finite ( notice also that @xmath552 are @xmath453-measurable ) . by induction in ( [ sec3xp ] ) , all @xmath553 almost surely and , therefore , all changes of density in ( [ sec3walpha ] ) satisfy @xmath554 almost surely . therefore , conditionally on @xmath453 , the distribution of all @xmath362 and @xmath382 are absolutely continuous with respect to each other and , therefore , we can write almost surely equality in ( [ sec5eq1 ] ) in terms of the random variables @xmath362 for @xmath549 $ ] . next , we will reformulate ( [ sec5eq1 ] ) using the assumption ( [ indass ] ) . to simplify the notation , let us denote for any @xmath555 , @xmath556 and define @xmath557 and @xmath558 similarly . then , we can rewrite ( [ xichi ] ) for @xmath312\preceq \alpha\in { \mathbb{n}}^r$ ] as @xmath559 } , z^i_{\preceq \alpha}\bigr),\,\ , a_i^{\alpha } = \chi_i \bigl ( \omega_{\preceq [ p+1 ] } , z^i_{\preceq \alpha}\bigr ) . \label{sec5xichi}\ ] ] since in the previous section we set @xmath521 , we have @xmath560 } , z^i_{\preceq \alpha}\bigr).\ ] ] because of the absence of the random variables @xmath291 for @xmath312\prec \alpha$ ] , the integration in @xmath362 in the recursive definition ( [ sec3xp ] ) will decouple when @xmath312\prec \alpha$ ] into integration over @xmath561 and @xmath562 . namely , let @xmath563 and , for @xmath312 \preceq \alpha$ ] , let us define by decreasing induction on @xmath36 , @xmath564 } , z^i_{\prec \alpha } \bigr ) = \frac{1}{\zeta_{|\alpha|-1 } } \log { \mathbb{e}}_{z_{\alpha}^i } \exp \zeta_{|\alpha|-1 } \chi_{i,|\alpha|}\bigl ( \omega_{\preceq [ p+1 ] } , z^i_{\preceq \alpha}\bigr ) . \label{chij}\ ] ] first of all , for @xmath312\prec \alpha$ ] , by decreasing induction on @xmath36 , @xmath565 } , z^i_{\preceq \alpha}\bigr ) \nonumber \\ \{\mbox{independence}\}= & \ \frac{1}{\zeta_{|\alpha|-1 } } \log \prod_{i\leq 2 } { \mathbb{e}}_{z_{\alpha}^i } \exp \zeta_{|\alpha|-1 } \chi_{i,|\alpha|}\bigl ( \omega_{\preceq [ p+1 ] } , z^i_{\preceq \alpha}\bigr ) \nonumber \\ \{\mbox{definition ( \ref{chij})}\}= & \ \sum_{i\leq 2 } \chi_{i,|\alpha|-1}\bigl ( \omega_{\preceq [ p+1 ] } , z^i_{\prec \alpha}\bigr ) . \label{xichii}\end{aligned}\ ] ] when we do the same computation for @xmath566 $ ] , the expectation @xmath567}}$ ] also involves @xmath317}$ ] , so we end up with @xmath568}\bigr ) = & \ \frac{1}{\zeta_p } \log { \mathbb{e}}_{\omega_{[p+1]}}\prod_{i\leq 2 } { \mathbb{e}}_{z_{[p+1]}^i } \exp \zeta_p \chi_{i , p+1}\bigl ( \omega_{\preceq [ p+1 ] } , z^i_{\preceq [ p+1]}\bigr ) \nonumber \\ \{\mbox{definition ( \ref{chij})}\ } = & \ \frac{1}{\zeta_p } \log { \mathbb{e}}_{\omega_{[p+1 ] } } \exp \zeta_p \sum_{i\leq 2 } \chi_{i , p}\bigl ( \omega_{\preceq [ p+1 ] } , z^i_{\preceq [ p]}\bigr ) . \label{sec5xp}\end{aligned}\ ] ] for @xmath312\preceq \alpha$ ] , let us define for @xmath569 , @xmath570 } , z^i_{\preceq \alpha}\bigr ) = \exp \zeta_{|\alpha|-1}\bigl(\chi_{i,|\alpha|}\bigl ( \omega_{\preceq [ p+1 ] } , z^i_{\preceq \alpha}\bigr ) - \chi_{i,|\alpha|-1}\bigl ( \omega_{\preceq [ p+1 ] } , z^i_{\prec \alpha}\bigr ) \bigr ) . \label{sec5walphai}\ ] ] comparing this with the definition ( [ ch31wp ] ) and using ( [ xichii ] ) we get that for @xmath312\prec \alpha$ ] , @xmath571 } , z^i_{\preceq [ j+1]}\bigr ) . \label{sec5walpha}\ ] ] for @xmath566 $ ] this is no longer true , but if we denote @xmath572 } , z_{\preceq [ p]}\bigr ) = \exp \zeta_p\bigl(\sum_{i\leq 2}\chi_{i , p}\bigl ( \omega_{\preceq [ p+1 ] } , z^i_{\preceq [ p]}\bigr ) - x_{p}(z_{\preceq [ p ] } ) \bigr ) \label{sec5q}\ ] ] then we can write @xmath573 } \bigr ) = { { { \sbox{\myboxa}{$\m@thw$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{-0.6mm}}_{p}}\bigl ( \omega _ { [ p+1 ] } , z_{\preceq [ p]}\bigr ) \prod_{i\leq 2 } w_p^i \bigl ( \omega_{\preceq [ p+1 ] } , z^i_{\preceq [ p+1]}\bigr ) . \label{sec5wp}\ ] ] if , similarly to ( [ sec5wa ] ) and ( [ sec5wpc ] ) , we denote @xmath574,{c}}^i = \prod_{\alpha\in p([p],{c } ) } w_\alpha^i \label{sec5wpci}\ ] ] then we can rewrite ( [ sec5wpc ] ) as @xmath548,{c } } = { { { \sbox{\myboxa}{$\m@thw$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{-0.6mm}}_{p}}\prod_{i\leq 2 } w_{[p],{c}}^{i}. \label{sec5wpr}\ ] ] notice that @xmath575 does not depend on @xmath576 for @xmath311\prec \beta$ ] , while @xmath577,{c}}^{1}$ ] and @xmath578 in ( [ sec5eq1 ] ) do not depend on @xmath579 for @xmath311\prec \beta$ ] and @xmath577,{c}}^{2}$ ] and @xmath580 do not depend on @xmath581 for @xmath311\prec \beta$ ] . this means that if we denote by @xmath313,i}}}$ ] the expectation in the random variables @xmath576 for @xmath311\prec \beta$ ] and denote @xmath582,i}}}\ , \xi_i^{{c } } w_{[p],{c}}^{i } \label{sec5eqsecond}\ ] ] then ( [ sec5eq1 ] ) can be rewritten as @xmath583 } } \eta_1^{{c } } \eta_2^{{c } } { { { \sbox{\myboxa}{$\m@thw$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{-0.6mm}}_{p}}= { \mathbb{e}}_{\omega_{[p+1 ] } } \eta_1^{{c } } { { { \sbox{\myboxa}{$\m@thw$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{-0.6mm}}_{p}}\ , { \mathbb{e}}_{\omega_{[p+1 ] } } \eta_2^{{c } } { { { \sbox{\myboxa}{$\m@thw$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{-0.6mm}}_{p}}\label{sec5eq2}\ ] ] almost surely , because after averaging @xmath313,i}}}$ ] in @xmath576 for @xmath311\prec \beta$ ] , the only random variable left to be averaged in @xmath313}}}$ ] is @xmath317}$ ] . so far we have just rewritten the equation ( [ sec5eq1 ] ) under the induction assumption ( [ indass ] ) . now , we will use this to prove the main result of this section . let us make the following simple observation : recalling the definition of @xmath508 in ( [ aieps ] ) , if we set all but finite number of random variables @xmath584 to zero , the equation ( [ sec5eq2 ] ) still holds almost surely . to see this , first of all , notice that because the random variables @xmath228 take any natural value with positive probability , we can set a finite number of them to any values we like in ( [ sec5eq2 ] ) . for example , for any @xmath585 , we can set @xmath586 for @xmath587 and set @xmath588 for @xmath589 . the remaining part of the last term in @xmath508 can be bounded uniformly by @xmath590 where , by the assumption ( [ gsvar ] ) , we have @xmath591 which implies that this sum goes to zero almost surely as @xmath592 goes to infinity . it follows immediately from this that we can set all but finite number of @xmath228 in ( [ sec5eq2 ] ) to zero . moreover , we will set @xmath593 , since the terms coming from the model hamiltonian will play no role in the proof - all the information we need is encoded into the perturbation hamiltonian . from now on we will assume that in ( [ sec5eq2 ] ) , for a given @xmath594 , @xmath595 in addition , let us notice that both sides of ( [ sec5eq2 ] ) are continuous functions of the variables @xmath596 and @xmath597 for @xmath598 for @xmath587 , @xmath569 . this implies that almost surely over other random variables the equation ( [ sec5eq2 ] ) holds for all @xmath597 and , in particular , we can set them to be equal to any prescribed values , @xmath599 the following is the main result of this section . [ sec5th ] the random variables @xmath600 do not depend on @xmath317}$ ] . here and below , when we say that a function ( or random variable ) does not depend on a certain coordinate , this means that the function is equal to the average over that coordinate almost surely . in this case , we want to show that @xmath601 } } \eta_i^{{c}}\ ] ] almost surely . * proof of theorem [ sec5th ] . * besides the poisson and gaussian random variables in ( [ fixpoisson ] ) , ( [ fixgs ] ) and the random variable @xmath317}$ ] over which we average in ( [ sec5eq2 ] ) , the random variables @xmath600 for @xmath569 depend on @xmath296 , @xmath602}$ ] , @xmath603 and @xmath604}^i$ ] , and @xmath575 depends on the same random variables for both @xmath569 . let us denote @xmath605}^i \bigr).\ ] ] we already stated that for almost all @xmath296 , @xmath602}$ ] , @xmath606 and @xmath607 , the equation ( [ sec5eq2 ] ) holds for all poisson and gaussian random variables fixed as in ( [ fixpoisson ] ) , ( [ fixgs ] ) . therefore , for almost all @xmath296 , @xmath602}$ ] the equation ( [ sec5eq2 ] ) holds for almost all @xmath606 , @xmath607 and for all poisson and gaussian random variables fixed as in ( [ fixpoisson ] ) , ( [ fixgs ] ) . let us fix any such @xmath296 , @xmath602}$ ] . then , we can write @xmath608 } ) \ \mbox { and}\ { { { \sbox{\myboxa}{$\m@thw$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{-0.6mm}}_{p}}= \psi(u_1,u_2,\omega_{[p+1]})\ ] ] for some functions @xmath358 and @xmath184 . these functions depend implicitly on all the random variables we fixed , and the function @xmath358 is the same for both @xmath609 and @xmath610 because we fixed all poisson and gaussian random variables in ( [ fixpoisson ] ) , ( [ fixgs ] ) to be the same for @xmath569 . the equation ( [ sec5eq2 ] ) can be written as ( in the rest of this proof , let us for simplicity of notation write @xmath322 instead of @xmath317}$ ] ) @xmath611 for almost all @xmath612 . we want to show that @xmath613 does not depend on @xmath322 . if we denote @xmath614 then , by ( [ fixpoisson ] ) and ( [ fixgs ] ) , we can bound @xmath508 in ( [ aieps ] ) by @xmath615 for @xmath569 . by induction in ( [ chij ] ) , @xmath616 and , by ( [ sec5xp ] ) , @xmath617 . therefore , from the definition of @xmath575 in ( [ sec5q ] ) , @xmath618 of course , @xmath619 . suppose that for some @xmath620 , there exists a set @xmath621 of positive measure such that the variance @xmath622 for @xmath623 given @xmath624 , let @xmath625 be a partition of @xmath626,d\omega)$ ] such that @xmath627 for all @xmath132 let @xmath628 for some @xmath217 , the measure of @xmath629 will be positive , so for some @xmath630 , @xmath631 the equations ( [ sec5eq5 ] ) and ( [ sec5eq6 ] ) imply that @xmath632 and , similarly , @xmath633 since @xmath619 and @xmath634 , the first inequality implies that @xmath635 which , together with the second inequality and ( [ sec5eq3 ] ) , implies @xmath636 the left hand side is a variance with the density @xmath184 and can be written using replicas as @xmath637 by ( [ sec5eq5 ] ) and the fact that @xmath638 , we can bound this from below by @xmath639 comparing lower and upper bounds , @xmath640 , we arrive at contradiction , since @xmath624 was arbitrary . therefore , @xmath641 for almost all @xmath642 and this finishes the proof . let us summarize what we proved in the previous section . we considered @xmath528 for some @xmath312\preceq \alpha_1,\ldots,\alpha_k \in { \mathbb{n}}^r$ ] and assumed that @xmath313,i}}}\prod_{\ell\leq k } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}$ ] does not depend on @xmath317}$ ] . then , as a consequence of this and the cavity equations , we showed that @xmath582,i}}}\ , \xi_i^{{c } } w_{[p],{c}}^{i } \label{sec6eqsecond}\ ] ] also does not depend on @xmath317}$ ] almost surely , where @xmath548,{c}}^i = \prod_{\alpha\in p([p],{c } ) } w_\alpha^i \label{sec6wpci}\ ] ] and @xmath643,{c } ) = \bigl\{\beta \ \bigr|\ [ p+1]\preceq \beta\preceq \alpha , \alpha\in{c } \bigr\}$ ] . moreover , this holds for poisson and gaussian random variables fixed to arbitrary values as in ( [ fixpoisson ] ) , ( [ fixgs ] ) . by the assumption ( m ) , the same statement holds in we replace @xmath537 by any subset @xmath644 . in this section , we will represent the expectation @xmath313,i}}}$ ] in ( [ sec6eqsecond ] ) with respect to @xmath645 for @xmath311\prec \alpha$ ] by using the property of the ruelle probability cascades in theorem [ th4label ] . essentially , the expectation in ( [ sec6eqsecond ] ) is of the same type as ( [ sec2expectw ] ) if we think of the vertex @xmath311 $ ] as a root . indeed , we are averaging over random variables indexed by the vertices @xmath311\prec \alpha$ ] which form a tree ( if we include the root @xmath311 $ ] ) isomorphic to a tree @xmath646 of depth @xmath647 . we can identify a vertex @xmath311\preceq \alpha \in { { \cal a}}$ ] with the vertex @xmath648 such that @xmath649\gamma$ ] ( for simplicity , we denote by @xmath311\gamma$ ] the concatenation @xmath650,\gamma)$ ] ) . similarly to ( [ indass ] ) , let us define for @xmath651 , @xmath652},\omega_{[p+1 ] } , ( \omega_\beta^i)_{\beta\preceq [ p]},(\omega_{[p]\beta}^i)_{*\prec \beta \preceq \gamma } \bigr ) . \label{sec6sialpha}\ ] ] notice a subtle point here : the random variables @xmath653 are not exactly the same as @xmath654 in ( [ indass ] ) for @xmath649\gamma$ ] . they are exactly the same only if @xmath312\preceq \alpha$ ] , and in this case we will often write @xmath655\gamma}.$ ] otherwise , if @xmath533\preceq \alpha$ ] for @xmath656 then in ( [ indass ] ) we plug in the random variable @xmath657}$ ] instead of @xmath317}$ ] as we did in ( [ sec6sialpha ] ) . the reason for this will become clear soon but , basically , we are going to represent the average @xmath313,i}}}$ ] in ( [ sec6eqsecond ] ) with respect to @xmath658\beta}^i$ ] for @xmath659 using the ruelle probability cascades while @xmath317}$ ] appears in ( [ sec6eqsecond ] ) on the outside of this average . similarly to ( [ aieps ] ) ( [ sec3xiialpha ] ) , let us define for @xmath660 , @xmath661 the reason why ( [ sec6aieps ] ) looks different from ( [ aieps ] ) is because we fixed poisson and gaussian random variables as in ( [ fixpoisson ] ) , ( [ fixgs ] ) , so @xmath662 and @xmath663 are defined in terms of @xmath664 and @xmath665 in ( [ fixgs ] ) . again , let us emphasize one more time that all these definition coincide with the old ones when @xmath312\preceq [ p]\gamma$ ] or , equivalently , when @xmath666\preceq \gamma.$ ] we will keep the dependence on the random variables @xmath667},\omega_{[p+1 ] } , ( \omega_\beta^i)_{\beta\preceq [ p]}$ ] implicit and , similarly to ( [ sec5xichi ] ) , we will write for @xmath651 , @xmath668\beta}^i)_{*\prec \beta \preceq \gamma}\bigr ) . \label{sec6xichi}\ ] ] let @xmath563 and define for @xmath669 by decreasing induction on @xmath670 , @xmath671\beta}^i)_{*\prec \beta \prec \gamma } \bigr ) = \frac{1}{\zeta_{p+|\gamma|-1 } } \log { \mathbb{e}}_{z_{[p]\gamma}^i } \exp \zeta_{p+|\gamma|-1 } \chi_{i,|\alpha|}\bigl ( ( z_{[p]\beta}^i)_{*\prec \beta \preceq \gamma}\bigr)\ ] ] and @xmath672\beta}^i)_{*\prec \beta \preceq \gamma } \bigr ) = \exp \zeta_{p+|\gamma|-1}\bigl(\chi_{i , p+|\gamma|}\bigl ( ( z_{[p]\beta}^i)_{*\prec \beta \preceq \gamma}\bigr ) - \chi_{i , p+ |\gamma|-1}\bigl ( ( z_{[p]\beta}^i)_{*\prec \beta \prec \gamma}\bigr ) \bigr).\ ] ] for @xmath666\preceq \gamma$ ] these are exactly the same definitions as in ( [ chij ] ) and ( [ sec5walphai ] ) , but here we extend these definition to all @xmath673 let @xmath674\beta}^i = ( \tilde{\omega}_{[p]\beta}^i)_{i\in{{\cal i}}_i}$ ] for @xmath675 be the array generated according to these changes of density along the tree @xmath676 as in section [ sec2ilabel ] . since @xmath311 $ ] acts as a root , we do not generate any @xmath677 for @xmath678 . similarly to ( [ sec4tildas ] ) , we can write for @xmath651 , @xmath679\beta}^i)_{*\prec \beta \preceq \gamma}\bigr),\,\ , \tilde{\xi}_i^\gamma = \xi_i \bigl ( ( \tilde{z}_{[p]\beta}^i)_{*\prec \beta \preceq \gamma } \bigr ) , \label{sec6tildas}\ ] ] where we continue to keep the dependence on @xmath667},$ ] @xmath317}$ ] and @xmath680}$ ] , as well as poisson and gaussian random variables we fixed above , implicit . given the vertices @xmath312\preceq \alpha_1,\ldots,\alpha_k \in { \mathbb{n}}^r$ ] let @xmath666 \preceq \gamma_1,\ldots,\gamma_k \in { \mathbb{n}}^{r - p}$ ] be such that @xmath681\gamma_\ell.$ ] then we can write @xmath682 in ( [ sec6eqsecond ] ) as @xmath683,i}}}\ , \xi_i^{\alpha_1}\cdots \xi_i^{\alpha_k } w_{[p],{c}}^{i } = { \mathbb{e}}_{*,i}\ , { { \tilde{\xi}}}_i^{\gamma_1}\cdots { { \tilde{\xi}}}_i^{\gamma_k } , \label{sec6rep1}\ ] ] where @xmath684 denotes the expectation in @xmath674\beta}^i$ ] for @xmath675 . below we will represent this quantity using the analogue of theorem [ th4label ] . let @xmath685 be the weights of the ruelle probability cascades corresponding to the parameters @xmath686 let @xmath687 be their rearrangement as in ( [ vs2 ] ) and , similarly to ( [ tildevs ] ) , define @xmath688 theorem [ th4label ] can be formulated in this case as follows . [ sec6th4label ] there exists a random bijection @xmath689 of the vertices of the tree @xmath676 , which preserves the parent - child relationship , such that @xmath690\rho(\gamma ) } \bigr)_{\gamma\in \gamma\setminus \{*\ } } \stackrel{d}{= } \bigl(\tilde{z}^i_{[p]\gamma } \bigr)_{\gamma\in \gamma\setminus \{*\ } } \label{sec6th4eq}\ ] ] and these two arrays are independent of each other . the expectation @xmath684 in ( [ sec6rep1 ] ) depends on @xmath691 only through their overlaps @xmath692 . above , we made the specific choice @xmath681\gamma_\ell$ ] , which implies @xmath693 now , consider the set of arbitrary configurations with these overlaps , @xmath694 let us denote by @xmath695 the expectation in @xmath687 in addition to @xmath674\beta}^i$ ] for @xmath675 in the definition of @xmath684 . we will also denote by @xmath695 the expectation in @xmath687 and @xmath696\beta}^i$ ] for @xmath675 . using theorem [ sec6th4label ] and arguing as in the proof of theorem [ sec4th ] , @xmath697 let us rewrite this equation using a more convenient notation . let @xmath698 be i.i.d . replicas drawn from @xmath699 according to the weights @xmath687 and let @xmath700 denote the average with respect to these weights . if we denote @xmath701 and @xmath702 then we can write @xmath703 and @xmath704 and we can rewrite ( [ sec6repres ] ) above as @xmath705 notice that this computation also works if we replace each factor @xmath706 in ( [ sec6repres ] ) by any power @xmath707 and , in particular , by setting @xmath708 or @xmath2 we get the following . for a subset @xmath335 , let us denote @xmath709 then @xmath710 furthermore , it will be convenient to rewrite the left hand side using ( [ sec6aialpha ] ) and ( [ sec6xiialpha ] ) as @xmath711 we showed as a consequence of the assumption ( m ) that all @xmath712 do not depend on @xmath317}$ ] and , therefore , we proved the following . [ sec6thend ] under the assumption ( m ) , for any subset @xmath335 , @xmath713 does not depend on @xmath317}$ ] almost surely . we begin by simplifying ( [ sec6aieps ] ) further , by taking @xmath714 and setting all other @xmath715 for @xmath716 except for one , @xmath717 , @xmath718 one can easily see that the definition of @xmath208 in ( [ thetadetx ] ) satisfies for @xmath719 @xmath720 and , therefore , we can rewrite @xmath721 at this moment , for simplicity of notation , we will drop some unnecessary indices . we will write @xmath722 instead of @xmath723 and , since @xmath196 is fixed for a moment , write @xmath724 instead of @xmath725 . also , we will denote @xmath726 then , we can write @xmath727 by theorem [ sec6thend ] , under the assumption ( m ) , the quantities in ( [ sec6thendeq ] ) do not depend on @xmath317}$ ] almost surely . in particular , as we discussed above , this almost sure statement can be assumed to hold for all @xmath728 by continuity . we will now take @xmath729 for @xmath730 and independent rademacher random variables @xmath731 and show that , by letting @xmath732 , we can replace the last sum in ( [ seca1 ] ) by some gaussian field in the statement of theorem [ sec6thend ] . let us denote @xmath733 then with the choice of @xmath734 we can use taylor s expansion to write @xmath735 the last term @xmath736 is uniform in all parameters , so it will disappear in ( [ sec6thendeq ] ) when we let @xmath223 go to infinity . for the first term , we will use the classical clt to replace it by gaussian and for the second term we will use the slln , which will produce a term that will cancel out in the numerator and denominator in ( [ sec6thendeq ] ) . however , before we do that , we will need to change the definition of the expectation @xmath695 slightly . recall that , by ( [ sec6sialpha ] ) , @xmath737 } , \omega_{[p+1 ] } , ( \omega_\beta)_{\beta \preceq [ p]}^{\ell , j , i } , ( \omega_{[p]\beta}^{\ell , j , i } ) _ { * \prec \beta \preceq \gamma } \bigr).\ ] ] in ( [ sec6thendeq ] ) we already average in the random variables @xmath658\beta}^{\ell , j , i}$ ] for @xmath738 but , clearly , the statement of theorem [ sec6thend ] holds if @xmath695 also includes the average in @xmath739}$ ] and the rademacher random variables @xmath731 in ( [ sec7etas ] ) . from now on we assume this . note that @xmath695 still does not include the average with respect to the random variables @xmath740}$ ] that appear in @xmath741 in ( [ seca1 ] ) , @xmath742 } , \omega_{[p+1 ] } , ( \omega_\beta^i)_{\beta\preceq [ p ] } , ( \omega_{[p]\beta}^{i } ) _ { * \prec \beta \preceq \gamma } \bigr ) . \label{sec7sbar}\ ] ] of course , one can not apply the clt and slln in ( [ sec6thendeq ] ) directly , because there are infinitely many terms indexed by @xmath651 . however , this is not a serious problem because most of the weight of the ruelle probability cascades @xmath743 is concentrated on finitely many indices @xmath744 and it is not difficult to show that ( [ sec6thendeq ] ) is well approximated by the analogous quantity where the series over @xmath744 are truncated at finitely many terms . moreover , this approximation is uniform over @xmath223 in ( [ sec7napprox ] ) . this is why the representation of @xmath712 in the previous section using the ruelle probability cascades plays such a crucial role . we will postpone the details until later in this section and first explain what happens in ( [ sec7napprox ] ) for finitely many @xmath744 . first of all , by the slln , for any fixed @xmath667}$ ] and @xmath317}$ ] , @xmath745 almost surely . of course , we can now simplify the notation by replacing @xmath746 with @xmath722 and replacing @xmath695 by the expectation @xmath342 with respect to @xmath314 for @xmath747 , @xmath748 lemma [ sec2ilem1 ] in section [ sec2ilabel ] ( for @xmath379 there ) yields that @xmath749 depends only on @xmath296 , @xmath750 and , since @xmath751 clearly does not depend on @xmath744 , we can take @xmath666\preceq \gamma$ ] , in which case @xmath752\gamma}_{i})^2 = r_{[p]\gamma,[p]\gamma } = q_r.\ ] ] this means that for any @xmath660 , @xmath753 almost surely . so , in the limit , these terms will cancel out in ( [ sec6thendeq ] ) at least when we truncate the summation over @xmath744 to finitely many @xmath744 , as we shall do below . next , let us look at the first sum in ( [ sec7napprox ] ) for @xmath754 for a finite set @xmath755 . by the classical multivariate clt ( applied for a fixed @xmath667}$ ] and @xmath317}$ ] ) , @xmath756 where @xmath757 is a centered gaussian random vector with the covariance @xmath758 first of all , as above @xmath759 to compute @xmath760 , we need to consider two cases . first , suppose that @xmath761 since @xmath760 clearly depends only on @xmath762 , we can suppose that @xmath666\preceq \gamma,\gamma'$ ] , in which case @xmath763\gamma}_{i}$ ] , @xmath764\gamma'}_{i}$ ] and @xmath765\gamma}_{i } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}^{[p]\gamma'}_{i } = r_{[p]\gamma,[p]\gamma ' } = q_{p+\gamma\wedge\gamma'}.\ ] ] in the second case , @xmath766 , when computing @xmath760 we can first average @xmath741 with respect to @xmath767\beta}^{i } ) _ { * \prec \beta \preceq \gamma}$ ] and @xmath768 with respect to @xmath767\beta}^{i } ) _ { * \prec \beta \preceq \gamma'}$ ] , since these are independent . however , by lemma [ sec2ilem1 ] , both of these averages do not depend on @xmath317}$ ] . this means that , after taking these averages , we can replace @xmath317}$ ] in ( [ sec7sbar ] ) by @xmath657}$ ] if @xmath769\preceq \gamma$ ] , and the same for @xmath770 . as a result , we can again write @xmath765\gamma}_{i } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}^{[p]\gamma'}_{i } = r_{[p]\gamma,[p]\gamma ' } = q_p = q_{p+\gamma\wedge\gamma'}\ ] ] almost surely . if we introduce the notation @xmath771 then we proved that the covariance is given by @xmath772 let us show right away that @xmath773 in particular , this means that the covariance in ( [ sec7cov ] ) is increasing with @xmath762 and the gaussian field @xmath774 is the familiar field that accompanies the ruelle probability cascades in the pure @xmath196-spin non - diluted models . of course , the only statement that requires a proof is the following . the inequality @xmath775 holds almost surely . * proof . * first of all , let us notice that , by lemma [ sec2ilem1 ] , we can also write the definition of @xmath776 in ( [ sec7q0 ] ) as @xmath777 for any vertex @xmath274 , where @xmath342 denotes the expectation in the random variables @xmath314 for @xmath778 let @xmath779 denote the expectation with respect to @xmath314 for @xmath780 , excluding the average in @xmath781 . again , by lemma [ sec2ilem1 ] , we can write @xmath782 as @xmath783 . if we take any @xmath784 such that @xmath785 then @xmath786 almost surely . therefore @xmath787 this finishes the proof . from now on let @xmath695 also include the expectation in the gaussian field @xmath774 ( conditionally on @xmath296 ) with the covariance ( [ sec7cov ] ) . let us denote by @xmath788 the quantity that will replace @xmath789 in ( [ seca1 ] ) in the limit @xmath732 . we are ready to prove the following . [ sec7th ] under the assumption ( m ) , for any subset @xmath335 , @xmath790 does not depend on @xmath317}$ ] almost surely . * we only need to show that the quantity in ( [ sec6thendeq ] ) with @xmath789 as in ( [ seca1 ] ) with the choice of @xmath791 as in ( [ sec7etas ] ) converges to ( [ sec7theq ] ) , where @xmath695 was redefined above ( [ sec7sbar ] ) to include the average over @xmath667}^{\ell , j , i}$ ] and rademacher random variables @xmath731 in ( [ sec7etas ] ) , as well as the gaussian field @xmath774 with the covariance ( [ sec7cov ] ) . let @xmath792 be the rpc weights @xmath687 arranged in the decreasing order . for some fixed large @xmath793 , let us separate the averages @xmath794 in the numerator and denominator in ( [ sec6thendeq ] ) into two sums over @xmath795 for @xmath796 and for @xmath797 ( for each replica ) . let us denote the corresponding averages by @xmath798 and @xmath799 . let @xmath800 note that @xmath801 , @xmath802 and @xmath803 this means that the difference between ( [ sec6thendeq ] ) and @xmath804 can be bounded by @xmath805 . by ( [ sec7napprox ] ) , the slln in ( [ sec7lln ] ) and clt in ( [ sec7clt ] ) , with probability one , ( [ sec7trunc ] ) converges to @xmath806 as @xmath732 . next , we show that @xmath807 is small for large @xmath808 , uniformly over @xmath223 . since @xmath809 $ ] , it is enough to show that @xmath196 is small and @xmath238 is not too small with high probability . to show that @xmath196 is small , we will use chebyshev s inequality and show that @xmath810 is small . by jensen s inequality , @xmath811 if we denote @xmath812 then , since the weights @xmath743 and the random variables in @xmath789 are independent , @xmath813 using that @xmath814 , we can bound @xmath815 with the choice ( [ sec7etas ] ) by @xmath816 using that , for a rademacher random variable @xmath817 , we have @xmath818 we get that @xmath819 and , therefore , @xmath820 we showed that @xmath821 , and this bound does not depend on @xmath223 . since @xmath822 is small for @xmath808 large , @xmath196 is small with high probability uniformly over @xmath823 on the other hand , to show that @xmath238 is not too small , we simply bound it from below by @xmath824 for @xmath744 corresponding to the largest weight , @xmath825 the weight @xmath826 is strictly positive and its distribution does not depend on @xmath223 . also , using ( [ sec7napprox ] ) , we can bound @xmath815 from below by @xmath827 for some absolute constant @xmath828 . even though the index @xmath744 here is random , because it corresponds to the largest weight @xmath826 , we can control this quantity using hoeffding s inequality for rademacher random variables conditionally on other random variables to get @xmath829 therefore , for any @xmath624 there exists @xmath830 ( that depends on @xmath831 , @xmath832 , @xmath833 and the distribution of @xmath826 ) such that @xmath834 all together , we showed that @xmath807 is small for large @xmath808 , uniformly over @xmath223 . to finish the proof , we need to show that ( [ sec7limm ] ) approximates ( [ sec7theq ] ) for large @xmath808 . clearly , this can be done by the same argument ( only easier ) that we used above to show that ( [ sec7trunc ] ) approximates ( [ sec6thendeq ] ) . theorem [ sec7th ] implies that , under the assumption ( m ) , @xmath835 does not depend on @xmath317}$ ] almost surely . this follows from ( [ sec7theq ] ) by multiplying out @xmath836 using that for @xmath105 , @xmath837 one can see from ( [ sec7a1 ] ) that @xmath838 and@xmath839 if for simplicity the notation we denote @xmath840 and @xmath841 then ( [ sec8eq1 ] ) can be written , up to a factor @xmath842 , as @xmath843 as before , the statement that this quantity does not depend on @xmath317}$ ] almost surely holds for all @xmath730 , @xmath844 and @xmath845 , by continuity . therefore , if we take the derivative with respect to @xmath360 and then let @xmath846 then the quantity we get ( up to a factor @xmath847 ) , @xmath848 does not depend on @xmath317}$ ] almost surely . this is a polynomial in @xmath849 of order @xmath850 and if we take the derivative @xmath851 we get that @xmath852 does not depend on @xmath317}$ ] almost surely . let us now take the expectation @xmath853 with respect to the gaussian field @xmath854 . by ( [ sec7cov ] ) , @xmath855 and , therefore , the quantity @xmath856 does not depend on @xmath317}$ ] almost surely . notice that because of the indicator @xmath857 , all the overlaps @xmath858 are fixed for @xmath859 , so the factor @xmath860 can be taken outside of @xmath861 and cancelled out , yielding that @xmath862 does not depend on @xmath317}$ ] almost surely . taking the derivative with respect to @xmath863 at zero gives that @xmath864 does not depend on @xmath317}$ ] almost surely . we proved this statement for a fixed but arbitrary @xmath716 , but it also holds for @xmath865 by setting @xmath866 in the previous equation . let us take arbitrary @xmath867 for @xmath868 and consider a continuous function @xmath182 on @xmath75 $ ] such that @xmath869 approximating this function by polynomials , ( [ sec9polyd ] ) implies that @xmath870 does not depend on @xmath317}$ ] for all @xmath871 almost surely . taking the derivative in @xmath867 shows that @xmath872 does not depend on @xmath317}$ ] almost surely and , therefore , @xmath873 does not depend on @xmath317}$ ] almost surely for all @xmath874 . let us now express this quantity as a linear combination over all possible overlap configurations that the new replica @xmath875 can form with the old replicas @xmath698 . given a @xmath876 overlap constraint matrix @xmath877 , let @xmath878 be the set of admissible extensions @xmath879 of @xmath880 to @xmath881 overlap constraint matrices . in other words , @xmath882 for @xmath883 , and there exists @xmath884 such that @xmath885 for @xmath886 if we denote @xmath887 and denote @xmath888 then we showed that @xmath889 does not depend on @xmath317}$ ] almost surely for all @xmath874 . since the rpc weights @xmath743 are independent of @xmath890 , we can rewrite this as @xmath891 does not depend on @xmath317}$ ] almost surely for all @xmath874 , where @xmath892 for any @xmath884 such that @xmath885 for @xmath886 recall that this statement was proved under the induction assumption ( m ) in section [ sec2ilabel ] , so let us express ( [ sec8at ] ) in the notation of section [ sec2ilabel ] and connect everything back to the assumption ( m ) , which we repeat one more time . given @xmath312\preceq \alpha_1,\ldots,\alpha_k \in { \mathbb{n}}^r$ ] , we assumed that : 1 . for any subset @xmath335 , @xmath313,i}}}\prod_{\ell\in s } { { { \sbox{\myboxa}{$\m@ths$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\hspace{0.1mm}}}_i^{\alpha_\ell}$ ] does not depend on @xmath317}$ ] . if similarly to ( [ sec8sgamma ] ) we denote , for @xmath274 , @xmath893 then the assumption ( m ) is , obviously , equivalent to 1 . for any subset @xmath335 , @xmath313,i}}}\prod_{\ell\in s}s_i^{\alpha_\ell}$ ] does not depend on @xmath317}$ ] . let us define @xmath666 \preceq \gamma_1,\ldots,\gamma_{k}\in { \mathbb{n}}^{r - p}$ ] by @xmath681\gamma_\ell$ ] , and let @xmath880 be the overlap matrix @xmath894 the assumption ( m ) depends on @xmath318 only through this matrix @xmath880 , so one should really view it as a statement about such @xmath880 . fix @xmath895 and consider any @xmath896 such that @xmath897 since @xmath898 this means that @xmath899 for some @xmath900 . since our choice of @xmath901 was such that @xmath902 for all @xmath883 , this also implies that @xmath903 for all @xmath904 . in particular , we can find @xmath666 \preceq \gamma_{k+1}\in { \mathbb{n}}^{r - p}$ ] such that @xmath905 for @xmath900 . let @xmath906\gamma_{k+1}.$ ] recall that , whenever @xmath649\gamma$ ] for @xmath666 \preceq \gamma\in { \mathbb{n}}^{r - p}$ ] , the definitions ( [ sec6sialpha ] ) and ( [ indass ] ) imply that @xmath907 . therefore , in this case , we can rewrite the definition of @xmath908 below ( [ sec8at ] ) as @xmath909,i}}}\prod_{\ell\leq k+1 } s_i^{\alpha_\ell}.\ ] ] let us summarize what we proved . [ sec8th ] if the matrix @xmath880 defined in ( [ sec8qk ] ) satisfies the assumption ( m ) then @xmath910 does not depend on @xmath317}$ ] almost surely for all @xmath911 . finally , we will now use theorem [ sec8th ] to prove our main goal , theorem [ sec6ith1 ] in section [ sec2ilabel ] . to emphasize that our inductive proof will have a monotonicity property ( m ) , we can rephrase theorem [ sec6ith1 ] as follows . [ sec9th1 ] under the assumption ( [ indass ] ) , for any @xmath315 , any @xmath312\preceq \alpha_1,\ldots,\alpha_k \in { \mathbb{n}}^r$ ] and any @xmath335 , the expectation @xmath313,i}}}\prod_{\ell\in s } s_i^{\alpha_\ell}$ ] with respect to @xmath316\preceq \beta}$ ] does not depend on @xmath317}$ ] almost surely . it is much easier to describe the proof if we represent a configuration @xmath312\preceq \alpha_1,\ldots,\alpha_k \in { \mathbb{n}}^r$ ] not by a matrix @xmath912 with @xmath681\gamma_\ell$ ] but by a subtree of @xmath46 growing out of the vertex @xmath312 $ ] with branches leading to the leaves @xmath318 , and with an additional layer encoding their multiplicities ( see fig . [ fig1 ] ) . if we think of @xmath312 $ ] as a root of this subtree being at depth zero , then the leaves are at depth @xmath913 however , the multiplicity of any particular vertex @xmath33 in the set @xmath914 can be greater than one , so we will attach that number of children to each vertex @xmath33 to represent multiplicities , so the depth of the tree will be @xmath647 . whenever we say that we remove a leaf @xmath33 from the tree , we mean that we remove one multiplicity of @xmath33 . notice that removing a leaf from the tree also removes the path to that leaf , of course , keeping the shared paths leading to other leaves that are still there . first we are going to prove the following property , illustrated in fig . [ fig2 ] , that we will call property @xmath916 for @xmath917 let us consider an arbitrary configuration of paths leading from the root @xmath312 $ ] to some set of vertices at depth @xmath918 . let us call this part of the tree @xmath919 , which is now fixed . we pick one designated vertex at depth @xmath918 ( right - most vertex at depth @xmath918 in fig . [ fig2 ] ) . to all other vertices at depth @xmath918 we attach arbitrary trees @xmath920 leading to some arbitrary finite sets of leaves in @xmath45 and their multiplicities . we will use the same generic notation @xmath920 to represent an arbitrary tree , even though they can all be different . the designated vertex has some fixed number of children , say @xmath921 , and to each of these children we also attach an arbitrary tree @xmath920 . property @xmath916 will be the following statement . 1 . fix any @xmath919 and the number of children @xmath921 of a designated vertex . suppose that all trees that we just described are good ( this means for all choices of trees @xmath920 , possibly empty ) . then any tree obtained by adding a single new path leading from a designated vertex at depth @xmath918 to some new vertex @xmath922 with multiplicity one ( as in fig . [ fig2 ] ) is also good . illustrating property @xmath916 . solid lines represent the subtree @xmath919 and @xmath921 children of the designated vertex at depth @xmath918 . each @xmath920 represents an arbitrary tree leading to some set of leaves with their multiplicities . dashed line represents a new path from the designated vertex to a new vertex @xmath922 , which has multiplicity one.,scaledwidth=40.0% ] * proof . * let us fix any particular choice of trees @xmath920 attached to non - designated vertices at depth @xmath918 and @xmath921 children of the designated vertex . by the assumption in property @xmath916 , this tree , as well as any tree obtained by removing a finite number of leaves , is good . this precisely means that the assumption ( m ) holds for this tree , or for the sets of leaves with their multiplicities encoded by this tree . let @xmath880 be the matrix described above theorem [ sec8th ] corresponding to this set of leaves . then , theorem [ sec8th ] implies that @xmath924 does not depend on @xmath317}$ ] . obviously , each @xmath925 corresponds to a new tree constructed by adding one more new vertex @xmath33 to our tree or increasing the multiplicity of some old vertex by one . moreover , if @xmath926 then the overlap @xmath927 with one of the old vertices @xmath928 should be greater or equal than @xmath918 . this means that this new vertex will be attached to the tree somewhere at depth @xmath918 or below . one of the possibilities is described in fig . [ fig2 ] , when @xmath33 is attached by a new path to the designated vertex at depth @xmath918 . all other possibilities attaching @xmath33 below one of the non - designated vertices at depth @xmath918 or below one of the @xmath921 children of the designated vertex would simply modify one of the trees @xmath920 in fig . but such a modification results in a good tree , by the assumption in property @xmath916 . since the sum in ( [ sec9lem1eq ] ) is a linear combination of all these possibilities , the term corresponding to adding a new path as in fig . [ fig2 ] must be good , which finishes the proof . next we will prove another property that we will denote @xmath929 for @xmath930 , described in fig . [ fig3 ] . as in fig . [ fig2 ] , we consider an arbitrary configuration @xmath919 of paths leading from the root @xmath312 $ ] to some set of vertices at depth @xmath918 and we pick one designated vertex among them . to all other vertices at depth @xmath918 we attach arbitrary trees @xmath920 , while to the designated vertex we attach a single path leading to some vertex @xmath274 with multiplicity one . property @xmath929 will be the following statement . 1 . suppose that the following holds for any fixed tree @xmath919 up to depth @xmath918 . suppose that any tree as in fig . [ fig3 ] is good , as well as any tree obtained by removing any finite number of leaves from this tree . then any tree obtained by replacing a single path below the designated vertex at depth @xmath918 by an arbitrary tree @xmath920 is also good . * proof . * first of all , notice that property @xmath931 follows immediately from property @xmath932 . in property @xmath932 , arbitrary trees @xmath920 below non - designated vertices at depth @xmath933 represent their arbitrary multiplicities , the trees @xmath920 below the children of the designated vertex are empty , and the multiplicity of the designated vertex is @xmath921 . property @xmath932 then implies that we can increase this multiplicity by one to @xmath934 starting from multiplicity one and using this repeatedly , we can make this multiplicity arbitrary . this is exactly the property @xmath931 . illustrating property @xmath935 . solid lines represent the subtree @xmath919 and one path from a designated vertex at depth @xmath918 to a leaf @xmath922 with multiplicity one . property @xmath935 allows to replace this single path by an arbitrary tree @xmath920.,scaledwidth=35.0% ] next , we are going to show that property @xmath936 implies property @xmath929 . the proof of this is illustrated in fig . [ fig4 ] . given any tree as in fig . [ fig3 ] , let us denote by @xmath937 the designated vertex at depth @xmath918 . consider the subtree @xmath938 up to depth @xmath939 it forms the same pattern , with the child of @xmath937 playing a role of the designated vertex at depth @xmath939 therefore , by property @xmath936 we can replace the single path below this vertex by an arbitrary tree @xmath920 . by property @xmath916 , if we attach another path to @xmath937 , the resulting new tree is good . then we can again treat the child of @xmath937 along this new path as a designated vertex at depth @xmath940 , apply property @xmath936 and replace the path below this vertex by an arbitrary tree . if we continue to repeatedly use property @xmath916 to attach another path to @xmath937 and then use property @xmath936 to replace the part of this path below depth @xmath940 by an arbitrary tree , we can create an arbitrary tree below @xmath937 , and this tree is good by construction . this is precisely property @xmath929 , so the proof is completed by decreasing induction on @xmath941 * proof of theorem [ sec9th1 ] . * by lemma [ sec2ilem1 ] , the tree consisting of one path from @xmath312 $ ] ( at depth zero ) to some vertex @xmath942 ( at depth @xmath933 ) with multiplicity one is good . using property @xmath943 implies that arbitrary finite tree is good , which finishes the proof . illustrating proof of property @xmath935 . first we replace single path in fig . [ fig3 ] by arbitrary tree below the child of the designated vertex using @xmath944 . then we iteratively add a new path using property @xmath916 and then replace this path below depth @xmath940 be arbitrary tree using @xmath944.,scaledwidth=40.0% ]

we suggest a possible approach to proving the mzard - parisi formula for the free energy in the diluted spin glass models , such as diluted @xmath0-spin or random @xmath0-sat model at any positive temperature . in the main contribution of the paper , we show that a certain small modification of the hamiltonian in any of these models forces all finite - rsb asymptotic gibbs measures in the sense of the overlaps to satisfy the mzard - parisi ansatz for the distribution of spins . unfortunately , what is still missing is a description of the general full - rsb asymptotic gibbs measures . if one could show that the general case can be approximated by finite - rsb case in the right sense then one could a posteriori remove the small modification of the hamiltonian to recover the mzard - parisi formula for the original model . key words : spin glasses , diluted models + mathematics subject classification ( 2010 ) : 60k35 , 60g09 , 82b44

consider a discrete time martingale @xmath3 adapted to a filtration @xmath4 whose increments are uniformly bounded by @xmath5 , i.e. @xmath6 , and such that @xmath7 . it is folklore that in many respects , such a martingale should be well approximated by brownian motion . in particular , one would expect that @xmath8 should be of order @xmath9 . our goal in this paper is to point out that this naive expectation is completely wrong . we will frame this in the language of controlled processes below , but a corollary of our main result , theorem [ theo - main ] below , is the following . [ cor - main ] for any @xmath2 there exist @xmath10 and @xmath11 so that for any @xmath12 there exists an @xmath4-adapted discrete time martingale @xmath3 with @xmath6 and @xmath13 such that @xmath14 corollary [ cor - main ] can be viewed as a localization lemma . a complementary delocalization estimate was obtained by de la rue @xcite . we provide a different proof to a strengthened version of his results . [ theo - gen ] for any @xmath15 $ ] and @xmath16 there exist @xmath17 and @xmath18 so that the following holds . if @xmath3 is a discrete time martingale ( with respect to a filtration @xmath4 ) satisfying @xmath19 $ ] and @xmath20 a.s . , then @xmath21 the heart of the proof of theorem [ theo - gen ] uses a sequence of entrance times to a space - time region , which may be of independent interest ( see figure [ fig-1 ] for a graphical depiction ) . our interest in these questions arose while one of us was working on @xcite . charlie smart then kindly pointed out @xcite that the continuous time results in @xcite and @xcite concerning the viscosity solution of certain optimal control problems could be adapted to the discrete time setting ( using @xcite ) in order to show an integrated version of corollary [ cor - main ] , namely that for any fixed @xmath22 a martingale @xmath23 as in the lemma could be constructed so that for all @xmath24 small , @xmath25 ( note that @xmath26 can be taken arbitrarily large , for @xmath27 fixed . the estimate is in contrast with the expected linear - in-@xmath24 behavior one might naively expect from diffusive scaling . ) this then raises the question of whether a local version of this result could be obtained , and our goal in this short note is to answer that in the affirmative . we phrase some of our results in the language of _ controlled random walks_. fix a parameter @xmath28 . consider a controlled simple random walk @xmath29 , defined as follows . let @xmath30 and let @xmath31 denote the sigma - field generated by the process up to time @xmath32 . @xmath33-admissible control _ is a collection of random variables @xmath34 satisfying the following conditions : 1 . \a ) @xmath35 $ ] , a.s .. 2 . \b ) @xmath36 is @xmath4-adapted . let @xmath37 denote the set of all @xmath33-admissible controls . for @xmath38 , the controlled simple random walk @xmath29 is determined by the equation @xmath39 of course , @xmath29 is an @xmath4-martingale . for @xmath40 , we recover the standard simple random walk . we prove the following . [ theo - main ] for any @xmath41 , there exists @xmath42 and @xmath43 such that for any @xmath44 @xmath45 and @xmath46 work related to ours ( in the context of the control of diffusion processes ) has appeared in @xcite ; more recently , the results in @xcite are related to the lower bound in theorem [ theo - main ] . theorem [ theo - gen ] ( which immediately implies the upper bound in theorem [ theo - main ] ) is obtained by observing that any martingale has to overcome a ( logarithmic number of ) barriers in order to reach the target region , and each such barrier can be overcome only with ( conditional on the history ) probability bounded away from @xmath5 . the lower bound in theorem [ theo - main ] , on the other hand , will be obtained by exhibiting an explicit control . throughout this subsection , @xmath15 $ ] is a fixed constant , and @xmath3 denotes a martingale adapted to a filtration @xmath4 , satisfying the condition @xmath47 we begin with an elementary lemma . [ lem-0 ] assume that @xmath48 , that holds , and that for some @xmath49 , @xmath50 almost surely . fix @xmath51 let @xmath52 . then , @xmath53 _ proof of lemma [ lem-0 ] . _ by , the process @xmath54 is a sub - martingale , hence @xmath55 where the bound on the increments of @xmath23 was used in the last inequality . it follows that @xmath56 , and therefore , @xmath57 where was used in the second inequality . on the other hand , using again that increments of @xmath23 are bounded by @xmath58 , @xmath59 which implies that @xmath60 . combining this with yields the lemma . we have the following corollary . [ lem-2 ] let @xmath61 and let @xmath62 be a positive integer so that @xmath63 . assume , @xmath48 , and that @xmath64 let @xmath65 . then , @xmath66 _ proof of lemma [ lem-2 ] . _ set @xmath67 , @xmath68 , and iterate lemma [ lem-0 ] @xmath62 times . combining lemma [ lem-0 ] and lemma [ lem-2 ] , one obtains the following . [ lem-3 ] let @xmath69 be a positive integer . set @xmath70 and let @xmath62 be a positive integer so that @xmath71 . assume , @xmath72 , and @xmath73 let @xmath74 , |x|\leq h/3\}\,.\ ] ] then , @xmath75 _ proof of lemma [ lem-3 ] . _ it is enough to consider @xmath76 . let @xmath77 . note that the condition on @xmath62 ensured that @xmath78 . by lemma [ lem-2 ] , @xmath79 on the other hand , by doob s inequality and , on the event @xmath80 , @xmath81 combining the last two displays completes the proof . we can now begin to construct the barriers alluded to above . fix @xmath12 and set @xmath82\cap { { \mathbb{z}}}$ ] , @xmath83 $ ] . write @xmath84 and @xmath85 . define the following nested subsets of @xmath86 : @xmath87 , d_{i}= r_{[2^{-i/2}\sqrt{n}]}\times b_{i , n}\,.\ ] ] let @xmath88 , i.e. @xmath89 [ fig-1 ] let @xmath90 and for @xmath91 set @xmath92 . a direct consequence of lemma [ lem-3 ] is the following . [ lem - tau ] there exists a constant @xmath93 so that on the event @xmath94 , and with @xmath95 , one has @xmath96 ( the choice of @xmath97 ensured that in the applications of lemma [ lem-3 ] for any @xmath95 , the condition holds . ) _ proof of theorem [ theo - gen ] . _ it is clearly enough to consider @xmath98 with arbitrary @xmath99 . adjusting @xmath100 if necessary , we may and will assume that @xmath101 . note that @xmath102 and therefore , by lemma [ lem - tau ] , @xmath103 this yields the theorem . the upper bound in is a consequence of theorem [ theo - gen ] . we thus need only to consider the lower bound in , and the claim . first note that the simple control @xmath104 already achieves the lower bound with exponent @xmath105 . thus , what we need to show is that for any @xmath106 there is a ( polynomially ) better control and that as @xmath107 we can achieve an exponent close to 0 . toward this end , we use two very simple controls , that are not approximation of the optimal control . see section [ sec - concl ] for further comments on this point . we begin with the following a - priori estimate . [ lem - ori2 ] for any @xmath106 there exist @xmath2 , @xmath10 , @xmath108 and @xmath109 such that for any @xmath110 there is a @xmath33-admissible control such that @xmath111 for any @xmath112 $ ] . _ proof of lemma [ lem - ori2 ] : _ the control we take is slow inside @xmath113 $ ] and fast outside , i.e. we take @xmath104 for @xmath114 and @xmath115 for @xmath116 . we claim that given any @xmath106 , using this control with @xmath2 and @xmath10 small enough and @xmath110 with @xmath117 large enough will satisfy the conclusion of the lemma with some @xmath109 . our control does not change with time , it is a reversible markov chain with weights @xmath118 and @xmath119 for @xmath120 and @xmath121 for @xmath122 . its reversing measure is thus @xmath123 for @xmath120 and @xmath124 for @xmath122 . using reversibility we get @xmath125 thus , @xmath126 @xmath127 + \sum_{x=-\beta k}^{\beta k } p_y(s_{\alpha k^2}^u = x)\ ] ] @xmath128)- q p_y(s_{\alpha k^2}^u \in [ -\beta k , \beta k])\big]\ ] ] now , the probability that a simple random walk will get to a distance of more than @xmath129 in @xmath130 steps tends to 0 as @xmath131 tends to 0 , uniformly in @xmath62 . obviously , this applies also for our controlled random walk ( which is sometimes lazy ) , hence by choosing small enough @xmath131 we can guarantee that for any @xmath132 and any @xmath133 $ ] we have @xmath134)>1-q$ ] . having fixed @xmath131 , we now claim that @xmath135 } p_y(s_{\alpha k^2}^u \in [ -\beta k , \beta k])= 0\,.\ ] ] indeed , by ( * ? ? ? * corollary 14.5 ) , there exists a constant @xmath136 so that @xmath137 for any two states @xmath138 and @xmath139 . ( the bound in @xcite is valid for any random walk on an infinite graph with bounded degree and bounded above and below conductances , see ( * ? ? ? * 40 ) ; note that while the bound is stated for discrete time chains , it can also be transferred without much effort to the continuous time setting . see e.g. ( * ? ? ? * theorem 2.14 and proposition 3.13 ) . ) plugging @xmath140 , we get @xmath141 < \frac{c(q)(2\beta k + 1)}{\sqrt{\alpha}k } \ , , \ ] ] which tends to 0 when @xmath142 and @xmath143 in the order prescribed in . thus , by choosing small enough @xmath27 and large enough @xmath117 we can have @xmath144)- q p_y(s_{\alpha k^2}^u \in [ -\beta k , \beta k ] ) > 1-q\ ] ] uniformly for all @xmath110 and we are done . _ proof of the lower bound in theorem [ theo - main ] : _ fix @xmath106 and choose @xmath145 and @xmath146 according to lemma [ lem - ori2 ] . let @xmath147 and let @xmath148 , for @xmath149 and @xmath150 . for time @xmath151 we use the control @xmath152 . notice that @xmath153 so standard estimates for lazy random walk show that there exists a constant @xmath11 , independent of @xmath154 , such that @xmath155 , for any @xmath156\subset [ -t^{1/2},t^{1/2}]$ ] . for any @xmath157 , from time @xmath158 to @xmath159 we use the strategy provided by lemma [ lem - ori2 ] for @xmath160 . applying lemma [ lem - ori2 ] repeatedly , we see that for any @xmath161 , at time @xmath158 we have @xmath162 for any @xmath163 $ ] . in particular , we have @xmath164 showing that @xmath165 . this completes the proof of . in preparation for the proof of , we provide an auxilliary estimate . [ lem - ori ] for any @xmath109 there exist @xmath166 and @xmath167 such that for any @xmath62 there is a @xmath33-admissible control with the property that for any @xmath168 $ ] we have @xmath169 ) > 1 - { { \varepsilon}}\ , .\ ] ] _ proof of lemma [ lem - ori ] : _ let @xmath166 be so that for a simple random walk on @xmath170 we have for any @xmath62 , @xmath171 where @xmath172 is the first hitting time of @xmath173 . having chosen @xmath166 , let @xmath167 be so big such that for a @xmath33-lazy random walk ( that is , a random walk with control @xmath104 ) we have for any @xmath62 , @xmath174 where @xmath175 is the first time of hitting either @xmath62 or @xmath176 . we now define the control to be fast until the walk hits @xmath177 and slow afterwards , i.e. we take @xmath115 for @xmath178 and @xmath104 for @xmath179 . if the starting location @xmath180 is in @xmath181 $ ] , then by ( [ ori1 ] ) with probability at least @xmath182 we hit @xmath177 before time @xmath183 . if that happens , then by ( [ ori2 ] ) with probability at least @xmath182 , the walk stays inside @xmath184 $ ] until time @xmath183 . we can now complete the proof of theorem [ theo - main ] . + _ proof of : _ fix @xmath109 and choose @xmath33 and @xmath166 according to lemma [ lem - ori ] . let @xmath185 and let @xmath186 , for @xmath187 . for time @xmath177 to @xmath188 , we have @xmath189)>c\ ] ] for some fixed @xmath11 , regardless of the control . for any @xmath157 , from time @xmath158 to @xmath159 we use the strategy provided by lemma [ lem - ori ] for @xmath190 . then with probability at least @xmath191 we have @xmath192 . this yields the required lower bound . motivated by the structure of the optimal control in the continuous time - and - space analogue of our control problem , see @xcite , one could attempt to improve on the lower bound in by using a bang - bang control of the type @xmath104 if @xmath193 $ ] and @xmath115 otherwise , where @xmath194 is a domain whose boundary is determined by an appropriate ( roughly parabolic ) curve . the analysis of that control is somewhat tedious , and proceeding in that direction we have only been able to show the lower bound in with @xmath165 when @xmath33 is sufficiently large . it would be interesting to check whether an analysis of the dynamic programming equation associated with the control problem , in line with its continuous time analogue in @xcite , could yield that estimate , and more ambitiously , show the equality of @xmath195 and @xmath196 in . one could also consider the dual problem of _ minimizing _ the probability of hitting @xmath177 at time @xmath44 , that is , in the setup of theorem [ theo - main ] , of evaluating @xmath197 one can adapt the proof of the lower bound in theorem [ theo - main ] ( replacing in the sub - optimal control `` fast '' by `` slow '' ) to obtain a polynomial upper bound in that has exponent larger than @xmath198 . similarly ( using the invariance principle for martingales ) , one shows that there is @xmath199 such tha the controlled walk with @xmath200 satisfies @xmath201 with positive ( depending only on @xmath33 and independent of @xmath62 ) probability , and from this a polynomal lower bound in follows . we omit further details . + * acknowledgement * we thank bruno schapira for pointing out @xcite to us , and charlie smart for his crucial comments @xcite . 99 k. alexander , _ controlled random walks with a target site _ , s. n. armstrong and m. trokhimtchouk , _ long time asymptotics for fully nonlinear homogeneous parabolic equations _ , calc . partial differential equations * 38 * ( 2010 ) , pp . 521540 . barenblatt and g. i. sivashinskii , _ self similar solutions of the second kind in nonlinear filtration _ , j. appl . 33 * ( 1969 ) , pp . g. barles and p. e. souganidis , _ convergence of approximation schemes for fully nonlinear second order equations _ , asymtotic anal . * 4 * ( 1991 ) , pp . t. de la rue , _ vitesse de dispersion pour une classe de martingales _ , annales de lihp probab . stat . * 38 * ( 2002 ) , pp . 465474 . t. kumagai . random walks on disordered media and their scaling limits.st . flour lecture notes ( 2010 ) , to appear lecture notes in mathematics , springer . current version available at + http://www.kurims.kyoto-u.ac.jp/ kumagai / stflour - tk.pdf . j. r. lee and y. peres , _ harmonic maps on amenable groups and a diffusive lower bound for random walks _ , arxiv:0911.0274 , annals probab . ( to appear ) . j. m. mcnamara , _ a regularity condition on the transition probability measure of a diffusion process _ , stochastics * 15 * ( 1985 ) , pp . c. k. smart , _ personal communication _ w. woess , _ random walks on infinite graphs and groups _ , cambridge university press ( 2000 ) .

we consider controlled random walks that are martingales with uniformly bounded increments and nontrivial jump probabilities and show that such walks can be constructed so that @xmath0 decays at polynomial rate @xmath1 where @xmath2 can be arbitrarily small . we also show , by means of a general delocalization lemma for martingales , which is of independent interest , that slower than polynomial decay is not possible . # 1to # 1to # 1

the formation of disk - dominated galaxies remains a challenge for modern cosmology @xcite . bright spiral galaxies with stellar masses around @xmath0 m@xmath1 at @xmath2 typically have bulge - to - disk mass ratios ( b / d ) of 0.2 to 0.4 @xcite . for instance , the milky way itself has a b / d ratio of 0.35 - 0.40 @xcite . luminosity ratios in near - infrared and optical bands are even lower @xcite . however , cosmological simulations predict much higher bulge fractions , and consequently a baryonic angular momentum much lower than observed . the hierarchical assembly of dark matter halos drives successive galaxy mergers , followed by a rapid growth of central bulges . indeed , a single major merger , even starting with a high gas fraction , will generally end - up with more baryons in the bulge than in the disk @xcite . successive minor mergers also drive bulge growth ( bournaud , jog & combes 2007 ) . it has been realized recently that the assembly of baryons onto galaxies is far from being only driven by mergers . rapid accretion of cold gas is another major mode of galaxy assembly @xcite . this cold accretion mode could apparently feed disk - dominated galaxies , but it makes the disks so massive and turbulent that they violently fragment into giant clumps ( bournaud & elmegreen 2009 ; dekel , sari & ceverino 2009 ; agertz , teyssier & moore 2009 ; burkert et al . clump coalescence will rapidly fuel the bulge ( @xcite ; bournaud , elmegreen & elmegreen 2007 ) , and signatures of this additional bulge formation process have been observed @xcite . disk internal secular evolution can also lead to the slow buildup of pseudo - bulges @xcite . a general result is that the rapid formation of massive bulges appears unavoidable because of the combination of mergers and disk instabilities . the most disky galaxies in cosmological models have b / d around 1 in the best cases and generally higher , in both the merger - driven and stream - fed dominant modes ( ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) ; ceverino , dekel & bournaud 2009 ) . the bulge fraction problem is related to the well - known spin crisis : galaxies could gain angular momentum from gravitational torquing by satellites ( porciani , dekel & hoffman 2002 ) , but the early build - up of bulges quenches this process . energy feedback from supernovae explosions can regulate star formation and keep the gas fraction higher , so that disk components survive mergers more easily . stream - fed disks with efficient supernovae feedback can then have realistic rotation velocities , in agreement with the observed tully - fisher relation @xcite , but the mass fraction in these fastly rotating disks remains too low compared to their bulges , meaning that the real angular momentum is low . high bulge fractions thus remain as a separate issue that supernovae feedback can not solve , even when many feedback models are tested in numerical simulations @xcite . apart for dwarf galaxies @xcite , the standard models apparently fail to form high - spin , disk - dominated galaxies . a stellar evolution process that is more general than supernovae feedback is the continuous return of gas from stars of any mass , in particular through stellar winds and planetary nebulae @xcite . the gas return fraction of a coeval stellar population over the hubble time can be as high as 50% for a chabrier imf , or around 40% for a scalo imf , with small dependencies on the metallicity ( jungwiert , combes & palou 2001 ; lia , portinari & carraro 2002 ; pozzetti et al . supernovae account only for about 10% of this return fraction , which is dominated by low- and intermediate - mass stars . half of the stellar mass forming at redshift @xmath3 will thus be returned in the form of gas by redshift 0 , which can cool down in the disk and form new stars that will in turn return gas , in a continuous recycling process . this is another type of feedback , which is more a _ mass _ feedback than the energy feedback from supernovae explosions usually considered . this continuous mass return has already been included in a few cosmological simulations ( e.g. , * ? ? ? * ; * ? ? ? * ) , but these simulations were limited to high redshift , or lacked resolution to clearly resolve disks and bulges , or the specific effects of gas return were not explored . here we show that this known process has unsuspected effects on regulating the growth of bulges and the survival of massive disks in spiral galaxies . using a cosmological simulation of the formation of a massive galaxy , which in the standard model ends - up with a b / d ratio larger than unity , we show that continuous gas recycling reduces the final b / d ratio to 40% , and the stellar light fraction in the bulge by a factor of 3 . the mass and disk rotation speed of this galaxy are consistent with observed scaling relations , and its remarkably flat rotation curve is typical for late - type , high - spin spirals . the mass return of stars regulates the bulge growth and angular momentum dissipation in such proportion that a late - type spiral ( sb - sc ) is formed instead of an early - type galaxy . this shows how strongly stellar evolution can affect the formation of galaxies , and help to solve the issue of massive late - type spiral galaxy formation . our model is based on a high resolution re - simulation of a galaxy identified in a large - scale cosmological simulation . the initial cosmological run was performed assuming standard @xmath4-cdm cosmological parameters using the ramses code @xcite and is described in @xcite . the chosen dark matter halo has a virial mass of 1.3@xmath5 m@xmath1 at redshift 0 , in a field environment . we avoid dense environments and halos undergoing many violent mergers so that the chosen halo is prone to hosting a disk - dominated spiral galaxy at @xmath2 . the mass assembly history of the re - simulated galaxy is shown on figure [ evol_mass ] , and results from a combination of galaxy mergers and diffuse infall from cosmic gas filaments . and @xmath6 . the subsequent evolution is much smoother , dominated by diffuse gas infall : the most important merger of this second phase takes place at @xmath7 with a mass ratio of 6:1.,width=302 ] using the zoom - in technique presented in @xcite , we re - simulate the evolution of the chosen galaxy from @xmath8 to @xmath2 inside a 400 kpc - large zoom area . the spatial resolution ( gravitational softening ) is 150 pc and the mass resolution ( particle mass ) is @xmath9 m@xmath1 for gas , @xmath10 m@xmath1 for stars , and @xmath11 m@xmath1 for dark matter . gas dynamics is modeled with a sticky - particle scheme and star formation is computed with a schmidt - kennicutt law with an exponent of 1.5 . the threshold for star formation is set to 0.03 m@xmath1pc@xmath12 , which corresponds to the minimal density for diffuse atomic clouds formation @xcite . the galaxy is initialized at @xmath8 as a very gas - rich disk , with a gas fraction of 0.5 with respect to total baryonic mass . the initial dark halo follows a burkert profile with a core radius of 5.9 kpc and a truncation radius of 13.8 kpc . gas and stars are in a disk with exponential scale - lengths of 540 and 230 pc , respectively . starting the simulation at @xmath8 captures 97% of the mass assembly history of this galaxy , as the total mass ( including dark matter ) at redshift 5 is only @xmath13 m@xmath1 . for this reason and because numerous mergers rapidly destroy the proto - disk into a spheroid around which the regular bulge and disk components will gradually grow ( figure [ snapshots ] ) , our choice of starting with a disk is not influential in the final @xmath2 properties of the galaxy . 0.5 cm [ snapshots ] we do not include the energy feedback from supernovae . there is no unique model for this , and feedback is anyway most efficient in dwarf galaxies where the structure is strongly influenced @xcite ; the effects in massive galaxies are more moderate ( e.g. , * ? ? ? * ) . not including feedback would be a serious problem at very high redshift ( @xmath14 ) because of the overcooling issue @xcite , or in very high resolution ( pc - scale ) simulations to avoid the collapse of all gas into very dense clumps that would never be disrupted . this is not resolved in our simulations at resolutions of 100 pc or more . at these scales , feedback mostly regulates star formation , and we calibrate the star formation efficiency in our simulations so that the gas consumption time matches typical observed values , so this effect of feedback is indirectly modeled . in turn , we implement the continuous gas return from high- , intermediate- and low - mass stars . to this aim , we follow the scheme proposed by @xcite : each stellar particle represents a population of stars born at the same time , whose return rate is given by : @xmath15 where @xmath16 is the age of the particle since its formation , @xmath17 is set to 4.86 myr to fit standard imf return function @xcite , and @xmath18 is such that the integrated return fraction of a stellar population over 10 gyr is 40% . the returned gas is directly set cold , immediately available for star formation . real stellar winds would provide warmer gas , but overestimating the delay before it can form new stars would artificially increase the gas fraction and the capacity of disks to survive mergers : to avoid such favorable biases , we let the returned gas immediately available for star formation . we performed a `` fiducial simulation '' without gas return from evolved stars , and a simulation with stellar mass loss . the structural properties were analyzed @xmath2 in both cases . we measured bulge - to - disk mass ratios with different techniques . first , we used azimuthally - averaged stellar density profiles : an exponential disk is fitted and the mass excess in the central regions corresponds to the bulge . we also performed a decomposition based on kinematics , and more precisely on the angular momentum of each star . we computed the total angular momentum of the gas disk in its inner 10 kpc , and set this as the @xmath19 axis . for each stellar particle , we then computed @xmath20 , the ratio of its angular momentum along the @xmath19 axis to the angular momentum it would have if it were on a circular orbit ( that is @xmath21 ) . the distribution of the values of @xmath22 shows two peaks : one around @xmath23 , that corresponds to bulge stars , and one around @xmath24 , that corresponds to disk stars . the limit between bulge and disk is set at the value of @xmath22 corresponding to the minimum of the distribution between these two peaks . finally , we also computed @xmath25-band surface luminosities using the spectral evolution model pegase.2 @xcite to derive bulge - to - disk ratios in the @xmath25-band , assuming a solar metallicity : the @xmath25-band mass - to - luminosity does not vary much across the radial metallicity gradient of typical @xmath2 spiral galaxies . the time evolution of the studied galaxy is shown on figure [ snapshots ] . interactions and mergers with companion galaxies rapidly transform the initial disk into a spheroid . at later times , star formation in gas left over after these early mergers and gas gradually brought in by cosmic gas flows and merging satellites builds a large disk , while interactions continue to grow a bulge - at a rate that depends on the model . at @xmath2 , both cases have a central bulge and an exponential disk with grand design spiral arms . a major difference , though , lies in the distribution of the @xmath2 stellar mass in the disk and bulge components . without stellar mass - loss , the bulge to disk mass ratio derived from the stellar mass profile is slightly above 1 , which is typical for massive `` disk '' galaxies in current @xmath4-cdm models ( see introduction ) . the disk component contains barely half of the stellar mass , even though we include both the thin and thick disks . when the continuous stellar mass loss is included , the bulge to disk mass ratio is reduced to 0.40 : the bulge is less massive , with a lower density and a smaller size ( right panels of figure [ snapshots ] ) , while the disk is denser with a similar scale length ( figure [ profiles ] and table [ tab ] ) . other definitions of b / d lead to the same conclusion . a kinematical decomposition even shows a more drastic effect , with b / d decreasing from 1.40 to 0.32 ( figure [ rotcurve ] and table [ tab ] ) . in the @xmath25 band luminosity , b / d decreases from 0.49 to 0.16 . similar values were obtained when the simulation was run with a twice lower spatial resolution and a eight times higher particle mass , suggesting reasonable convergence of the simulation . ccc & fiducial run & with mass loss + stellar disk mass ( m@xmath1 ) & @xmath26 & @xmath27 + b / d ( stellar mass profile ) & 1.01 & 0.40 + b / d ( kinematics ) & 1.40 & 0.32 + b / d ( @xmath25 band ) & 0.49 & 0.16 + disk exponential scale length ( kpc ) & 4.5 & 5.0 + optical radius @xmath28 ( kpc ) & 22.0 & 19.8 + bulge half - mass radius ( kpc ) & 1.3 & 1.6 + absolute @xmath29 band magnitude & -22.8 & -22.9 + hi linewidth ( km s@xmath30 ) & 583 & 591 + axis to the angular momentum they would have on circular orbits : bulge stars are found around @xmath23 while disk stars are found around @xmath24 . the bulge component is significantly reduced under the effect of stellar mass loss.[rotcurve],title="fig:",width=302 ] axis to the angular momentum they would have on circular orbits : bulge stars are found around @xmath23 while disk stars are found around @xmath24 . the bulge component is significantly reduced under the effect of stellar mass loss.[rotcurve],title="fig:",width=302 ] the effect of stellar mass loss on the @xmath2 structural properties of the galaxy is thus major , although we studied quite a massive galaxy for which preserving disks is harder than in lower - mass systems . continuous stellar mass loss transforms a bulge - dominated system , typically classified as s0/sa in the hubble classification @xcite into a disk - dominated sb / sc - like galaxy , with a bulge fraction typical for the milky way and the majority of massive spirals in the nearby universe . as a result from the different mass distribution , the rotation curve at @xmath2 has a different profile ( figure [ rotcurve ] ) . in the fiducial case , the rotation curve is peaked a low radii before decreasing and reaching a flat plateau . this is typical for bulge - dominated , s0-sa galaxies ( e.g. , m31 chemin , carignan & foster 2009 ; ngc 3031 , 4736 , or 2841 in @xcite ) . the regulation of bulge growth by continuous stellar mass - loss results in a flater rotation curve , which is typical of disk - dominated spiral galaxies like ngc 2903 , 3198 , 3621 @xcite . the comparison of rotation curve with typical cases also corresponds to a shift by two categories in the hubble classification scheme , from early - type disky galaxies to late - type disk - dominated galaxies , in agreement with the direct estimates of bulge fractions . we also derived the equivalent of an observed atomic gas ( hi ) linewidth : we select gas in regions denser than one atom per cubic centimeter in order to include the warm / cold gas disk without the hot halo , simulate the velocity spectrum for the galaxy , and compute its fwhm averaged on several edge - on projections . the @xmath29 band absolute magnitude to hi linewidth ratio ( see table [ tab ] ) puts both systems very close to the observed tully - fisher relation for local spiral galaxies as observed by @xcite . the tully - fisher relation is nevertheless not the best direct estimate of the baryonic angular momentum of a galaxy since it measures the disk rotation speed at large radii without indicating whether or not a large fraction of the stellar mass lies in a concentrated , low angular momentum bulge . a similar effect has been also found in two other simulations . we measured bulge and disk mass following the decomposition based on kinematics . in one case ( a galaxy with m@xmath31 m@xmath32 at @xmath8 and 1.3@xmath33 m@xmath32 at @xmath2 ) , b / d decreases from 1.24 to 0.75 when mass loss is included , in the other case ( with m@xmath34 m@xmath32 at @xmath8 and @xmath35 m@xmath32 at @xmath2 ) , it decreases from 0.94 to 0.49 . our simulation of cosmological galaxy formation shows that the continuous gas return from high- , intermediate- and low - mass stars over cosmic times can strongly regulate the growth of bulge and the dissipation of angular momentum in disk galaxies . there are at least two ways in which stellar mass loss can support the survival of large , massive disks . first , it can keep the gas fraction higher in young galaxies before a merger occurs , which helps disks survival against destruction into bulges @xcite . second , the bulge , stellar halo and thickened disk components release fresh gas after interactions and mergers : this decreases their mass and increases that of the thin gas disk . the disk re - formed by gas returned by a bulge or halo should have a low angular momentum , but can be torqued by satellites into a large , high - spin disk that gradually forms new stars ( an example of such an interaction is shown at @xmath36 on figure [ snapshots ] ) . the radial migration of disk stars can also arise from resonant interactions with spiral arms : @xcite have shown , in a milky - way like galaxy , that up to 50% of stars in the solar neighborhood could have migrated from the inner disk . in our model , the large disk scale length is preserved by the combination of stellar mass - loss and radial migration following external interactions and/or internal evolution . overall , the effect of stellar mass loss on disk survival and regulation of bulge growth is major . it reduces the bulge - to - disk ratio by a factor up to @xmath37 3 in both the stellar mass and light . this factor is what was typically missing in cosmological models to account for the formation of massive , disk - dominated , late - type spiral galaxies . this does not imply that all galaxies would end - up with a spiral - like morphology at redshift zero , and we briefly presented cases of galaxies with other merger and star formation histories where the decrease of b / d is weaker . gas return promotes disk survival against mergers and disk instabilities , but will preserve a disk - dominated galaxy only if the mergers are not too numerous , do not happen too late , and the internal instabilities not too violent . for instance , a galaxy undergoing a major merger at low redshift would still end - up as a @xmath2 elliptical or lenticular , because of a stellar population globally too old to return a significant amount of gas . such @xmath2 ellipticals are indeed found in our cosmological simulations with the same technique ( e.g. in * ? ? ? * ) . gas consumption , stripping and strangulation in groups , will also support the survival of early - type elliptical and s0s . thus , gas return from stellar populations can explain the origin of late - type galaxies , without challenging the formation of early - type galaxies in systems that undergo later mergers or more violent internal instabilities . other sorts of feedback processes , like the energy released by supernovae or the radiation pressure from young massive stars ( murray , quataert & thompson 2009 ) could further regulate the bulge growth especially in low - mass galaxies , potentially forming almost bulgeless galaxies . however , we propose that the formation of massive late - type galaxies in the @xmath4-cdm universe is explained not by an unknown combination of minor factors , but mostly by one major effect , namely by the gas return from stellar populations all across their initial mass function . this process can largely help to solve one of the main challenges in the origin of modern galaxies , namely the origin of late - type , disk - dominated spiral galaxies like the milky way .

spiral galaxies have most of their stellar mass in a large rotating disk , and only a modest fraction in a central spheroidal bulge . this challenges present models of galaxy formation : galaxies form at the centre of dark matter halos through a combination of hierarchical merging and gas accretion along cold streams . cosmological simulations thus predict galaxies to rapidly grow their bulge through mergers and instabilities , and to end - up with most of their mass in the bulge and an angular momentum much below the observed level , except in dwarf galaxies . we propose that the continuous return of gas by stellar populations over cosmic times could help to solve this issue . a population of stars formed at a given instant typically returns half of its initial mass in the form of gas over 10 billion years , and the process is not dominated by supernovae explosions but by the long - term mass - loss from low- and intermediate - mass stars . using simulations of galaxy formation , we show that this gas recycling can strongly affect the structural evolution of massive galaxies , potentially solving the bulge fraction issue , as the bulge - to - disk ratio of a massive galaxy can be divided by a factor of 3 . the continuous recycling of baryons through star formation and stellar mass loss helps the growth of disks and their survival to interactions and mergers . instead of forming only early - type , spheroid - dominated galaxies ( s0 and ellipticals ) , the standard cosmological model can successfully account for massive late - type , disk - dominated spiral galaxies ( sb - sc ) .

while all attention is now focused on the large hadron collider ( lhc ) as a possible _ gold mine _ of physics beyond the standard model ( sm ) , one should not lose sight of other territories rich with new physics , e.g. , lepton flavor violating ( lfv ) rare decays , which could provide complementary information . ever since neutrino flavor mixing was established , interests for observing flavor violation in charged lepton decays have boomed . while in the neutrino sector flavor violation could be rather large ( maximal between @xmath18 and @xmath19 ) , in the charged lepton decays there is no sign of flavor violation as yet . the sm contributions to charged lfv decays are quite small , orders of magnitude below the current experimental sensitivity , due to the smallness of neutrino mass . hence , any observation of lfv processes in the charged lepton sector , which are being probed with ever increasing sensitivity , would unambiguously point to non - standard interactions . indeed , such indirect observations taken in isolation may not imply much on the exact nature of new physics . but a study of possible correlations of its effects on different independently measured charged lfv observables might provide a powerful cross - check and lead to identification of new physics through lhc / lfv synergy . in this paper , we consider @xmath0-parity violating ( rpv ) supersymmetry @xcite and perform a correlation analysis of its numerical impact on different lfv @xmath6 decays . we also study at tandem the rpv contribution to @xmath5 , an observable which continues to provide a @xmath20 room for new physics despite significantly improved theoretical and experimental accuracies . @xmath0-parity is a discrete symmetry , which is defined as @xmath21 , where @xmath22 , @xmath23 , and @xmath24 are the baryon number , lepton number and spin of a particle , respectively . @xmath0 is @xmath25 for all sm particles and @xmath26 for their superpartners . the usual assumption of @xmath22 and @xmath23 conservation in supersymmetric models are not supported by any deep underlying principle . the @xmath23-violating @xmath27-type superpotential is written as @xmath28 , where @xmath29 stands for su(2 ) doublet lepton superfields , @xmath30 for su(2 ) doublet quark superfields , @xmath31 for su(2 ) singlet down - type quark superfields , and @xmath32 are generation indices . there are 27 such @xmath27 couplings , on each of which and also on many of their combinations exist strong constraints @xcite . we select _ only two _ of them , namely @xmath33 and @xmath34 , and consider only them to be large and the rest to be either vanishing or negligibly small . _ why @xmath33 and @xmath34 _ ? it has been observed that these two couplings ( each with a magnitude of @xmath35 and for superparticle masses around 300 gev ) can justify the recently observed large @xmath3 @xmath36 branching ratios that the sm can not explain @xcite . on top of that , if we turn on even a small @xmath37 , then together with @xmath38 , one can _ also _ explain the large phase in @xmath39@xmath40 mixing @xcite . turning our attention to the neutrino sector , we recall that generation of neutrino masses and mixing by @xmath27-type couplings usually require a specific combination of indices , namely , the @xmath41 product couplings @xcite . but if we take @xmath33 and @xmath34 ( together with @xmath42 as the only non - vanishing and large rpv couplings , a phenomenologically acceptable pattern of neutrino masses and mixing emerge at two - loop level @xcite . to sum up , we consider four non - vanishing rpv couplings : two of them @xmath33 and @xmath34 large ( @xmath35 ) and relevant for the present analysis , the other two @xmath43 and @xmath44 ( from charged - current universality @xcite ) not to be used in the present analysis but implicitly present to justify our choices of @xmath33 and @xmath34 through the correlated phenomena mentioned above . now , motivated by the observation of maximal mixing between @xmath18 and @xmath19 , we make a further assumption @xmath45 . keeping all these in mind , we outline our agenda as follows : consider @xmath33 and @xmath34 as the only two _ relevant _ rpv couplings ( the other two , viz . @xmath46 and @xmath47 , optional and small ) , assume them to be real , set their magnitudes equal and just enough to explain the @xmath3 anomaly , predict its effect on @xmath5 , and estimate its numerical impact on different lfv @xmath6 decays ( @xmath48 , @xmath49 , @xmath50 , @xmath51 ) . for simplicity , we assume all squark and slepton masses to be degenerate , and denote the common mass by @xmath52 . we derive various one - loop effective flavor violating vertices , which we have often referred to as _ form - factors_. we display their exact as well as approximate expressions . while for numerical plots we use the exact formulae , the approximate expressions serve to provide an intuitive feel of the numerical impact . the branching fraction of the leptonic decay @xmath3 ( @xmath54 ) is given by @xmath55 where @xmath56 is the lifetime of @xmath8 . the decay constant is defined as @xmath57 , where @xmath58 is the momentum of @xmath8 . the branching ratio has a helicity suppression factor characterized by @xmath59 on account of a spin - zero particle decaying into two spin - half particles . monte - carlo simulations of qcd on lattice predict @xmath60 mev @xcite . the experimental average is somewhat higher : @xmath61 mev @xcite . the enhancements are @xmath62 in the muon channel , @xmath63 in the tau channel , and @xmath64 on average . on the other hand , the lattice estimate and the experimentally obtained value for @xmath65 seem to be in perfect agreement around 206 mev @xcite . the latter suggests that the discrepancy in @xmath53 may very well be influenced by new physics contributing in a _ flavor specific way _ to @xmath8 decay . note that @xmath3 in the sm proceeds at tree level and it is cabibbo - allowed . hence , loop suppressed new physics is an unlikely candidate to account for the discrepancy . leptoquark or charged higgs interactions have been advocated in this context as they provide new tree amplitudes for the above decay @xcite . our candidate is supersymmetric rpv interaction and our chosen couplings , @xmath33 and @xmath34 , contribute to @xmath66 _ via _ @xmath67-exchanged _ tree _ graphs @xcite . the net contribution to the @xmath68 channel can be obtained by replacing @xmath69 in eq . ( [ dlnu ] ) by @xmath70 for @xmath71 , we must do the replacements @xmath72 in eq . ( [ dsmunurpv ] ) . the effective vertex of photon with any charged fermion is given by @xmath73 u(p ) \ , .\ ] ] the muon magnetic moment for @xmath74 is given by @xmath75 . at tree level , @xmath76 and @xmath77 . quantum correction yields @xmath78 , while @xmath79 remains unity at all order due to charge conservation . since @xmath80 , it follows that @xmath81 . as per current estimation @xcite , the room for new physics is given by @xmath82 the coupling @xmath83 induces a contribution to @xmath84 , which proceeds through the diagrams in fig . [ gminustwo ] . the quarks and squarks inside the loop have been labeled by generic symbols @xmath85 and @xmath86 respectively , which can take two sets : @xmath87 and @xmath88 . the loop integrals would depend on @xmath89 and @xmath90 . as mentioned earlier and assumed throughout our analysis , @xmath91 . we obtain @xmath92 ~\simeq ~ 3\frac{|\lambda'_{223}|^2 m_\mu^2}{16\pi^2{\tilde m}^2 } \left(\frac{1}{6}\right ) \ , .\end{aligned}\ ] ] the @xmath93-functions used throughout our analysis are given by @xmath94 \\[2ex ] \bar{\xi}_n(r ) & = & \displaystyle { 1\over{r } } \xi_n\left({1\over r}\right)\ . & & \end{array } \label{xi}\ ] ] the decay @xmath96 proceeds through photon and @xmath97 penguins ( fig . [ peng_tau3mu ] ) and box graph ( fig . [ box_tau3mu ] ) . we consider each of them below . here flavor violation is induced by @xmath33 and @xmath34 _ via _ loops with quarks and squarks in internal lines . the amplitude of the photon exchanged diagrams for @xmath98 decay can be written as @xmath99u_\tau(p ) \frac{e^2}{q^2}\bar u_\mu(p_2)\gamma^\mu v_\mu(p_3)-(p_1\leftrightarrow p_2 ) \ , , \ ] ] where @xmath85 is the photon momentum . the form - factors @xmath100 and @xmath101 are induced by the flavor - changing @xmath102 couplings . each penguin diagram will have a quark ( @xmath85 ) and a squark ( @xmath86 ) inside the loop . there are two such sets : @xmath87 and @xmath103 . we obtain @xmath104 \nonumber \\ & \simeq & \frac{3\lprod}{16\pi^2 } \left(\frac{1}{9 \tilde{m}^2 } \right ) \left[5 + 4 \ln\left(\frac{m_c}{\tilde{m } } \right ) + 2 \ln\left(\frac{m_b}{\tilde{m } } \right ) \right ] \ , .\end{aligned}\ ] ] the magnetic form - factor is given by @xmath105 ~\simeq ~ 3\frac{\lprod}{32\pi^2{\tilde m}^2 } \left(\frac{1}{6}\right ) \ , .\end{aligned}\ ] ] the @xmath97-mediated penguin amplitude for the process @xmath106 is given by @xmath107u_\tau(p ) \frac{1}{m_z^2 } \bar u_\mu(p_2)\left[\gamma^\mu ( a_l^\ell p_l+a_r^\ell p_r)v_\mu(p_3)\right ] - ( p_1\leftrightarrow p_2 ) \ , .\ ] ] the @xmath97 boson couplings with the left- and right - chiral fermions are given by @xmath108 the @xmath102-induced contribution to the form - factor @xmath109 proceeds through two sets of penguins : @xmath87 and @xmath103 , yielding @xmath110 \simeq \frac{g}{\cos\theta_w } \left(\frac{3\lambda'^*_{223}\lambda'_{323}}{32\pi^2}\right ) \left[\frac{m_b^2}{\tilde{m}^2 } \left(1 + 2\ln\frac{m_b}{\tilde{m}}\right ) \right ] \ , .\end{aligned}\ ] ] the @xmath83 and @xmath111 couplings also induce a box graph for @xmath112 with internal quark and squark lines . again , two sets of box diagrams contribute @xmath113 and @xmath88 . the amplitude is given by @xmath114 \left[\bar u_\mu(p_2)\gamma_\mu p_lv_\mu(p_3)\right]-(p_1\leftrightarrow p_2 ) \ , .\ ] ] for the sake of convenience , we normalize @xmath115 with a prefactor @xmath116 , though no gauge interaction is actually involved : @xmath117 \ , , ~{\rm where}~ f(r)= \frac{1-r^2 + 2r\ln(r)}{(1-r)^3 } \ , .\ ] ] the total decay amplitude of this process is the sum of the penguin and box contributions , given by @xmath118 . the branching ratio of @xmath95 is given in terms of the different form - factors @xcite : @xmath119 \ , , \end{aligned}\ ] ] where @xmath120 is the total decay width of @xmath6 . our form - factors ( @xmath121 ) are all real . the expressions of @xmath122 and @xmath123 are given by , @xmath124 we have shown in fig . [ taumug ] how @xmath33 together with @xmath34 drive the magnetic transition @xmath49 . the amplitude for this transition is given by @xmath126 where @xmath127 is the photon polarization . the expression for @xmath101 can be found in eq . ( [ ar ] ) . in the amplitude we have neglected a similar term proportional to @xmath128 . the branching ratio for this radiative decay mode is given by ( neglecting any @xmath128-dependent term ) @xmath129 the semileptonic decay @xmath131 with @xmath132 decay is mediated by a @xmath97-penguin and a box graph , as shown in figs . ( [ taumup]a ) and ( [ taumup]b ) , respectively . photon penguin can not contribute as it can not provide the axial current for the quarks to condense to a meson . the @xmath97-boson mediated penguin amplitude for @xmath133 is given by @xmath134 u_\tau \frac{1}{m_z^2}\bar{u}_q\left[\gamma^\mu ( a^q_l p_l+a^q_r p_r)\right]v_q \ , , \ ] ] where @xmath135 and @xmath136 are given in eq . ( [ zcoup ] ) . the relevant @xmath85 for the formation of @xmath137 and @xmath138 are @xmath139 and @xmath140 . the form factor @xmath109 is already given in eq . ( [ fl ] ) . the couplings @xmath33 and @xmath34 also induce @xmath133 through box graphs . because of the specific @xmath27-indices , @xmath85 can only be @xmath140 . the box graph contains two fermion lines and two scalar lines . there are two types of box diagrams : ( i ) the fermions are the same ( @xmath141 quark ) , but the scalars are different ( @xmath142 and @xmath143 ) ; ( ii ) the scalars are same ( @xmath67 ) , but the fermions are different ( @xmath144 and @xmath145 ) . the sum of box amplitudes is given by , @xmath146 \left[\bar{u}_s\gamma_\mu p_l v_s\right ] \ , .\ ] ] the form - factor @xmath147 is given by @xmath148 \simeq { 3 \over { 32 \pi^2 \tilde{m}^2 } } \sum_{i=2,3 } \lambda^{\prime}_{323}\lambda^{\prime * } _ { 223 } \left|\lambda^{\prime}_{i23}\right|^2 \ , , \ ] ] where @xmath149 has already been expressed in eq . ( [ bl ] ) , while @xmath150 is given by @xmath151 using eqs . ( [ zpenguin - mup]-[fprime ] ) we obtain the branching ratio , @xmath152 the decay constants involving @xmath137 and @xmath138 are given by @xmath153 the numerical values of the involved parameters are given by @xcite ; @xmath154 in table 1 we have displayed the present experimental status of different branching ratios of our concern . . [ cols="^,^",options="header " , ] we divide this section in three parts : ( i ) we briefly mention about the existing studies on @xmath0-parity conserving supersymmetric contribution to lfv @xmath6 decays , ( ii ) remark on the previous works on @xmath0-parity violating contributions to lepton flavor violation , and finally ( iii ) highlight the new things that we have done in this work . ( i ) lfv decays have been analyzed in supersymmetric scenarios with conserved @xmath0-parity but with different sets of supersymmetry breaking parameters . in a class of scenarios where minimal supersymmetry is augmented by three right - handed neutrino superfields for generating neutrino masses _ via _ see - saw mechanism , it has been shown @xcite that large neutrino yukawa couplings induce large flavor violation in the slepton sector which is ultimately transmitted to the lfv observables . the general conclusion is that light supersymmetry ( @xmath155 gev ) is disfavored . large lfv branching ratios ( with large @xmath156 ) can be obtained when light neutrino masses are hierarchical . in general , @xmath157 is the most sensitive lfv channel , but to explore the higgs sector @xmath50 and @xmath158 channels are more effective . it has been shown that in a general _ unconstrained _ minimal supersymmetric framework @xcite , for low @xmath159 , the branching ratio in the @xmath160 channel is @xmath161 and in the @xmath162 channel less than @xmath163 . on the other hand , for large @xmath156 and for small pseudo - scalar mass ( @xmath164 ) , the higgs mediated contributions are extremely dominant . in the latter case , indeed with strong fine - tuning of parameters , @xmath165 is enhanced to @xmath166 and @xmath167 to even larger values . in supersymmetric models embedded in minimal so(10 ) group @xcite , the lfv branching ratios are , however , several orders of magnitude below the present experimental sensitivities . ( ii ) rpv induced lfv processes have been studied in the past in different contexts @xcite . except @xmath168 , all other lfv processes considered there proceed at _ tree level _ with appropriately chosen rpv couplings . the choices of such couplings are , in general , different in different processes . their primary intentions were to put upper limits on different single and product couplings by confronting lfv observables with experimental results . ( iii ) _ what are the new things that we have done in this paper _ ? we made an economical choice of rpv couplings ( @xmath33 and @xmath34 only ) , motivated _ primarily _ by their ability to explain the large @xmath3 ( @xmath169 ) branching ratios . we set these two couplings equal , a choice inspired by maximal @xmath18-@xmath19 mixing . we have kept the sparticle mass fixed at 300 gev . explanation of @xmath170 branching ratios require @xmath171 at 90% c.l . on the other hand , @xmath172 with an upper limit of @xmath173 on its branching ratio at 90% c.l . offers the most sensitive lfv probe of the rpv dynamics , and sets _ an improved upper limit _ @xmath174 at 90% c.l . enhanced theoretical and experimental accuracies in the @xmath3 channels might eventually release the tension between the apparently conflicting requirements . putting @xmath175 , we obtain @xmath176 , and @xmath177 . the correlation plots capture the underlying dynamics . to sum up , instead of considering just one experimental observation at a time , be it an anomaly or an excess _ vis - - vis _ the sm expectation , providing a _ raison dtre _ for one set of new interactions , we have studied the possibility of correlated enhancements in a variety of lfv channels using just two rpv couplings . we demonstrated our results through ` observable versus observable ' plots . * acknowledgments : * gb thanks the cern theory division for hospitality and acknowledges a partial support through the project no . 2007/37/9/brns of brns ( dae ) , india . sn s work is supported by a european community s marie - curie research training network under contract mrtn - ct-2006 - 035505 ` tools and precision calculations for physics discoveries at colliders ' . 99 g. r. farrar and p. fayet , phys . b * 76 * ( 1978 ) 575 ; 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with just two @xmath0-parity violating couplings , @xmath1 and @xmath2 , we correlate several channels , namely , @xmath3 ( @xmath4 ) , @xmath5 , and some lepton flavor violating @xmath6 decays . for @xmath7 and for a common superpartner mass of 300 gev , which explain the recently observed excesses in the above @xmath8 decay channels , we predict the following @xmath0-parity violating contributions : @xmath9 , @xmath10 , @xmath11 , and @xmath12 . we exhibit our results through observable versus observable correlation plots . ` pacs nos : 12.60.jv , 13.35.dx ` + ` key words : r - parity violation , lepton flavor violation ` sinp / tnp/2009/25 * correlated enhancements in @xmath13 , @xmath14 of muon , and lepton flavor violating @xmath6 decays with two @xmath0-parity violating couplings * + gautam bhattacharyya @xmath15 , kalyan brata chatterjee @xmath16 , and soumitra nandi @xmath17 + +

the idea suggested by skyrme @xcite that baryon physics could emerge as solitons from an effective lagrangian of meson fields remains one of the most original and successful attempts for the description of the low - energy regime of the theory of strong interactions ( qcd ) . although it predates qcd and was almost eclipsed by it , the proposal gained strong support when it was realized that , in the large @xmath4 limit , qcd is equivalent to an effective theory of mesons @xcite . perhaps the most important feature of the skyrme model in that regard is that the soliton solutions which arise are characterized by a conserved topological charge , the winding number , which skyrme identified as the baryon number . in other words , in this scheme , the baryons as well as nuclei are simply topological solitons . in its original formulation , the skyrme model succeeds in predicting the properties of the nucleon within a precision of 30% . this is considered a rather good agreement for a two - parameter theory @xcite . however , a number of generalizations of the model have been proposed to improve this concordance with baryon and nuclear physics . they mostly exploit our ignorance of the exact form of the low - energy effective lagrangian of qcd for example , the structure of the mass term @xcite , the contribution of other vector mesons @xcite or simply the addition of higher - order terms in derivatives of the pion fields @xcite . unfortunately , for now , qcd alone only gives hints that such extensions should appear and the complete determination of the effective skyrme - like lagrangian remains a most serious challenge . despite such efforts , one of the recurring problems of skyrme - like lagrangians is that they almost inevitably lead to large binding energy for nuclei already at the classical level . a solution may be at hand by constructing effective lagrangians with soliton solutions that saturate the bogomolnyi bound , i.e. so - called bogomolnyi - prasad - sommerfield type ( bps ) skyrmions , since their classical static energy grows linearly with the baryon number @xmath5 ( or atomic number ) much like the nuclear mass . support for this idea comes from a recent result from sutcliffe @xcite who found that bps - type skyrmions seem to emerge for the original skyrme model when a large number of vector mesons are added . the additional degrees of freedom cause the mass of the soliton to decrease down to the saturation of the bogomolnyi bound . a different and more direct approach was proposed by adam , sanchez - guillen , and wereszczynski ( asw ) @xcite by means of a prototype model consisting of only two terms : one of order six in derivatives of the pion fields @xcite and a second term , called the potential , which is chosen to be the customary mass term for pions in the skyrme model @xcite . the model leads to bps - type compacton solutions with size and mass growing as @xmath6 and @xmath5 respectively , a result in general agreement with experimental observations . however , the connection between the asw model and pion physics , or the skyrme model , is more obscure due to the absence of the nonlinear @xmath1 and so - called skyrme terms which are of order 2 and 4 in derivatives , respectively . following this picture , some of us @xcite have reexamined a more realistic generalization of the skyrme model which includes terms up to order six in derivatives @xcite in the sector where the nonlinear @xmath1 and skyrme terms are small . in that limit and for an appropriate choice of mass term , it is possible to find well - behaved analytical solutions for the static solitons . since they saturate the bogomolnyi bound , their static energy is directly proportional to @xmath5 and we recover some of the results in ref . in fact , these solutions allow computing analytically the mass of the nuclei ( static and rotational energy ) in the regime where quadratic and quartic terms are small perturbations . adjusting the four parameters of the model to fit the resulting binding energies per nucleon with respect to the experimental data of the most abundant isotopes leads to an impressive agreement . these results support the idea of a bps - type skyrme model as the dominant contribution to an effective theory for the properties of nuclear matter . however , a few issues remain to be addressed before such a model is considered viable . we shall concentrate on two of them in this work . first , as for most extensions of the skyrme model , the bps - type models in refs . @xcite and @xcite generate shell - like configurations for the energy and baryon densities as opposed to what experimental data suggests , i.e. almost constant densities in the nuclei . we show here that it is possible to construct an effective lagrangian which leads to nonshell configurations and still preserves the agreement with nuclear data . the second issue concerns the inclusion of the coulomb energy and the isospin symmetry breaking term in the calculation of nuclear masses . in the context of the skyrme model , these contributions have been thoroughly studied for @xmath7 @xcite but are usually neglected , to a first approximation , for higher @xmath5 since they are not expected to overcome the binding energies which are usually large and also because finding the configurations is already numerically challenging so that only small @xmath5 solutions are known ( e.g. approximate toroidal , tetrahedral , cubic configurations for @xmath8 standard skyrmions , respectively ) . however , for our type of near - bps model , they may have a significant impact on the predictions given the already good agreement with data . moreover , it turns out that the calculation of the coulomb energy is simplified by the axial symmetry of the solutions and is calculable for all @xmath5 . we propose to study the model based on the lagrangian density @xmath9 with @xmath10 \label{l2}\\ \mathcal{l}_{4 } & = \beta\text{tr}\left ( \left [ l_{\mu},l_{\nu}\right ] ^{2}\right ) \label{l4}\\ \mathcal{l}_{6 } & = -\frac{3}{2}\frac{\lambda^{2}}{16^{2}}\text{tr}\left ( \left [ l_{\mu},l_{\nu}\right ] \left [ l^{\nu},l^{\lambda}\right ] \left [ l_{\lambda},l_{\mu}\right ] \right ) \label{l6}\ ] ] where @xmath11 is the left - handed current of the meson fields represented by the @xmath12 matrix @xmath13 which obey the nonlinear condition @xmath14 the constants @xmath15 @xmath16 , @xmath17 and @xmath18 are left as free parameters of the model although we shall be interested in the regime where @xmath16 and @xmath19 are small . the original skyrme model was built out of only the nonlinear @xmath1 term , @xmath20 and the skyrme term , @xmath21 . one often adds the so - called mass term , @xmath22 to take into account chiral symmetry breaking and generate a pion mass term for small fluctuations of the chiral field in @xmath23 . finally , the term of order six in derivatives of the pion fields @xmath24 is equivalent to @xmath25 that was first proposed by jackson et al . @xcite to allow for the possibility of @xmath26-meson interactions . here , we define the topological ( baryon ) current density @xmath27 @xmath28 the boundary condition at infinity must be constant to ensure that solutions for the skyrme field have finite energy but it also characterizes solutions by a conserved topological charge , @xmath29 the static energy arising from @xmath24 comes from the square of the baryon density@xmath30 so in a sense , it is the analog of the coulomb energy @xmath31 except that instead of following the @xmath32 law , the interaction is replaced by a @xmath33function@xmath34 in other words , the baryonic charge interacts locally . historically , @xmath35 and @xmath24 were introduced to provide a more general effective lagrangian than the original skyrme model and indeed , the lagrangian in ( [ model0to6 ] ) represents the most general @xmath12 model with at most two time derivatives . as an effective theory based on the @xmath36 expansion of qcd , there no reason to believe that higher - order derivatives should be absent however , since one generally relies on the standard hamiltonian interpretation for the quantization procedure , higher - order time derivatives are usually avoided . as a result , the model has been studied extensively but remarkably , this was done only for values of parameters @xmath15 @xmath16 , @xmath17 and @xmath18 close to that of the original skyrme model . presumably these choices were made so that @xmath37 and @xmath21 would continue to have a significant contribution to the mass of the baryons and thereby preserve the relative successes of the skyrme model in predicting nucleon properties and their link to pion physics ( @xmath16 is proportional to the pion decay constant @xmath38 ) . yet this sector of the theory fails to provide an accurate description of the binding energy of heavy nuclei . noting that this caveat may come from the fact that the solitons of the skyrme model do not saturate the bogomolnyi bound , asw proposed a model @xcite ( equivalent to setting @xmath39 whose solutions are bps - type solitons and have lower binding energies . a more realistic approach was proposed in ref . @xcite to analyze the full lagrangian ( [ model0to6 ] ) in the sector where @xmath16 and @xmath19 are relatively small treating these two terms as perturbations . however , in spite of a very good agreement with experimental nuclear masses , there remains an obstacle to the acceptance of such model . nuclear matter is believed to be uniformly distributed inside a nucleus whereas the solutions of the aforementioned models display shell - like configuration for the baryon and energy densities . part of this work is to demonstrate that it is possible to construct an effective lagrangian which leads to nonshell configuration and still preserves and even improves the agreement with nuclear mass data . we may write the general static solution as @xmath40 where @xmath41 is the unit vector @xmath42 let us consider the model in ( [ model0to6 ] ) in the limit where @xmath16 and @xmath19 are small . for that purpose , we introduce the axial solutions for the @xmath43 case@xmath44 @xmath45 where @xmath46 is an integer . the static energy arising from [ model0to6 ] becomes @xmath47 \nonumber\\ & + \left . 16\beta\frac{\sin^{2}f}{r^{2}}\left [ \left ( n^{2}+1\right ) f^{\prime2}+n^{2}\frac{\sin^{2}f}{r^{2}}\right ] \right ) \label{estat}\ ] ] here @xmath48 and the topological charge simplifies to @xmath49 minimizing of the static energy for @xmath43 leads to the differential equation for @xmath50 @xmath51 the change of variable @xmath52 allows this last expression to be written in a simple form @xmath53 -\frac{\partial v}{\partial f}=0\ ] ] that can be integrated @xmath54 regrouping the terms , we get @xmath55 where @xmath56 is an integration constant . finally , the expression for @xmath57 can be found analytically provided the integral on the left - hand side is an invertible function of @xmath58 . the potential ( or so - called mass term ) @xmath2 in ( [ gdeu ] ) is a key ingredient in the determination of the solution here . unfortunately , its exact form is unknown and indeed , has been the object of several discussions @xcite . for simplicity , it is often assumed to be@xmath59 = 1-\cos f.\ ] ] this form was considered in asw for @xmath43 in the context of bps - skyrmions and solving ( [ gdeu ] ) for @xmath58 led to a bps - compacton @xmath60 for @xmath61 , $ ] where @xmath62 is a constant depending on the parameters @xmath18 , @xmath15 and @xmath46 . note that @xmath63 diverges as @xmath64 and vanishes at @xmath65 . since this solution saturates the bogomolnyi bound , the static energy is proportional to the baryon number @xmath66 . a more general choice was introduced in ref . @xcite : @xmath67 \label{ck}\ ] ] this form allows one to recover the chiral symmetry breaking pion mass term @xmath68 in the limit of small pion field fluctuations @xmath69 and to find a relation between the pion mass @xmath70 and the parameter @xmath15 @xmath71 the case considered in ref . @xcite is a particular example of such potential with@xmath72 and @xmath73 assuming the axial solution ( [ axialsolution ] ) , the potential simplifies to @xmath74 and upon integration ( [ gdeu ] ) , we get the solution @xmath75 with @xmath62 . here , we use the absolute value in order to eliminate of the sign ambiguity of the arccos function . in order to set the baryon number to @xmath76 and the integration constant @xmath77 we fix the boundary conditions @xmath78 and @xmath79 for positive and negative baryon number respectively . note that the exponential fall off of @xmath58 at large @xmath80 helps prevent some quantities such as the moments of inertia from becoming infinite . unfortunately , the bps - type models in refs . @xcite and @xcite both lead to shell configurations for the baryon and energy densities which disagrees with experimental results . this is often the case for skyrme models and it is clear from expressions ( [ estat ] ) and ( [ bint ] ) that this behavior can be traced back to the form of the profile @xmath81 or more precisely to the derivative @xmath82 which tends to zero near @xmath65 for such models . let us consider the more appropriate solution of the form @xmath83 with @xmath84 and similar boundary conditions @xmath78 and @xmath79 here , since @xmath85 @xmath86 neither the baryon density @xmath87 nor the static energy density vanishes near @xmath65 . we find by inspection of ( [ gdeu ] ) , that this solution emerges from a potential similar to ( [ vr3 ] ) , namely @xmath88 the logarithmic dependence in the denominator of this expression could be problematic since @xmath89 at @xmath65 and @xmath90 at @xmath91 but the limits for @xmath2 are well defined and finite , i.e. @xmath92 respectively . it is interesting to note that according to ( [ equipartition ] ) , the square root of the potential@xmath93 corresponds to the baryon radial density ( the term in parenthesis ) up to a multiplicative constant . thus , in order to obtain a nonshell baryon density , it suffices to construct a potential @xmath2 that does not vanish at small @xmath94 such a potential would also imply that @xmath95 our choice of potential clearly verifies this requirement but this relation also explains why the earlier bps - type models could not generate a nonshell configuration , namely @xmath96 and @xmath97 in that limit . the expression ( [ vr2 ] ) only applies to the axial solution ( [ axialsolution ] ) and we need to write a more general form for @xmath2 in terms of @xmath98 if this is to be used in the expression for the lagrangian . a simple but not unique approach to construct the potential is to identify @xmath99 to the expression @xmath100 where @xmath101 is the identity matrix . then , a convenient expression for @xmath2 is given by @xmath102 \nonumber\ ] ] comparing this expression to ( [ ck ] ) allows retrieving each coefficient@xmath103 such that @xmath104 inserting expression ( [ bhmsoln ] ) in ( [ estat ] ) , we get the static energy of the soliton in the small @xmath16 and @xmath19 approximation@xmath105 with@xmath106 where @xmath62 sets the scale of the solution and @xmath107 is the riemann @xmath108 function . the terms @xmath2 and @xmath109 are proportional to the baryon number @xmath66 as one expects from solutions that saturate the bogomolnyi bound whereas the small perturbations @xmath110 and @xmath111 have a more complex dependence . part of this behavior , the overall factor @xmath112 is due to the scaling . the additional factor of @xmath113 comes from the axial symmetry of the solution ( [ axialsolution ] ) . note that it is also easy to calculate analytically the root mean square radius of the baryon density @xmath114 which is consistent with experimental observation for the charge distribution of nuclei @xmath115 . in order to represent physical nuclei , we have taken into account their rotational and isorotational degrees of freedom and quantize the solitons . the standard procedure is to use the semiclassical quantization which is described in the next section . skyrmions are not pointlike particles . so we resort to a semiclassical quantization method which consists in adding an explicit time dependence to the zero modes of the skyrmions and applying a time - dependent ( iso)rotations on the skyrme fields by @xmath12 matrix @xmath116 and @xmath117 @xmath118 where @xmath119 is the associated @xmath120 rotation matrix . the approach assumes that the skyrmion behave as a rigid rotator . upon insertion of this ansatz in the time - dependent part of the full lagrangian ( [ model0to6 ] ) , we can write the ( iso)rotational lagrangian as @xmath121 where @xmath122tr@xmath123 and @xmath124tr@xmath125 the moment of inertia tensors @xmath126 is given by@xmath127 \left [ l_{p},t_{j}\right ] \right ) \nonumber\\ & + \left . \frac{9\lambda^{2}}{16^{2}}\text{tr}\left ( \left [ t_{i},l_{p}\right ] \left [ l_{p},l_{q}\right ] \left [ l_{q},t_{j}\right ] \right ) \right ] \label{minertia}\ ] ] where @xmath128 $ ] . the expressions for @xmath129 and @xmath130 are similar except that the isorotational operator @xmath131 is replaced by a rotational analog @xmath132 as follows : @xmath133 following the calculations in @xcite for axial solution of the form ( [ axialsolution ] ) , we find that all off - diagonal elements of the inertia tensors vanish . furthermore , one can show that @xmath134 and @xmath135 can be obtained by setting @xmath136 in the expression for @xmath137 . similar identities hold for @xmath130 and @xmath129 tensors . the axial symmetry of the solution imposes the constraint @xmath138 which is simply the statement that a spatial rotation by an angle @xmath139 about the axis of symmetry can be compensated by an isorotation of @xmath140 about the @xmath141 axis . it follows from expressions ( [ minertia])-([vij ] ) that @xmath142 for @xmath143 and @xmath144 . the general form of the rotational hamiltonian is given by @xcite @xmath145 \label{hrot}\ ] ] where ( @xmath146 ) @xmath147 the body - fixed ( iso)rotation momentum canonically conjugate to @xmath148 ) @xmath149 . the expression for the rotational energy of the nucleon @xmath7 simplifies due to the spherical symmetry @xmath150 it is also easy to calculate the rotational energies for nuclei with winding number @xmath143@xmath151\ ] ] with@xmath152 these momenta are related to the usual space - fixed isospin ( @xmath153 ) and spin ( @xmath154 ) by the orthogonal transformations @xmath155@xmath156 according to ( [ eq : i ] ) and ( [ eq : j ] ) , we see that the casimir invariants satisfy @xmath157 and @xmath158 so the rotational hamiltonian is given by @xmath159 . \label{erot}\ ] ] we are looking for the lowest eigenvalue of @xmath160 which depends on the dimension of the spin and isospin representation of the eigenstate @xmath161 . for @xmath162 we can show that @xmath163 is negative and we shall assume that this remains true for small values of @xmath16 and @xmath19 . then , for a given spin @xmath164 and isospin @xmath165 , @xmath166 must take the largest possible eigenvalue @xmath167 since @xmath157 and @xmath168 the state with highest weight is characterized by @xmath169 and @xmath170 and since nuclei are build out of @xmath5 fermions we must have an isospin @xmath171 on the other hand , the axial symmetry of the static solutions implies that @xmath172 where @xmath173 but for even @xmath5 nuclei , @xmath174 must be an integer and @xmath175 so @xmath176 = 0\ ] ] similarly for half - integer spin nuclei , @xmath177 must be a half - integer so the only possible value is @xmath178 summarizing , if we assume for simplicity that the @xmath16 and @xmath19 terms only generate small perturbations , the largest possible eigenvalue @xmath174 is @xmath179{l}$0\qquad$for $ a=$ even\\ $ \frac{1}{2}\qquad$for $ a=$ odd \end{tabular } \ \ \ \ \ \ \ \ \ \ \right . . \label{kappa}\ ] ] the lowest eigenvalue of the rotational hamiltonian @xmath160 for a nucleus is then given by @xcite @xmath180 \label{erotijk}\ ] ] the spin of the most abundant isotopes is fairly well known . the isospins are not so well known so we resort to the usual assumption that the most abundant isotopes correspond to states with lowest isorotational energy . since @xmath181 , the lowest value that @xmath165 can take is simply @xmath182 where @xmath183 for example , the deuteron corresponds to @xmath184 @xmath185 and @xmath186 so the rotational energy reduces to @xmath187 the explicit calculations of the rotational energy of nuclei then require only three moments of inertia which can be found analytically : @xmath188 \nonumber\end{aligned}\ ] ] @xmath189 \nonumber\end{aligned}\ ] ] and @xmath190 . so far , both contributions to the mass of the nucleus , @xmath191 and @xmath192 are charge invariant . since this is a symmetry of the strong interaction , it is reflected in the construction of the lagrangian ( [ model0to6 ] ) and one expects that the two terms form the dominant portion of the mass . however , isotope masses differ by a few percent so this symmetry is broken for physical nuclei . in the next section , we consider two additional contributions to the mass , the coulomb energy associated with the charge distribution inside the skyrmion and an isospin breaking term that may be attributed to the up and down quark mass difference . even if we thought of a nucleus as a simple collection of individual protons and neutrons , there would be a repulsive electromagnetic force between protons and the process would require energy to bring these charges together . the result is an increase in the mass of the object by an amount corresponding to the coulomb energy . such an effect is of course also present in the skyrmions description of nuclei since the static configuration has non - vanishing charge density . the electromagnetic and isospin breaking contributions to the mass have been thoroughly studied for @xmath7 , mostly in the context of the computation of the proton - neutron mass difference @xcite , but are usually neglected , to a first approximation , for higher @xmath5 since they are not expected to overcome the large binding energies predicted by the model . there are also practical reasons why they are seldom taken into account . the higher baryon number configurations of the original skyrme model are nontrivial ( toroidal shape for @xmath193 , tetrahedral for @xmath194 , etc . ) and finding them exactly either requires heavy numerical calculations ( see for example @xcite ) or some kind of clever approximation like rational maps @xcite . moreover , the computation of the coulomb energy is more challenging in general since it involves two integrations over volume . one can also argue that the coulomb energy of skyrmions is somewhat reduced by shell - like configurations of the charge densities as opposed to what it would be for a nearly constant spherical density found in electron scattering experiments . in our case however , we are interested in a more precise calculation of the nuclei masses and an estimate of the coulomb energy is desirable , and even more so in our model which generates nonshell configurations . it turns out that the analytical form of the chiral angle @xmath81 in ( [ bhmsoln ] ) and the axial symmetry of the solution simplify the computation of the coulomb energy . let us first consider the charge density inside skyrmions . following adkins et al . @xcite , we write the electromagnetic current @xmath195 with @xmath196 the baryon density and @xmath197 the vector current density , so the conserved electric charge is given by @xmath198 with @xmath199 the eigenvalue of third component of isospin in the body - fixed frame . the vector current is then defined as the sum of the left and right handed currents @xmath200 which are invariant under @xmath201 transformations of the form @xmath202 more explicitly , we get@xmath203 where @xmath204 and @xmath205 are the moment of inertia densities in ( [ minertia])-([vij ] ) . in the quantized version , @xmath206and @xmath207 are expressed in terms of the conjugate operators @xmath146 and @xmath208 here we only need the relation @xmath209 since the off - diagonal elements of @xmath126 and @xmath129 vanish when the solution is axially symmetric and also @xmath144 , we have @xmath210 inserting @xmath211 in ( [ j3v ] ) , the isovector electric current density reduces to @xmath212 where @xmath213 may be interpreted here as a normalized moment of inertia density for the third component of isospin . finally , the electric charge density is given by @xmath214 where we have replaced @xmath174 by @xmath215 using the fact that the charge density of body - fixed and space - fixed frame only differs by a rotation . the coulomb energy stored in a charge distribution @xmath216 takes the usual form ( [ ecoulomb ] ) . in practice , unless one considers very simple configurations , it is not possible to find an analytical expression for the coulomb energy . nonetheless , it is often helpful to expand @xmath216 in terms of normalized spherical harmonics to take care of the angular integrations @xmath217 following the approach described in @xcite , we define the quantities @xmath218 which , at large distance , are equivalent to a multipole moments of the distribution . then , each moment contributes to the coulomb energy by an amount @xmath219 and the total coulomb energy associated to the distribution is given by@xmath220 in our case , the angular dependence of the charge density is rather simple . the first part is a spherically symmetric contribution@xmath221 whereas the only non - trivial piece comes from the third moment density @xmath222 and is proportional to @xmath223 @xmath224 the summation ( [ ylm ] ) consists of only two terms @xmath225 the expressions for moments @xmath226 and @xmath227 are found by integrating ( [ qlm ] ) analytically . finally , we obtain the coulomb energy by computing numerically the last remaining integral @xmath228 the coulomb energy alone can not explain the isotope mass difference . this is particularly evident for @xmath7 where the proton mass is known to be smaller than that of the neutron although the coulomb energy alone would suggest otherwise . on the other hand , isospin is not an exact symmetry , a fact that may be traced back to the up and down quark mass difference . several attempts have been made to modelize the isospin symmetry breaking term within the skyrme model @xcite . here we shall assume for simplicity that this results in a contribution proportional to the third component of isospin@xmath229 with the parameter @xmath230 fixed by setting the neutron - proton mass difference to its experimental value . since both of them have the same static and rotational energies,@xmath231 and@xmath232 summarizing , the mass of a nucleus reads@xmath233 where we have written the explicit dependence of each piece in terms of the relevant nuclear quantum numbers of the nuclei . the prediction depends on the parameters of the model @xmath15 @xmath234 and @xmath235 the values of the parameters @xmath15 @xmath236 and @xmath18 remain to be fixed . let us first consider the case where @xmath237 this should provide us with a good estimate for the values of @xmath238 and @xmath18 required in the 4-parameter model ( [ model0to6 ] ) and , after all , it corresponds to the limit where the minimization of the static energy leads to the exact analytical solution ( [ bhmsoln ] ) . we need two input parameters to set @xmath239 and @xmath235 for simplicity , we choose the mass of the nucleon and that a nucleus @xmath240 with zero ( iso)rotational energy ( i.e. a nucleus with zero spin and isospin ) and neglect for now the coulomb and isospin breaking energies . the total energy of these two states is according to ( [ estatn ] ) and ( [ erotijk ] ) @xmath241 solving for @xmath18 and @xmath239 we get @xmath242@xmath243 as an example , let us examine the case where the nucleus @xmath244 is helium-4 , the first doubly magic number nucleus with zero spin and isospin . setting the mass of the nucleon as the average mass of the proton and neutron i.e. @xmath245 mev and that of helium-4 nucleus to @xmath246 mev , we get the numerical value @xmath247 mev@xmath248 , @xmath43 and @xmath249 mev@xmath250 which we shall refer as set i. the masses of the nuclei including static , ( iso)rotational , coulomb , and isospin breaking contributions are then computed using ( [ etot ] ) . table i shows the relative deviation of the predicted with regard to experimental values of nuclear masses of a few isotopes ( set i ) . the predictions are accurate to @xmath251 or better even for heavier nuclei . part of this accuracy is probably due to the fact that the static energy of a bps - type solution is proportional to @xmath5 so if it dominates , the nuclear masses should follow approximately the same pattern . however , the predictions remain surprisingly good for a 2-parameter model@xmath252 perhaps more relevant are the predictions of the binding energy per nucleon ( @xmath3 ) . the results are presented in fig . [ figbovera ] set i ( solid line ) and can be compared to the experimental values ( black circles ) . we consider here only a subset of the table of nuclei in @xcite composed of the most abundant 144 isotopes . we observe a sharp rise of the binding energy per nucleon at small @xmath5 followed by a slow linear increase for larger nuclei . the overall accuracy is of the order of @xmath253 which is rather good considering the fact that the calculation involves the mass difference between the nucleus and its constituents . experimentally the charge radius of the nucleus is known to behave approximately as @xmath254 with @xmath255 fm . on the other hand , it is possible to calculate the root mean square radius for the baryon density [ see eq ( [ rrms ] ) ] which leads to@xmath256 for the charge radius @xmath257 , the dependence on @xmath5 is more complex since it involves an additional isovector contribution ( [ rhocharge])@xmath258 where @xmath259 is the charge of the nucleus . we get the expression@xmath260\end{aligned}\ ] ] where @xmath135 also depends on @xmath5 and is obtained by substituting @xmath261 in ( [ u112 ] ) . our computation verifies that the charge radius obeys roughly the proportionality relation @xmath262 but overestimates the experimental value of @xmath263 by approximately a factor of 2 . let us now release the constraints on @xmath16 and @xmath19 and allow for small perturbations from the nonlinear @xmath1 and skyrme term . in order to estimate the magnitude of the parameters @xmath16 and @xmath19 in a real physical case , we perform two fits : set ii optimizes the four parameters @xmath264 , @xmath265 @xmath19 and @xmath18 to better reproduce the masses of the nuclei while set iii tries to reach the best agreement with respect to the binding energy per nucleon , @xmath3 . both fits are performed with data from the same subset of the most abundant 144 isotopes as before . a summary of the results is presented in table i while fig . [ figbovera ] displays the general behavior of @xmath3 as a function of the baryon number for sets i , ii , iii , and experimental values . @xmath266{|c|c|c|c|c|}\hline\hline \multicolumn{5}{|c|}{table i : prediction versus experimental nuclear masses}\\\hline\hline \ & $ \quad$set~i$\quad$ & $ \quad$set~ii$\quad$ & $ \quad$set~iii$\quad$ & experiment\\\hline\hline $ \mu$ $ ( 10^{4}$ mev$^{2})$ & $ 1.490\ 80 $ & $ 1.505\ 71 $ & $ 1.729\ 55 $ & \ \\ $ \alpha$ $ ( 10^{-3}$ mev$^{2})$ & $ 0 $ & $ 5.881\ 18 $ & $ 22.0821 $ & \ \\ $ \beta$ $ ( 10^{-6}$ mev$^{0})$ & $ 0 $ & $ -1.84877 $ & $ -5.80989 $ & \ \\ $ \lambda$ $ ( 10^{-3}$ mev$^{-1})$ & $ 6.413\ 62 $ & $ 6.339\ 73 $ & $ 5.536\ 91 $ & \ \\ $ f_{\pi}$ $ ( $ mev$)$ & $ 0 $ & $ 0.307 $ & $ 0.594 $ & $ 186$\\ $ m_{\pi}$ $ ( $ mev$)$ & --- & $ 208\ 530 $ & $ 82\ 300 $ & $ 138$\\ $ e$ $ ( 10^{4})$ & --- & $ -185\ 000 $ & $ -5380 $ & \ \\ $ r_{0}$ ( fm ) & $ 2.637 $ & $ 2.617 $ & $ 2.385 $ & $ 1.23$\\\hline\hline nucleus x\ & \multicolumn{3}{|c|}{$\frac{e_{x}-e_{\exp}}{e_{\exp}}$ } & $ \ e_{\exp}\text{(mev)}\ $ \\\hline\hline nucleon & input & $ -0.0008 $ & $ 0.0020 $ & $ 938.919$\\ $ ^{2}$h & $ -0.0032 $ & $ -0.0048 $ & $ -0.0020 $ & $ 1875.61$\\ $ ^{3}$h & $ -0.0042 $ & $ -0.0057 $ & $ -0.0030 $ & $ 2808.92$\\ $ ^{4}$he & input & $ -0.0017 $ & $ -0.0009 $ & $ 3727.38$\\ $ ^{6}$li & $ -0.0017 $ & $ -0.0034 $ & $ -0.0010 $ & $ 5601.52$\\ $ ^{7}$li & $ -0.0014 $ & $ -0.0031 $ & $ -0.0008 $ & $ 6533.83$\\ $ ^{9}$be & $ -0.0006 $ & $ -0.0023 $ & $ -0.0001 $ & $ 8392.75$\\ $ ^{10}$b & $ -0.0004 $ & $ -0.0021 $ & $ -0.00001 $ & $ 9324.44$\\ $ ^{16}$o & $ 0.0010 $ & $ -0.0008 $ & $ 0.0009 $ & $ 14\ 895.1$\\ $ ^{20}$ne & $ 0.0010 $ & $ -0.0007 $ & $ 0.0008 $ & $ 18\ 617.7$\\ $ ^{40}$ca & $ 0.0016 $ & $ 0.0001 $ & $ 0.0006 $ & $ 37\ 214.7$\\ $ ^{56}$fe & $ 0.0018 $ & $ 0.0001 $ & $ 0.0004 $ & $ 52\ 089.8$\\ $ ^{238}$u & $ 0.0004 $ & $ 0.00001 $ & $ 0.0006 $ & $ 221\ 696$\\\hline \end{tabular } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ] ] as a function of the baryon number @xmath5 : the experimental data ( black circles ) are shown along with predicted values ( solid lines ) for parametrization of set i ( @xmath43 ) , set ii ( best fit for nuclear masses ) , and set iii ( best fit for @xmath3 ) , respectively.,scaledwidth=65.0% ] we find that the two new sets of parameters are very close to set i. the nonlinear @xmath1 and skyrme parameters @xmath16 and @xmath19 are small in magnitude but in order to make a relevant comparison , it is best to look at the relative importance of the contributions in ( [ model0to6 ] ) and how they scale with respect to the parameters of the model , namely@xmath266{rccccccc } & $ \lambda\mu$ & $ : $ & $ \alpha\nu^{-1/3}$ & $ : $ & $ \beta\nu^{1/3}$ & $ : $ & $ \lambda\mu$\\ $ \text{set~i\qquad}$ & $ 95.61 $ & $ : $ & $ 0 $ & $ : $ & $ 0 $ & $ : $ & $ 95.61$\\ $ \text{set~ii\qquad}$ & $ 95.46 $ & $ : $ & $ 4.408\times10^{-5}$ & $ : $ & $ -2.255\times10^{-4}$ & $ : $ & $ 95.46$\\ $ \text{set~iii\qquad}$ & $ 95.74 $ & $ : $ & $ 1.418\times10^{-4}$ & $ : $ & $ -7.904\times10^{-4}$ & $ : $ & $ 95.74$\end{tabular } \ \ \ \ \ \ ] ] for @xmath267 and @xmath24 respectively . clearly , the nonlinear @xmath1 and skyrme terms are extremely small compared to that of @xmath268and @xmath269 i.e. by at least 6 orders of magnitude . this provides support to the assumption that ( [ bhmsoln ] ) is a good approximation to the exact solution . the overall factor @xmath270 remains approximately the same for all the sets but @xmath3 turns out to be somewhat sensitive to these small variations because it involves a mass difference . even more sensitive to small change in parameters is the charge radius @xmath263 with @xmath271 decrease between set ii and set iii ( table i ) which suggests that the predicted value of @xmath263 should be taken as an estimate rather than a firm prediction . comparing set ii and set iii to the original skyrme model with a pion mass term , we may identify@xmath272 and using ( [ mumpi ] ) we find@xmath273 these quantities , @xmath274 and @xmath70 take values ( see table i ) which are orders of magnitude away for those obtained for the skyrme model but this is expected since we have assumed from the start that @xmath16 and @xmath19 are relatively small . we find also that the skyrme term has the wrong sign so it would destabilize the soliton against shrinking if it was not for the contribution of order six in derivatives which ensures stability against scale transformations . indeed the term of order six was even introduced at one point to resolve some problems with this sign @xcite . in principle a negative coefficient for the skyrme term could become problematic since the energy may no longer be bounded from below . one can argue that for our set of parameters , the relative weight of the @xmath21 piece with respect to that of @xmath35 or @xmath24 is so small , i.e. approximately @xmath275 and is at least partially canceled by that of the nonlinear @xmath1 term @xmath37 so that the energy would remain bounded from below . to substantiate this point on the relative contribution of each term , it is useful at this point to invoke some relevant links noticed by manton @xcite between an effective @xmath12 scalar lagrangian and the strain tensor in the theory of elasticity . as in nonlinear elasticity theory , the energy density of a skyrme field depends on the local stretching associated with the map @xmath276 this is related to the strain tensor at a point in @xmath277 which is defined as @xmath278\end{aligned}\ ] ] where @xmath279 refers to the cartesian space coordinates . @xmath280 is a @xmath281 symmetric matrix with three positive eigenvalues @xmath282 and @xmath283 . three fundamental invariants emerges from @xmath280 in this simple geometrical interpretation due to manton . they correspond to the lagrangians @xmath284 and @xmath24 and lead to the following energy densities , respectively : @xmath285 where we wrote for simplicity @xmath286 and to the baryon density @xmath287 assuming without loss of generality that @xmath288 we find@xmath289 for a total energy density@xmath290 so , negative energy density contributions would come from regions where @xmath291 , in other words , where the baryon density @xmath292 in ( [ bxyz ] ) is very small . even for @xmath293 and @xmath16 negligibly small , the energy density should be dominated by the potential term @xmath294 . if we consider the integrated energy density subject to the condition that the total baryon number @xmath5 is a positive integer , then we expect the energy to be bounded from below for our set of parameters . clearly for our axial solution , the @xmath37 and @xmath21 pieces of the lagrangian do not play the same significant role in the stabilization of the soliton as they do in the case of the skyrme model . the properties of the soliton are almost completely determined by the values of @xmath239 and @xmath18 so @xmath38 and @xmath70 may not be so closely related to the nucleon mass scale as for the original skyrme model . perhaps the explanation for such a departure is that the parameters of the model are merely bare parameters and they could differ significantly from their renormalized physical values . in other words , we may have to consider two quite different sets of parameters : a first one , relevant to the perturbative regime for pion physics where @xmath38 and @xmath70 are close to their experimental value and a second one , that applies to the nonperturbative regime in the case of soliton . unfortunately , one of the successes of the original skyrme model is that it established a link between pion physics with realistic values for @xmath38 and @xmath70 and baryon masses . such a link here is more obscure . on the other hand , the model in ( [ model0to6 ] ) ( in the regime where @xmath16 and @xmath19 are small ) improves the prediction with regard to the properties of the nuclei of nuclear masses . let us look more closely at the results presented in fig . [ figbovera ] . these are in the form of the ratio of the binding energy per nucleon ( @xmath3 ) as a function of the baryon ( or atomic ) number @xmath5 . the experimental data ( black circles ) are shown along with predicted value ( solid lines ) for parametrization of set i , set ii and set iii . set i is the least accurate when it comes to reproducing the experimental data , especially in the heavy nuclei sector . yet , the agreement remains within a @xmath251 of the experimental masses which is much better than with the original skyrme model . moreover , since the ratio @xmath3 depends on the difference between the mass of a nucleus and that of its constituents , it is sensitive to small variation of the nuclear masses so the results for @xmath3 may be considered as rather good . the second fit ( set ii ) , which is optimized for nuclear masses , overestimates the binding energies of the lightest nuclei while it reproduces almost exactly the remaining experimental values ( @xmath295 . finally , the least square fit based on @xmath3 ( set iii ) is the best fit overall but in order to provide a better representation for light nuclei , it abdicates some of the accuracy found in set ii for @xmath296 . this apparent dichotomy between light and heavy nuclei may be partly attributed to the ( iso)rotational contribution to the mass . the size of nuclei grows as @xmath6 and their moments of inertia increase accordingly . also , the spin of the most abundant isotopes remains small while isospin can have relatively large values due to the growing disequilibrium between the number of proton and the number of neutron in heavy nuclei . our numerical calculations reveal that the total effect leads to a ( iso)rotational energy @xmath297 1 mev for @xmath298 for all sets of parameters considered and its contribution to @xmath3 decreases rapidly as @xmath5 increases . on the contrary for @xmath299 the rotational energy is responsible for a larger part of the binding energy which means that @xmath3 should be sensitive to the way the rotational energy is computed . so clearly , the shape of the baryon density will have some bearing on the predictions for the small @xmath5 sector . as a function of the baryon number @xmath5 for set iii : the experimental data ( black circles ) are shown along with contributions due to @xmath300 @xmath301 and @xmath302 = @xmath303 @xmath304 ( solid lines).,scaledwidth=65.0% ] since part of this work is to propose a model with nonshell configuration , it is relevant to compare our result with a similar analysis @xcite which involves a typical shell - like configuration . for this purpose we repeated our calculation omitting the coulomb and isospin breaking term . it turns out that both models are equally successful at reproducing data . minimizing the square root of the mean squared deviation of @xmath3 from its experimental value gives almost identical results for both models , @xmath305 mev per nucleon , despite generating completely different baryon and energy configurations . in fact , the absence of a variation in @xmath1 signals somehow the equal inability for both models to provide an accurate description of light and heavy nuclei sectors at the same time . one could improve the agreement by fitting separately the parameters @xmath238 and @xmath18 in the two sectors @xmath306 and @xmath307 but this would means introducing an arbitrary baryon number dependence on the parameter which could only be justified by introducing some kind of dynamical effect on @xmath308 and @xmath235 the second motivation for this work regards this addition of the coulomb and isospin breaking effect into the nuclear masses . these are often neglected in the context of the skyrme model although they must inevitably be taken into account for a complete description of the nucleus . the coulomb energy and isospin breaking term represent small corrections to the nuclear mass ( of the order of @xmath309 and @xmath310 respectively ) however our results show that the coulomb effect is much more significant in the calculation of the binding energy . the change in @xmath3 is depicted in fig . [ figcoulomb ] for set iii in the separation between the red and blue lines ( with and without coulomb term , respectively ) . the effect increases almost linearly with the baryon number up to approximately @xmath311 mev per nucleon for the heaviest nuclei . it represents roughly half the coulomb effect estimated in the liquid drop model @xmath312 where the value of @xmath313 mev / nucleon . on the other hand , the isospin breaking contribution due to @xmath314 remains very small . despite the magnitude of these corrections , it turns out that the optimization of the model parameters only yield , a slight improvement of the overall agreement with @xmath315 mev per nucleon . to summarize , we have proposed a 4-terms model as a generalization of the skyrme model . by choosing an appropriate form for the potential @xmath2 , we allowed for near - bps solitons with nonshell configurations for the baryon density in order to achieve a more realistic description of nuclei as opposed to the more complex configurations found in most extensions of the skyrme model ( e.g. @xmath193 toroidal , @xmath194 tetrahedral , @xmath316 cubic , ... ) . moreover , we introduced additional contributions to the mass of the nuclei coming from the coulomb energy and an isospin breaking term . fitting the model parameters , we find a remarkable agreement for the binding energy per nucleon @xmath3 with respect to experimental data . these results suggest that nuclei could be considered as near - bps skyrmions . on the other hand , there remain some caveats . first , the skyrme model provides a simultaneous description for perturbative pion interactions and nonperturbative baryons physics with single realistic values for @xmath38 and @xmath70 and baryon masses . the connection between the two sectors here seems to be much more intricate . also , a much better agreement could be reached if one could construct a solution that would describe equally well the light and heavy nuclei . finally , one would like ultimately to reproduce the observed structure of the nucleus , i.e. a roughly constant baryon density becoming diffuse at the nuclear surface which is characterized by a skin thickness parameter . a more appropriate choice of potential may be instrumental in achieving some of these goals . skyrme , proc . a , 260:127 - 138 , 1961 ; t.h.r . skyrme , proc . lond . a , 247:260278 , 1961 ; t.h.r . skyrme , nucl . 31:556 - 569 , 1962 ; t.h.r . skyrme , proc . a , 247:260278 , 1958 .

the relatively small binding energy in nuclei suggests that they may be well represented by near - bps skyrmions since their mass is roughly proportional to the baryon number @xmath0 for that purpose , we propose a generalization of the skyrme model with terms up to order six in derivatives of the pion fields and treat the nonlinear @xmath1 and skyrme terms as small perturbations . for our special choice of mass term ( or potential ) @xmath2 , we obtain well - behaved analytical bps - type solutions with nonshell configurations for the baryon density , as opposed to the more complex shell - like configurations found in most extensions of the skyrme model . along with static and ( iso)rotational energies , we add to the mass of the nuclei the often neglected coulomb energy and isospin breaking term . fitting the four model parameters , we find a remarkable agreement for the binding energy per nucleon @xmath3 with respect to experimental data . these results support the idea that nuclei could be near - bps skyrmions . nomdufichiernotesarticle = pubnotes2012prd1v3.tex

exotic atoms have a long history , and have stimulated interesting developments in quantum dynamics of systems involving both long - range and short - range forces . for the exploratory studies presented here , the refinements of effective theories @xcite are not required , and we shall restrict ourselves to the schrdinger framework , as reviewed , e.g. , in @xcite for the three - dimensional case , and extended in @xcite for the two - dimensional one . in units simplifying the treatment of the pure coulomb case , an exotic atom can be modeled as @xmath0 where @xmath1 is the inter - particle distance and @xmath2 the short - range correction , with a variable strength for the ease of the discussion . when the above spectral problem is solved , the most striking observations are : 1 . the energy shift @xmath3 , as compared to the pure - coulomb energies @xmath4 , is often rather small , but is usually not given by ordinary perturbation theory : for instance , an infinite hard - core of small radius corresponds to a small energy shift but to an infinite first order correction . 2 . each energy @xmath5 , as a function of the strength parameter @xmath6 , is almost flat in a wide interval of @xmath6 , with a value close to some @xmath7 , one of the pure coulomb energies . then @xmath8 is well approached by a formula by deser et al . , and trueman @xcite , @xmath9 where @xmath10 is the scattering length in the short - range interaction @xmath2 alone . 3 . if @xmath11 is attractive , when the strength @xmath6 approaches one of the positive critical values at which a first or a new bound state appears in the spectrum of @xmath12 alone , the energy @xmath5 quits its plateau and drops dramatically from the region of atomic energies to the one of deep nuclear energy . it is rapidly replaced in the plateau by the next level . this is known as _ level rearrangement _ @xcite . the above patterns are more general , and hold for for any combination of a long - range and a short - range interaction , say @xmath13 . the deser - trueman formula is generalized as @xmath14 where @xmath15 is the normalized wave - function in the external potential @xmath16 . if @xmath16 supports only one bound state , then the rearrangement `` extracts '' states from the continuum , instead of shifting bound radial excitations . an illustration is given in fig . [ fig : rearr2bs ] , with a superposition of two exponential potentials of range parameters @xmath17 and @xmath18 , namely @xmath19 where @xmath20 is tuned to start binding at unit strength , for a reduced mass @xmath21 . is due to the variational approximation . the dashed line indicates the approximation corresponding to the deser - trueman formula . the dotted vertical lines show the coupling thresholds for the short - range part of the interaction . ] there are several possible improvements and alternative formulations of . for instance , @xmath10 , the bare scattering length in @xmath2 can be replaced by the `` long - range corrected '' scattering length where the solutions of the radial equation for @xmath2 are matched to the eigenfunction of the external potential . see , e.g. , @xcite , and refs . there . in the physics of cold atoms , one is more familiar with the approach by busch et al . it deals with the case of an harmonic oscillator modified at short distances , but the derivation can be generalized as follows . let @xmath22 the @xmath23-wave reduced radial wavefunction for @xmath24 that is regular at large @xmath1 , at energy @xmath25 , with some normalization , e.g. , @xmath26 for @xmath27 . the levels in @xmath16 correspond to the quantization condition @xmath28 , where @xmath29 is an eigenenergy of @xmath16 , for instance the ground state . when a point - like interaction of scattering length @xmath10 is added , then the boundary condition is modified into @xmath30 which can be expanded near @xmath29 to give @xmath31 now , the equivalence of and , rewritten as , where @xmath32 is the normalized version of @xmath33 , comes from the relation @xmath34 which is easily derived from the wronskian identity , widely used in some textbooks @xcite , here applied to energies @xmath35 and @xmath29 . the generalization to a number of dimension @xmath36 is straightforward for @xmath37 . for @xmath38 , the first plateau is avoided , as the short - range potential , if attractive , develops its own discrete spectrum for any @xmath39 . the case of @xmath40 is more delicate : see , e.g. , @xcite . our aim here is to present a first investigation of the three - body analog of exotic atoms . we consider three identical bosons , though we have in mind some less simple systems for future work . we address the following questions : is there a pattern similar to the level rearrangement ? is there a generalization of the deser - trueman formula ? what are the similarities with the case where the long - range interaction is replaced by an overall harmonic confinement ? the paper is organized as follows . in sec . [ sec : three - body ] , we give some basic reminders about the spectrum of a three - boson systems from the borromean limit of a single bound state to the regime of stronger binding , with a word about the numerical techniques the results corresponding to a superposition @xmath41 are displayed in sec . [ sec : results ] . an interpretation is attempted in sec . [ sec : interpretation ] , with the a three - body version of the deser - trueman formula . section [ sec : outlook ] is devoted to our conclusions . if two bosons interact through an attractive potential , or a potential with attractive parts , @xmath42 , a minimal strength is required to achieve binding , say @xmath43 . a collection of values of @xmath44 can be found , e.g. , in the classic paper by blatt and jackson @xcite . in the following , we shall normalize @xmath45 so that @xmath46 . if one assumes that @xmath47 is attractive everywhere , once two bosons are bound , the @xmath48-boson system is also bound . ( the case of potentials with a strong inner repulsion would require a more detailed analysis which is beyond the scope of this preliminary investigation . ) this means that for a single monotonic potential @xmath49~,\ ] ] where @xmath50 , the minimal coupling to achieve three - body binding , @xmath51 , is less than 1 , in our units . this is implicit in the seminal paper by thomas @xcite . the inequality @xmath52 is now referred to as `` borromean binding '' , after the study of neutron halos in nuclear physics @xcite . one gets typically @xmath53 , i.e. , about 20% of borromean window @xcite . in short , the three - boson spectrum has the following patterns : * for @xmath54 , no binding * for @xmath55 , a single borromean bound state , and , for @xmath6 close to 1 , a second three - body bound state just below the two - body energy , * for @xmath56 , two bound states below the @xmath57 break - up . * in addtion , very near @xmath58 , the very weakly bound efimov states . the excited state can be considered as the first member of the sequence of the efimov states occurring near @xmath59 . however , it differs from the other efimov states in the sense that when the coupling is varied , it remains below the two - body break - up threshold , at least for the simple monotonic potentials we consider here . this is schematically illustrated in fig . [ fig : levels3b ] . the two - body energy is know analytically for a single exponential potential . the three - body energies have been calculated with variational method based on either exponential of gaussian wavefunctions , say@xmath60 + \cdots\right]~,\ ] ] where the dots stand for terms deduced by permutation . for @xmath61 , we wrote our own code : the range parameters are chosen in a geometric progression @xmath62 . the lowest term can be linked to the energy @xmath35 by the relation @xmath63 suggested by the feshbach - rubinow equation @xcite . for @xmath64 , we used the code made available by suzuki and varga @xcite . we also did some checks based on the hyperspherical expansion . . the excited 3-body state ( dashed ) is always found below the 2-body energy . ] [ fig : levels3b ] we now replace by a superposition @xmath65 where @xmath11 is significantly shorter ranged than the pair potential entering @xmath16 . in practice , we will choose the long - range part as of unit range , and @xmath66 , with @xmath67 ranging from 20 to 100 . note that the computations become rather delicate for @xmath18 , and would require dedicated techniques for larger @xmath67 . in fig . [ fig:2e ] are shown the spectra for @xmath68 , i.e. , twice the two - body critical coupling , and @xmath69 and @xmath70 , with a magnification of the rearrangement region in the latter cases . and 100 , as a function of the strength of the short - range potential . red : two - body energy , blue : three - body levels.,title="fig : " ] + and 100 , as a function of the strength of the short - range potential . red : two - body energy , blue : three - body levels.,title="fig : " ] + and 100 , as a function of the strength of the short - range potential . red : two - body energy , blue : three - body levels.,title="fig : " ] the unit of energy is irrelevant , as it can be modified by an overall rescaling of the distances . comments are in order : * as in fig . [ fig : rearr2bs ] , a convex behavior as a function of @xmath6 is observed for the excited energy - levels , i.e. , @xmath71 with @xmath72 . this is permitted , provided that the sum of the first energies remains concave @xcite . * as in the two - body case , the transition is sharper when the range of the additional potential becomes shorter . * there is a clear rearrangement , in the sense that for @xmath73 , the excited state falls suddenly near the unperturbed ground - state energy . * however , there is no second plateau for the excited state , just somewhat a smoothing of the fall - off for @xmath74 . * for large enough @xmath67 , one observes that when the second level experiences rearrangement , a third state emerges furtively from the two - body threshold , and disappears for larger coupling @xmath6 . this means that as in the efimov effect , the number of normalizable three - body bound states below the two - body threshold is not a monotonic function of the strength @xmath6 . let us first concentrate on the region of small @xmath6 . one can estimate the energy shifts @xmath75 corresponding to a collection of external potentials @xmath76 and short - range potentials @xmath77 , and study empirically the properties of the matrix @xmath78 . in the two - body case , one finds that the @xmath79 sub - determinants vanish almost exactly . this is compatible with a factorization @xmath80 as a product of a long - range term depending only on @xmath45 and a short - range term depending only on @xmath12 . this is achieved by the deser - trueman formula , with @xmath81 being the square of the wave function at @xmath82 ( times @xmath83 ) and @xmath84 the scattering length . in the three - body case , it is observed that the @xmath79 sub - determinants still nearly vanish , especially for the smaller values of the short - range strength @xmath6 , but that the @xmath85 sub - determinants vanish even better ( of course we compared the determinants divided by the typical values of a product of 2 or 3 @xmath8 ) . this is compatible with @xmath8 being a sum of two factorized contributions , @xmath86 that we conjecture as the generalization of the deser - trueman formula . it is now rather natural to guess that the first contribution is a simple extension of and reads @xmath87 where @xmath88 is a short - hand notation for the two - body correlation factor @xmath89 . it is checked that this term dominates for small shifts , i.e. , for small @xmath6 , see fig . [ fig : trueman3 ] . however , this term , if alone , would induce a fall - off the atomic energies toward the nuclear region only for @xmath90 , i.e. , for @xmath91 , the coupling threshold for two - binding . , and another of range parameter @xmath92 and strength @xmath6 . ] the second contribution in eq . should thus account for the genuine three - body effects . so @xmath93 is a kind of generalized scattering length that blows up when @xmath6 approaches the coupling threshold @xmath94 for three - body binding . as the theory of three - body scattering is a little intricate , we postpone the precise definition of @xmath93 to some further study . as for the long - range factor @xmath95 of this second term , the simplest guess is to assume that it is proportional to the square of the wavefunction at @xmath96 , or in terms of the jacobi variables @xmath97 and @xmath98 describing the relative motion , @xmath99 , but this is seemingly not the case . our study of generalized exotic atoms is related to the efimov physics , _ _ si parva licet componere magnis__. in particular , the authors of refs . @xcite , and probably some others , have studied how the efimov effect is modified if each atom is submitted to an individual harmonic confinement . they also found that near a point where the two - body scattering length becomes infinite , there is a finite number of three - body bound states , instead of an infinite number in absence of confinement . the third three - body bound state in the last two plots of fig . [ fig:2e ] can be interpreted as an efimov state of the short - range potential , modified by the long - range potential . the lowest states of three - bosons have been calculated with a superposition of long - range and short - range attractive potentials . when the strength @xmath6 of the latter is increased , starting from @xmath100 , the 3-body energies decreases very slowly , and can be well approximated by a straightforward generalization of the deser - trueman formula involving only the 2-body scattering length . however , when @xmath6 approaches 0.8 ( in units where @xmath58 is the coupling threshold for binding in the short - range potential alone ) , there is a departure for the deser - trueman formula , which can be empirically accounted for by the product of a short - range and a long - range factor . the short - range factor is the three - body analog of the scattering length and becomes very large when @xmath101 which corresponds to the occurrence of a borromean bound state in the short - range potential alone . many developments are required . what is the precise definition of the three - body short - range factor ? what is the corresponding long - range factor ? what is the minimal ratio of range parameters required for the occurrence of the third stable three - body state ? when does a fourth state show up ? what are the analogs for @xmath102 bosons ? we also aim at studying some asymmetric systems . for instance , a prototype of @xmath103 could be built , with a coulomb interaction , that is known to produce a stable ion , below the threshold for breakup into a @xmath104 atom and an isolated proton @xcite . then the strong interaction between the two protons and the strange meson @xmath105 could be mimicked by a simple potential of range about @xmath106fm , to study how the existence of a nuclear bound state @xmath103 modifies the atomic spectrum .

we study systems of three bosons bound by a long - range interaction supplemented by a short - range potential of variable strength . this generalizes the usual two - body exotic atoms where the coulomb interaction is modified by nuclear forces at short distances . the energy shift due to the short - range part of the interaction combines two - body terms similar to the ones entering the trueman - deser formula , and three - body contributions . a phenomenon of level rearrangement is observed , similar to the zeldovich effect , by the onset of an additional stable level which is eventually absorbed by the two - body threshold energy , and can be interpreted as an efimov - like state of the short - range potential .

there are 35 @xmath4-nuclei on the neutron deficient side of the valley of stability between @xmath5se and @xmath6hg , which are shielded against production by the neutron - capture processes and are produced in the @xmath4-process . the main production of these @xmath4-nuclei occurs via ( @xmath7 ) , ( @xmath8 ) and ( @xmath9 ) reactions , and subsequent beta decays in the so - called @xmath3-process . therefore , it is essential to determine cross sections of photodisintegration reactions or their inverse reactions for @xmath4-process network calculations @xcite . thousands of nuclear reactions are involved in network calculations for @xmath4-process nucleosynthesis . however , only very few of the required cross sections have been measured by experiments , and thus most of them rely solely on predictions of the hauser - feshbach ( hf ) model codes , e.g. , non - smoker @xcite and talys @xcite , which often have very large uncertainties from nuclear input parameters @xcite . following current @xmath4-process network calculations @xcite , the pattern of the solar abundance for about 60% of the @xmath4-nuclei is reproduced within a factor of 3 . however , for the @xmath10mo and @xmath11ru isotopes , an underproduction of a factor of 20 - 50 has been calculated for the @xmath3-process in core collapse supernova models with massive stars of 13 - 25 m@xmath12 @xcite . this deficiency has motivated the search for additional production mechanisms , e.g. , the @xmath13-process @xcite or the @xmath14@xmath4-process @xcite , but also intensified efforts to remove the uncertainty in required nuclear physics parameters by measuring reaction cross sections . these cross sections are most sensitive to nuclear parameters , i.e. , the @xmath3-ray strength function , nuclear level density and optical potential , in the hf model @xcite . however , these critical parameters are not well constrained by experiments , and hence there are large differences between predictions using different parameters . most of the existing experimental data for the @xmath4-process were measured in direct kinematics using stable isotope targets @xcite . however , a direct measurement on unstable nuclei is still a major challenge @xcite . in this work , we present a novel method using a heavy - ion storage ring developed to measure cross sections of low - energy nuclear reactions , e.g. , ( @xmath1 ) reactions , in inverse kinematics for nuclear astrophysics . this method offers some key advantages over traditional methods . it can efficiently use the beam and is well suited for measurements on unstable nuclei with half - lives longer than several minutes . this method was suggested already several decades ago @xcite . however , it was not realized until the latest achievements in producing , cooling , decelerating , and storing of heavy ions , as well as developments in the nuclear detection system at the experimental storage ring ( esr ) of gsi @xcite . this novel technique provides a unique condition for the direct measurement of ( @xmath1 ) reactions around the energy range of astrophysical interest @xcite and has been successfully demonstrated for the first time by measuring the @xmath0ru(@xmath1)@xmath2rh cross section between 9 mev / u and 11 mev / u . a preliminary data analysis at 11 mev / u has been reported in conference proceedings @xcite . here we report a full analysis with all corrections made . experimental results of the present work allow us to constrain the most important parameters in the hf model , and thus provide a reliable prediction for this reaction over a wide energy range . during this experiment , @xmath0ru ions from the linear accelerator ( unilac ) were first accelerated to 100 mev / u in the heavy - ion synchrotron ( sis ) and then stripped to the bare charge state of 44 + using a 11 mg/@xmath15 carbon stripper foil . the fully stripped ions were injected into the esr and slowed down to 9 mev / u , 10 mev / u and 11 mev / u , respectively , by ramping the magnetic fields and the frequency of the radio - frequency ( rf ) system synchronously . however , large beam losses occurred during this deceleration phase mainly due to imperfections of the ramping parameters . the @xmath0ru@xmath16 ions were cooled by the electron cooler before and after the slowing down phase to a small diameter ( about 5 mm ) and momentum spread ( around 10@xmath17 ) . after the final slowing down phase , about 5@xmath18@xmath19 @xmath0ru@xmath16 ions were stored in the esr with a lifetime of several hundred seconds when the hydrogen target was switched off . finally , a windowless hydrogen microdroplet target of high density @xcite was switched on and the decelerated @xmath0ru@xmath16 ions were focused onto this target for nuclear reactions . the great advantage of this storage ring method is that unreacted @xmath0ru@xmath16 ions were recycled and repeatedly impinged on the hydrogen target for reactions . considering the revolution frequency of about 400 khz for @xmath0ru@xmath16 ions circulating in the ring and the thickness of the h@xmath20 target of about @xmath21 particles/@xmath15 , a luminosity of about 2@xmath18@xmath22 /@xmath15/s has been achieved . at about 10 mev / u , the main reaction channels of @xmath0ru@xmath16 with the hydrogen target include the atomic electron capture ( ec ) reactions and different nuclear reactions . the former contain mainly two parts , namely the non - radiative electron capture ( nrc ) and the radiative electron capture ( rec ) accompanied by the emission of a photon . there are large uncertainties of about 30% according to ref . @xcite in accurately determining the absolute beam intensities , target densities and the beam - target overlap at the esr . to remove these large uncertainties , both the k - shell rec ( k - rec ) products ( photons ) and the ec products were registered by our detectors , which allowed us to absolutely determine the ( @xmath1 ) cross section by using two normalization methods . [ fig : exp_setup_esr ] shows the experimental setup from the hydrogen target to all used detectors at the esr . target and the detectors used in this experiment are marked . both the multi - wire proportional chamber ( mwpc ) and the double - sided silicon strip detectors ( dsssds ) were placed in pockets with 25 @xmath23 m thick windows . a schematic view of the dsssds is also shown . the unreacted @xmath0ru@xmath16 ions ( magenta solid line after the target ) were recycled and focused on the h@xmath20 target repeatedly during the measurement phase.,width=325 ] the x - rays emitted from the atomic reactions were registered by a ge detector mounted close to the target interaction area at the observation angle of 90@xmath24 with respect to the beam axis . the detection efficiency of this ge detector was calibrated with mixed @xmath3-ray sources in the energy range between 10 kev and 140 kev . in the k - rec energy region , an intrinsic efficiency of about 88% has been reached for this detector with a solid angle ( @xmath25 ) of about 2.5@xmath18@xmath26 sr . the down - charged @xmath0ru ions ( @xmath0ru@xmath27 , @xmath0ru@xmath28 , @xmath0ru@xmath29 , etc . ) produced by ec were recorded by a position sensitive multi - wire proportional chamber ( mwpc ) @xcite mounted in the vacuum chamber of the first dipole magnet behind the target , as indicated in fig . [ fig : exp_setup_esr ] . this detector was operated with a mixture of argon , co@xmath20 and heptane ( 80:20:1.5 ) gas at standard atmospheric pressure in a pocket with a stainless - steel window . the detection efficiency of this detector is better than 99% for ions above @xmath3010 mev / u and a position resolution ( fwhm ) of 1.9 mm has been reached @xcite . however , a fraction of electron capture events were lost at the beam energy of 9 mev / u due to significant energy losses in the pocket window with a thickness of about 25 @xmath23 m and the gas with a thickness of around 24 mm before the mwpc . at about 10 mev / u , there are only four open nuclear reaction channels : @xmath0ru(@xmath31)@xmath0ru elastic scattering , @xmath0ru(@xmath1)@xmath2rh , @xmath0ru(@xmath32)@xmath33tc and @xmath0ru(@xmath34)@xmath0rh . the above reaction products with different mass - over - charge ratios ( @xmath35 ) were separated by the magnetic field ( @xmath36 ) of the dipole magnet behind the target . for instance , the dispersion of the magnets displaced @xmath2rh@xmath37 from unreacted @xmath0ru@xmath16 by about 142 mm at the position of our detector , see dash - dotted line and solid line in fig . [ fig : exp_setup_esr ] . as will be shown below , the ( @xmath1 ) reaction products can be discriminated from other nuclear reaction products due to their relatively small momentum spread . the @xmath35 is always larger for atomic ec reaction products since nuclear reaction products are bare . therefore , orbits of the former are always on the outer side of the esr , see dashed line in fig . [ fig : exp_setup_esr ] , and bare nuclear reaction products on the inner side are not contaminated with atomic ones . the separated nuclear reaction products were detected by the position sensitive double - sided silicon strip detectors ( dsssds ) behind the quadrupole triplet magnets , as shown in fig . [ fig : exp_setup_esr ] . the dsssds consist of two silicon detectors with a @xmath3021.5 mm inactive gap between them , see fig . [ fig : exp_setup_esr ] . each silicon detector with an active area of 50@xmath1850 mm@xmath38 has 16 strips in both @xmath39- and @xmath40-directions . the strip pitch is 3.1 mm and the strip length is 49.5 mm . the dsssds were placed in a pocket separated from the vacuum of the esr by a 122@xmath1844 mm@xmath38 stainless - steel window with a thickness of about 25 @xmath23 m . the thickness of this window limited the reaction energy to above 9 mev / u in this experiment , since heavy ions with less energy would already be stopped in the window . recently , an improved detector omitting the window has been mounted @xcite , which will allow us to measure nuclear reactions around 5 mev / u in future experiments . for each beam energy setting , the dsssds were moved to two different positions along the @xmath39 direction , e.g. , 0 mm and 25 mm , to combine two measured spectra into a common @xmath39 position spectrum without any gap , see black points at 11 mev / u in fig . [ fig : exp_sim_11mev ] . to identify the ( @xmath1 ) reaction products unambiguously and to study their transmission efficiency , a geant4 @xcite simulation has been performed using a numerical model of the experimental setup shown in fig . [ fig : exp_setup_esr ] . fig . [ fig : exp_sim_11mev ] compares the simulated total @xmath39 position spectrum including all nuclear reaction events ( red line ) with the experimental data ( black points ) registered by the dsssds at 11 mev / u . the experimental data , which have an uncertainty of about 10% , can be well reproduced and the ( @xmath1 ) reaction events can be disentangled clearly from the background produced by ( @xmath31 ) , ( @xmath32 ) , and ( @xmath34 ) reactions , based on geant4 simulations . the angular distribution of ( @xmath31 ) scattering products in the center - of - mass ( cm ) system , which serves as an input for the simulation , is from the prediction by talys @xcite while other events are assumed to be isotropic . the magnetic fields in the simulation were set to the experimental values . the minimum chi - square method has been utilized to obtain the best simulation spectrum for the experimental data and determine the number of ( @xmath1 ) products @xmath41 . the simulation uncertainty can be obtained by simulations varying the sensitive parameters , e.g. , magnetic fields , sizes of beam pipes and sizes of chambers , within their uncertainties . position distribution of nuclear reaction products ( black points ) registered by the dsssds at 11 mev / u is compared with the simulated @xmath39 position distribution ( red line ) . events from different reaction channels , i.e. , ( @xmath31 ) , ( @xmath32 ) , ( @xmath1 ) , and ( @xmath34 ) reactions , can be disentangled based on the geant4 simulation . the ( @xmath1 ) reaction products have a narrower distribution than other nuclear reaction products due to their smaller momentum spread.,width=347 ] the charge - state spectra measured by the mwpc at 11 mev / u are presented in figure [ fig : subfig:3a ] . five different charge states of @xmath0ru are produced by the electron capture reactions . the most prominent peak is from @xmath0ru@xmath27 ions produced by single ec . other peaks are caused by the capture of more electrons . the black line shows the spectrum measured when the h@xmath20 target is switched on while the red line indicates the background measured when the h@xmath20 target is off . hence , the single ec events caused by the h@xmath20 target can be determined by subtracting the corresponding background from the interaction of the @xmath0ru@xmath16 beam with the residual gas in the ring . the x - ray spectrum measured by the 90@xmath24 ge detector at 11 mev / u is given in figure [ fig : subfig:3b ] . the k- , l- and m - rec peaks are caused by the radiative captures into the k- , l- and m - shells of ru , respectively . the k@xmath42 , k@xmath43 and k@xmath44 peaks originate from cascades after electron captures into higher shells of ru . positions of these peaks are in very good agreement with theoretical predictions of ref . @xcite . according to our experimental data with the h@xmath20 target switched off , the background spectrum registered by the 90@xmath24 ge detector is almost linear and there is no peak structure in the k - rec energy region . therefore , the number of k - rec events induced by the h@xmath20 target can be extracted from the sum of all events in the k - rec energy region , subtracting a linear background . in this experiment , the ( @xmath1 ) cross section can be normalized by two methods using ( 1 ) the theoretical single ec cross section , and ( 2 ) the calculated k - rec cross section at 90@xmath24 . the single ec cross section and the k - rec cross section can be predicted very well by different theoretical models . for the single ec at about 10 mev / u , the cross section can be calculated by the schlachter scaling rule @xcite . according to this scaling rule , the cross section of the single ec can be calculated by the relation @xcite : @xmath45 where @xmath46 is the projectile charge state , @xmath47 is 1 for the hydrogen target , and @xmath48 is the projectile energy in kev / u . the calculated single ec cross section for our experiment is given in table [ tab : table1 ] . its uncertainty is estimated to be about 20% . the single ec cross section above 10 mev / u can also be normalized to the theoretical k - rec cross section , see below , by a validated normalization method reported in ref . @xcite when both single ec and k - rec events were recorded by our detectors . using this normalization method , the single ec cross section has been calculated by eq . ( 2.2 ) and eq . ( 2.3 ) in ref . @xcite ( see also eq . ( [ eq : normalization - diff - k - rec ] ) given below with the ( @xmath1 ) replaced by the single ec ) . single ec cross sections determined by two different methods are in good agreement within uncertainties , see table [ tab : table1 ] . this agreement also indicates the schlachter scaling rule works well for @xmath0ru@xmath16 colliding with h@xmath20 at about 10 mev / u in our experiment . .[tab : table1]theoretical cross sections of the single ec and corrected differential k - rec cross sections at 90@xmath24 for @xmath0ru@xmath16 + h@xmath20 collisions between 9 mev / u and 11 mev / u . single ec cross sections are also normalized to corrected k - rec cross sections at 10 mev / u and 11 mev / u . [ cols="^,^,^,^",options="header " , ] the experimental rates have been theoretically extrapolated by normalizing the @xmath0ru(@xmath1)@xmath2rh cross sections calculated by non - smoker to the experimental data of j. bork _ et al . _ , see ref . @xcite . in the normalization , a factor of about 0.5 has been applied for non - smoker calculations . the talys rate using parameters constrained in the previous section can excellently reproduce the experimental rate for @xmath0ru(@xmath1)@xmath2rh between 2 gk and 2.5 gk , as displayed in fig . [ fig : exp_model_reaction_rates ] . hence , the talys rate constrained by two experimental data sets at different energy regions is recommended for @xmath0ru(@xmath1)@xmath2rh , as listed in table [ tab : table3 ] . on the contrary , both non - smoker and bruslib overestimate the rate at temperatures below 3 gk , particularly at low temperatures . a good agreement is reached between predictions by non - smoker and talys between 3.5 gk and 9 gk . however , non - smoker underestimates the rate above 9 gk , which is caused by the underestimation of the cross section above about 10 mev . above 3 gk , the rate extrapolated by non - smoker is lower than the talys rate since the former are determined by normalizing the non - smoker rate to the experimental rate around 2 gk . in addition , bruslib significantly underestimates the rate above 4 gk . for instance , bruslib underestimates the rate by a factor of 20 above 8 gk , which again indicates that our experimental data are important to constrain the theoretical rate . it should be stressed that the bruslib rate is calculated by the talys code , where the input parameters are not constrained by experiments , see ref . @xcite for details . for network calculations , the recommended rate has been parameterized in the reaclib format @xcite using the formula @xmath49 , \label{eq : rate_fit}\end{aligned}\ ] ] where @xmath50 is the temperature in gk . recommended reaclib parameters are listed in table [ tab : table4 ] . the fit using these parameters agrees very well with the recommended rate within 10% between 1 gk and 8 gk . in summary , a novel technique via the collision of stored heavy ions with a hydrogen target has been developed at the esr , which provides unrivalled opportunities for the direct measurement of ( @xmath1 ) reactions around the energy range of astrophysical interest , particularly for previously unreachable radioactive ions . this method has been successfully demonstrated for the first time by measuring the @xmath0ru(@xmath1)@xmath2rh cross sections between 9 mev and 11 mev . the present experimental results allowed us to pin down the @xmath3-ray strength function , which is a critical parameter in the hf model , as well as the nuclear level density model . after this , another important parameter , the proton potential , has also been constrained by combining our results with some additional data at lower energies . talys , constrained by two experiments in different energy regions , can excellently predict the stellar rates for @xmath0ru(@xmath1)@xmath2rh over a large temperature range for @xmath4-process network calculations . further measurements of ( @xmath1 ) reactions at lower energies around the gamow window via this method using our improved detector are planned in future experiments at heavy ion storage rings . besides , ( @xmath51 ) reactions will also be measured by this method when a helium target is utilized . 43ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * ( ) @noop * * , ( ) @noop `` , '' @noop `` , '' @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop `` , ''

this work presents a direct measurement of the @xmath0ru(@xmath1)@xmath2rh cross section via a novel technique using a storage ring , which opens opportunities for reaction measurements on unstable nuclei . a proof - of - principle experiment was performed at the storage ring esr at gsi in darmstadt , where circulating @xmath0ru ions interacted repeatedly with a hydrogen target . the @xmath0ru(@xmath1)@xmath2rh cross section between 9 and 11 mev has been determined using two independent normalization methods . as key ingredients in hauser - feshbach calculations , the @xmath3-ray strength function as well as the level density model can be pinned down with the measured ( @xmath1 ) cross section . furthermore , the proton optical potential can be optimized after the uncertainties from the @xmath3-ray strength function and the level density have been removed . as a result , a constrained @xmath0ru(@xmath1)@xmath2rh reaction rate over a wide temperature range is recommended for @xmath4-process network calculations .

the 30@xmath0 anniversary of the publication by the european muon collaboration ( emc ) @xcite on the modification of the per - nucleon cross section of nucleons bound in nuclei , named _ emc effect _ , is a good opportunity to review the status of the effect and to summarize the available experimental and theoretical information . the emc effect can be interpreted as a modification of quark and gluon distributions in bound nucleons by the nuclear environment . the dependences of these nuclear modifications on kinematics and various nuclear properties like mass , density or radius are meanwhile rather well known , nevertheless the origin of the effect is still not fully understood . recent data , discussed in section [ xlargerone ] , provide new important input . in my opinion further measurements of the effect in electron - nucleus , neutrino - nucleus and proton - nucleus scattering are needed to determine nuclear quark and gluon distributions ( npdfs ) down to very low parton momentum fractions @xmath1 and to constrain the initial state for the @xmath2 program at rhic and lhc , or for the correct interpretation of , e.g. , ( future ) @xmath3 experiments . nuclear modifications of parton distributions have been studied mainly by measurements of cross sections in deep - inelastic lepton - nucleus scattering . for electromagnetic interactions of charged leptons with nuclear targets and in the approximation of one - photon exchange the cross section reads @xmath4 } \right ] \;. \label{gl : discross}\ ] ] here @xmath5 represents the squared four - momentum of the virtual photon that mediates the interaction with coupling strength @xmath6 and @xmath7 can be interpreted as the fraction of the longitudinal nucleon momentum carried by the struck quark , in a frame where the nucleon moves with infinite momentum in the direction opposite to that of the virtual photon . the variable @xmath8 denotes , in the target rest frame , the virtual - photon energy @xmath9 with respect to the lepton - beam energy @xmath10 . at leading order in qcd the structure function @xmath11 is defined as the sum of the momentum distributions @xmath12 and @xmath13 of quarks and anti - quarks of flavor @xmath14 weighted by @xmath1 and @xmath15 , where @xmath16 is the quark charge ( in units of the elementary charge @xmath17 ) : @xmath18 \;. \label{gl : f2quarks}\ ] ] the quantity @xmath19 - 1 = \frac{{\rm f}_{\rm l}}{2x{\rm f}_1 } \label{gl : r}\ ] ] is the ratio of the longitudinal to transverse virtual - photon cross sections . in the quark - parton model , @xmath20 for the interaction of the virtual photon with a point - like spin-1/2 particle . quark transverse momenta , quark masses and gluon radiation cause @xmath21 to deviate from zero . if @xmath21 is independent of the nuclear mass number @xmath22 ( see the discussion in section [ raminusrd ] ) , then the ratio of cross sections for two different nuclei is equal to the ratio of their structure functions @xmath11 . subsequently , we will always discuss the ratio of structure functions ( cross sections ) per nucleon for a nucleus with mass number @xmath22 ( i.e. , @xmath22 nucleons ) and the deuteron @xmath23 . the latter is , to a good approximation , equal to the proton - neutron averaged structure function @xmath24 . the @xmath1 dependence of the structure functions @xmath25 and @xmath26 is different ( for free nucleons they are approximately related by @xmath27 ) . results for the nuclear structure function @xmath28 ( cross section @xmath29 ) for nuclei with @xmath30 protons and @xmath31 neutrons will always be corrected for neutron excess by @xmath32 , \label{gl : f2a}\ ] ] where it is assumed that proton and neutron structure functions are modified equally by the nuclear environment . thus , @xmath28 is the structure function per nucleon for a hypothetical isoscalar nucleus with an equal number ( @xmath33 ) of protons and neutrons . the historical result of the emc effect @xcite ( updated results were published in @xcite ) is presented in the left panel of fig . [ ab : history ] . it shows the ratio of the structure function @xmath11 per nucleon for iron and deuterium , both uncorrected for fermi motion , as a function of @xmath1 . the shaded area indicates the range for the errors on the slope of a linear fit to the data , the point - to - point systematic uncertainties are somewhat larger . in addition there is an overall uncertainty of @xmath34 . ( @xmath35 ) as a function of @xmath1 . left panel : the original emc result @xcite ; right panel : the slac result @xcite together with low-@xmath36 slac data for cu / d @xcite and the photoproduction result at @xmath37 gev@xmath38 @xcite . also shown is one expectation for the effect of fermi motion on @xmath39 in absence of other nuclear effects @xcite . , title="fig : " ] ( @xmath35 ) as a function of @xmath1 . left panel : the original emc result @xcite ; right panel : the slac result @xcite together with low-@xmath36 slac data for cu / d @xcite and the photoproduction result at @xmath37 gev@xmath38 @xcite . also shown is one expectation for the effect of fermi motion on @xmath39 in absence of other nuclear effects @xcite . , title="fig : " ] the ratio is seen to be different from unity . it falls from @xmath40 at @xmath41 to a value of @xmath42 at @xmath43 and does nt follow the expectations from fermi - motion calculations . this result demonstrated for the first time that the structure function @xmath11 is modified when nucleons are embedded in a nucleus . at the time when these data were presented the effect appeared quite astounding . especially members of the high - energy physics community could hardly believe that at momentum transfers several orders of magnitude larger than typical nuclear binding energies quark distributions should be affected by the nuclear environment . however , ` quarks in nuclei ' and phenomena that could possibly modify the quark distributions in bound nucleons were already discussed by the nuclear - physics community for some time before this discovery . ( for a first review of such ideas see ref . the emc result was quickly confirmed by the slac - mit - rochester group that recovered and reanalyzed the data stemming from the aluminum and steel cell walls of the liquid hydrogen and deuterium targets from the experiments e49b and e87 . the result for @xmath35 @xcite is shown in the right panel of fig . [ ab : history ] together with earlier data from slac - e61 at lower @xmath36 for cu / d in the range @xmath44 @xcite and a data point at @xmath37 gev@xmath38 from an experiment that had investigated shadowing in photoproduction @xcite . ( similar data were published for aluminum @xcite ) . at high @xmath1 , these data were in good agreement with the emc data , but the rather large value of @xmath45 for the two low-@xmath1 emc points was in disagreement with the low-@xmath36 slac data which indicated that for @xmath1 below @xmath46 the ratio decreases again with decreasing @xmath1 . ( indeed , it was found out later that the low-@xmath1 emc points suffered from correlated tracking inefficiencies affecting the deuterium but not the iron data ) . this exciting result caused enormous activities in both experiment and theory , resulting at present in about 1000 citations of the emc publication @xcite . the experimental data ( with the exception of the recent jlab data presented in sect . [ xlargerone ] ) and many of the theory papers have been discussed in great detail some time ago in the excellent review by p. norton @xcite . therefore , i will only summarize some of the key results and the main theoretical ideas for the interpretation of the effect . after the discovery , nuclear effects have been studied experimentally in charged lepton - nucleus scattering by the muon experiments bcdms @xcite , emc - na38 @xcite , emc @xcite and nmc @xcite at cern and e665 @xcite at fnal , in electron scattering at slac @xcite , desy @xcite and jlab @xcite , in neutrino - nucleus scattering @xcite and in the drell - yan process @xcite . i will only discuss the most important results . at large @xmath1 the most precise data are those from the slac experiment e139 @xcite that measured the cross section ratio @xmath47 for 8 nuclei ranging from @xmath48he to @xmath49au . the results of an updated analysis with an improved treatment of radiative corrections @xcite are shown in fig . [ ab : slace139 ] . as a function of @xmath1 for various nuclei measured by slac - e139 @xcite . ] for all nuclei one observes , in the region @xmath50 , a reduction of the per - nucleon cross section @xmath29 compared to the ` free nucleon ' one , @xmath51 . the @xmath1 dependence of this reduction has a very characteristic universal shape with a minimum near @xmath52 . the effect is already present for helium , its magnitude increases with the atomic mass number @xmath22 . the @xmath22 dependence will be discussed in more detail in section [ nuclprop ] . the e139 data provide precise information for @xmath53 . the region of lower @xmath1 is covered by the data from slac - e61@xcite , from the hermes experiment @xcite , where the 27.6 gev electron beam of hera was scattered from internal gas targets of various nuclear species , and the muon experiments . as an example , the @xmath1 dependence of the cross section ratio @xmath54 measured in electron scattering by e139 @xcite and hermes @xcite is presented in fig . [ ab : globalx ] . also shown are data from jlab - e03103 @xcite taken with a beam energy of 5.8 gev . as a function of @xmath1 from hermes @xcite , slac - e139 @xcite , and jlab - e03103 @xcite . open squares denote @xmath55 below 2 gev@xmath38 , where @xmath56 is the invariant mass of the photon - nucleon system . ] this figure nicely summarizes the universal @xmath1 dependence of the nuclear effects . it can be subdivided into four @xmath1 regions ( plus a fifth one at @xmath57 which will be discussed separately in section [ xlargerone ] ) : * the shadowing region ( @xmath58 ) , where the structure function ratio is smaller than unity and decreases with decreasing @xmath1 down to the value measured in photoproduction . here , the dominant contribution to the cross section is due to sea quarks . the essential longitudinal distances @xmath59 probed in the deep - inelastic interaction ( see section [ shad ] ) are @xmath60 fm , much bigger than the size of a nucleon ; * the anti - shadowing region ( @xmath61 ) , where the ratio shows a small increase of a few percent over unity ; * the region ( @xmath62 ) , where the ratio is smaller than unity with a minimum near @xmath52 . here , the sea - quark distribution is essentially negligible and the ratio reflects the behavior of the valence - quark distributions ; * the region ( @xmath63 ) , where the ratio increases rapidly with increasing @xmath1 . this behavior is dominantly a kinematic effect since the free - nucleon cross section vanishes for @xmath64 . it is partly also due to the fermi motion of the bound nucleons in the nucleus . in deep - inelastic scattering from stationary targets , the kinematic region of very low @xmath1 can only be accessed with muon beams , since those can be produced with much higher energies than electron beams . the first of such measurements was performed by emc - na28 , using a muon detection system at small scattering angles down to 2 mrad and nuclear targets of c and ca @xcite . this experiment demonstrated that shadowing persist also at high values of @xmath36 . the low-@xmath1 region was then explored in detail by nmc with nominal incident muon energies of 90200 gev @xcite and at even lower values of @xmath1 by e665 at fnal with a mean incident muon energy of 470 gev . nmc had the main objective to study the nuclear modification of the structure function @xmath65 with high precision . cross section ratios were measured for nine nuclear species . in one set of measurements @xcite @xmath48he , @xmath66li , @xmath67c and @xmath68ca were compared to deuterium . all these nuclei are isoscalar and no correction for neutron excess ( equ . ( [ gl : f2a ] ) ) is necessary . in another set of measurements @xcite , @xmath69be , @xmath70al , @xmath68ca , @xmath71fe , @xmath72sn and @xmath73pb were compared to carbon . one important peculiarity of the nmc experiment was the multiple target arrangement . targets of different materials were placed in a row at longitudinally well separated locations along the spectrometer axis and exposed simultaneously to the beam . two such rows , differing in the ordering of materials , were placed on a common platform . the rows of targets were positioned in the beam in turn by lateral displacement of the platform at approximately 3060 min intervals . with this arrangement beam flux and spectrometer acceptance corrections canceled in the determination of cross section ratios . these high - precision measurements are the low-@xmath1 counterpart of the e139 data at large @xmath1 . for he , c and ca measured at low @xmath1 by nmc @xcite and e665 @xcite . note that for both experiments @xmath74 gev@xmath38 for @xmath1 values below @xmath75 . ] as an example the structure function ratios of @xmath48he , @xmath67c and @xmath68ca to deuterium measured by nmc @xcite and e665 @xcite are shown in fig . [ ab : lowxdata ] . the e665 data nicely extrapolate to the shadowing results measured in photoproduction @xcite . the nmc and e665 data do not agree very well , but the e665 data move downwards by a few percent , when another method of radiative corrections is applied @xcite . as already seen in fig . [ ab : globalx ] , the ratio is smaller than unity below @xmath76 and decreases with decreasing @xmath1 . the effect is already visible for @xmath48he , it increase with @xmath22 . as stated in section [ kin ] , @xmath47 is only equal to @xmath77 , if the quantity @xmath21 is independent of the nuclear mass number @xmath22 . the nuclear dependence of @xmath21 has been studied by slac - e140 @xcite , nmc @xcite and hermes @xcite . all measurements are consistent with @xmath21 being independent of @xmath22 . a reanalysis of all slac data with an improved radiative - corrections calculation procedure @xcite resulted in @xmath78 , and the authors conclude that possible contributions to @xmath21 from nuclear higher - twist effects and possible spin-0 constituents in nuclei are not different from those in free nucleons . this conclusion is supported by the more recent hermes measurement @xcite . averaging over all measurements of @xmath79 for light and medium heavy nuclei ( @xmath80he , @xmath48he , @xmath67c , @xmath81n ) hermes obtained an average value for @xmath82 of @xmath83 . the result is unchanged if also the data on the heavier nuclei are included in the average . already from the good agreement between the muon and the electron data one can conclude that the @xmath36 dependence of the nuclear effects is very small , since , at the same value of @xmath1 , their average @xmath36 typically differs by more than an order of magnitude . dependence of the logarithmic slope @xmath84 . left panel : ca / d @xcite , right panel : sn / c @xcite . ] the @xmath36 dependence was studied in some detail by nmc @xcite , fnal - e665 @xcite and hermes @xcite . as an example , the left panel of fig . [ ab : q2dep ] shows the @xmath1 dependence of the logarithmic slope @xmath85 from fits of the form @xmath86 to the nmc data @xcite . in the covered @xmath36 range there is little indication for a @xmath36 dependence of the ratio . the nmc sn / c data @xcite , shown in the right panel of fig . [ ab : q2dep ] , are the only exception where one observes an indication of a small @xmath36 dependence at small values of @xmath1 . the available measurements in neutrino / antineutrino - nucleus scattering @xcite suffer from their large statistical and systematic uncertainties and do not provide additional information . apart from @xcite all these measurements have been performed with bubble chambers . precise measurements of neutrino and antineutrino scattering from deuterium and heavy nuclear targets would be very helpful for a separation of nuclear effects in sea - quark and valence - quark distributions and a determination of nuclear parton distribution functions ( npdfs ) . but such measurements have to wait for the future realization of a very - high - luminosity neutrino factory . the nuclear modification of anti - quark distributions can also be investigated by the drell - yan process @xcite in proton - nucleus scattering : @xmath87 . in this process a quark ( anti - quark ) with the four - momentum fraction @xmath88 from the beam proton and an anti - quark ( quark ) of the target nucleon with @xmath89 annihilate electromagnetically into a virtual photon , which immediately decays into a charged - lepton pair : @xmath90 . the longitudinal momentum of the @xmath91 pair in the proton - nucleon center - of - mass system is approximately given by @xmath92 and its invariant mass by @xmath93 . here , @xmath94 is the nucleon - nucleon center - of - mass energy . the drell - yan cross section reads @xmath95 \ ; , \label{gl : drellyan}\ ] ] where @xmath96 is a factor representing the deviation from the simple parton model due to qcd corrections . by a suitable choice of the kinematics of the @xmath97 pair the quark and anti - quark distributions in the target can be determined separately . if one requires for instance @xmath98 , then the second term in ( [ gl : drellyan ] ) can be neglected and the cross section ratio for two nuclei @xmath99 and @xmath100 is to a good approximation equal to the ratio of anti - quark distributions @xmath101 in the target . @xcite ( left ) and @xmath102 @xcite ( right)as a function of @xmath89 . ] in the left panel of fig . [ ab : dy ] , @xmath47 for four nuclei ( c , ca , fe , w ) measured by fnal - e772 @xcite is shown as a function of @xmath89 , whereas the right panel shows @xmath102 for ( fe , w ) measured by fnal - e866 @xcite , both with a proton - beam energy of 800 gev . at the smallest @xmath89 values the data show an indication of shadowing , while above @xmath103 the ratios are consistent with unity ( with the exception of the c / d data ) leading to the conclusion that anti - shadowing is very likely not a sea - quark effect or caused by nuclear pions . this aspect will be studied in detail by the experiment e906/seaquest at the 120 gev beam of the fnal main injector @xcite . unfortunately the drell - yan cross section is very small and it is very difficult to study the process with colliding ion beams at much higher center - of - mass energies . nevertheless such measurements in proton - deuteron and proton - nucleus collisions at rhic or the lhc would be very desirable for the determination of npdfs at low values of @xmath1 . * dependence on nuclear mass @xmath104 . * from figs . [ ab : slace139 ] and [ ab : lowxdata ] it is obvious that the nuclear effects increase continuously with nuclear mass number @xmath104 . this aspect has been studied in detail by e139 @xcite and nmc @xcite . as a function of nuclear mass @xmath105 at low @xmath1 from nmc @xcite ( left ) and @xmath47 at high @xmath1 from slac - e139 @xcite(right ) . the lower right panel shows the coefficient @xmath106 from a fit of the form @xmath107 . ] the left panel of fig . [ ab : adep ] shows the nmc results @xcite for the dependence of @xmath108 on @xmath104 for the two bins @xmath109 and @xmath110 , and the upper plot in the right panel shows the e139 results @xcite for @xmath47 in the bin @xmath111 . obviously the nuclear effects increase to a good approximation linearly with log@xmath104 . small deviations from this linear dependence are observed in the left panel for he and li and in the right panel for he and c. obviously there are other nuclear properties than a affecting the nuclear dependence . the lines are results of fits of the form : @xmath112 . the coefficient @xmath113 determined from the e139 data is shown in the lower right panel as a function of @xmath1 . if we rewrite @xmath106 as @xmath114 , then @xmath115 can be interpreted as the effective number of nucleons participating in the interaction . * dependence on nuclear density @xmath116 . * one important aspect for the understanding of the nuclear medium effects is their dependence on the nuclear density @xmath116 . in fig . [ ab : rhodep ] the data presented in fig . [ ab : adep ] are shown as a function of @xmath116 in four bins of @xmath1 . as a function of nuclear density @xmath116 at low @xmath1 from nmc @xcite and @xmath47 at high @xmath1 from slac - e139 @xcite . ] here @xmath117 is given by @xmath118 , with @xmath119 , and @xmath120 is taken from ref . @xcite . at large @xmath1 the cross section ratio approximately scales with @xmath116 , with the exception of the rather special nuclei @xmath48he and @xmath69be . the deviation of @xmath69be from this linear behavior is , however , much smaller than in the analysis of jlab - e03103 @xcite , where a density based on ab initio few - body calculations @xcite , scaled by @xmath121 is used . the small-@xmath1 data are not well described by a linear function of nuclear density . * dependence on nuclear radius @xmath122 . * the deviations from a linear behavior observed in fig . [ ab : rhodep ] indicate that besides the nuclear density other parameters like the nuclear radius or the nuclear surface may play a role . the nuclei @xmath48he , @xmath66li , @xmath67c and @xmath68ca have been used by nmc @xcite to possibly differentiate between effects originating from the nuclear density or from the nuclear radius . these nuclei differ primarily either in radius ( @xmath122 ) or density ( @xmath116 ) . in particular , @xmath123li ( @xmath124 ) and @xmath67c ( @xmath125 ) have nearly equal radii but different densities , whereas @xmath48he ( @xmath126 ) and @xmath67c have nearly equal densities but different radii . @xmath67c and @xmath68ca ( @xmath127 ) differ more in radius than in density . the analysis presented in @xcite demonstrates that there is a rather complicated interplay between the dependences on radius and density : the depletion at low @xmath1 is larger in @xmath67c than in @xmath123li , showing that at the same radius the effect increases with @xmath116 ; the comparison of the pairs ( @xmath48he,@xmath67c ) and ( @xmath68ca,@xmath67c ) indicates that at similar densities the effect increases with radius ; the depletion in @xmath68ca is twice as large than in @xmath123li , implying that the depletion increases with both radius and density ; and the li / he ratio is consistent with unity over the common @xmath1 range , indicating that the opposing dependencies on radius and density tend to cancel . furthermore nmc has shown that at low @xmath1 the data are best described by a fit of the form @xmath128 @xcite . it is beyond the scope of this lecture note to address the multitude of possible explanations for the observed nuclear medium effects ( for a rather detailed discussion see ref . @xcite ) . instead , only the main ideas of some classes of models will be summarized without a discussion why and where they fail to reproduce the data correctly . the first class of models deals with the shadowing region . the term ` shadowing ' has been introduced to explain the reduction of the nuclear cross sections in photoproduction , but has then also been used in the discussion of the low-@xmath1 modification of the nuclear cross section in inelastic lepton - nucleon scattering . there are two possibilities to explain this phenomenon . in the first approach the parton distributions in bound nucleons remain unchanged compared to those in the free nucleon . the interaction is viewed in the rest system of the nucleus . the nuclear effects are attributed to a modification of the interaction of the virtual photon with the atomic nucleus by fluctuations of the virtual photon into quark - antiquark pairs . such a pair then interacts with the nucleus via the _ strong _ interaction . since the strength of the latter is much larger than the electromagnetic one , the interaction does no longer happen incoherently with all the nucleons in the nucleus but preferentially with those at the front surface . the nucleons being in the ` shadow ' of the nucleons at the front surface then do not or much less contribute to the cross section . quantitatively this happens , when the fluctuation length @xmath129 is larger than the mean free path @xmath130 of the quark - antiquark pair , i.e. , for @xmath1 below @xmath131 . in the second approach the effect is attributed to a modification of the quark and gluon distributions in the nucleus . the interaction of the virtual photon with the nucleus is viewed in a fast moving system , where the nucleon with diameter @xmath23 is lorentz contracted to a disc of thickness @xmath132 and the mean nucleon distance @xmath133 to @xmath134 . the longitudinal position of its constituents , however , has an uncertainty of @xmath135 . at small values of @xmath1 this uncertainty @xmath59 can be much larger than @xmath136 and in a nucleus much larger than @xmath137 . for @xmath138 there will be a spatial overlap of sea quarks and gluons of different nucleons . the smaller @xmath1 , the larger is the number of nucleons sharing their contents of sea - quarks and gluons . the effect increases with increasing mass number @xmath104 and saturates for @xmath139 , where @xmath140 is the diameter of the nucleus in its rest frame . thus , the density of gluons and sea quarks at the position of a nucleon in a nucleus can be much larger than for a free nucleon . due to this ` overcrowding ' the probability for an interaction between sea quarks and gluons is increased and by pair annihilation their density is reduced again , resulting in the observed reduction of the nuclear structure function . momentum conservation requires that this reduction of the number of partons at low values of @xmath1 gets compensated by an enhancement at larger @xmath1 , i. e. , anti - shadowing . both approaches for shadowing are equivalent . they describe the same phenomenon but viewed in a different reference frame . more details can be found , e. g. , in refs . @xcite . in the second class of models the structure function of a nucleus @xmath22 is described as the incoherent sum over contributions of all kind of clusters @xmath141 with structure functions @xmath142 convoluted with the probability @xmath143 to find a certain cluster @xmath141 of momentum @xmath8 in the nucleus : @xmath144 examples for such clusters are the nucleon itself , undisturbed or with a reduced mass due to nuclear binding or with an increased size due to different boundary conditions in the nuclear environment , extra pions being responsible for nuclear binding , @xmath145-isobars , multi - quark clusters like bags of 6 quarks , 9 quarks or 12 quarks ( i.e. , @xmath6 particles ) or the whole nucleus as one big bag with free quark and color flow throughout the whole nuclear volume . there is a lot of freedom in these approaches , concerning as well the choice of @xmath143 as the parameterization of @xmath146 that both are badly known . consequently , it is not very surprising that they succeed to reproduce a portion of the data rather well , at least in the medium @xmath1 range . in the third class of approaches the emc effect is explained by a change of either the @xmath36 scale or the @xmath1 scale for the nuclear structure function compared to the free nucleon s one . * @xmath36 rescaling . * in @xmath36 rescaling models , first proposed in refs . @xcite and @xcite , the emc effect is related to a change of confinement size inside the nucleus . the _ a dependence _ of quark and gluon distributions for bound nucleons and the _ @xmath36 evolution _ of these distributions for free nucleons both have the same origin . they are caused by the color forces between quarks and gluons which ensure confinement and are the origin of scaling violations , i.e. , the increase of the structure function with @xmath36 at low values of @xmath1 and the decrease with increasing @xmath36 at large values of @xmath1 . the qualitative argument is as follows : the strength of the strong force between quarks is not only determined by the transverse resolution @xmath147 at which they are probed , but also by the radial extension @xmath148 of the volume in which they are confined . therefore , the relevant parameter for the strength of the strong coupling constant @xmath149 is not just @xmath36 but @xmath150 . ( this is similar to the situation in nuclear physics where the form factors for spherical nuclei have the identical oscillating pattern when plotted against @xmath151 ) . if the confinement size is modified inside the nucleus , either due to a ` swelling ' of nucleons , the formation of multi - quark bags , short - range nucleon - nucleon correlations or free quark and color flow throughout the whole nucleus , then , as a consequence , quark and gluon distributions obtained for nuclei @xmath105 and @xmath152 are related by @xmath153 @xmath154 is a rescaling parameter determined by the two confinement scales @xmath148 and @xmath155 . in the so - called dynamical rescaling models it is given by @xmath156 where @xmath157 is a low - momentum cut - off for radiating gluons . consequently , at the same value of @xmath36 , @xmath158 for small values of @xmath1 and @xmath159 for large values of @xmath1 , if @xmath160 . * @xmath1 rescaling . * in the @xmath1-rescaling models , first proposed in @xcite and then refined by numerous authors , the depletion of the nuclear structure function at medium @xmath1 is explained by conventional nuclear binding and fermi - motion corrections . the @xmath1 dependence can be reasonably well reproduced if , for a nucleus , the scaling variable @xmath1 is replaced by a modified one @xmath161 . for a nucleon @xmath162 moving with momentum @xmath163 in a nucleus the variable @xmath7 has to be replaced by @xmath164,\ ] ] where @xmath165 is the removal energy of the nucleon ( @xmath166 mev ) and @xmath167 is the momentum of the virtual photon . consequently , at the same kinematics of the scattered lepton , the effective @xmath168 at which the structure function is probed in a nucleus , is larger than @xmath1 for a free nucleon . at large @xmath1 the structure function @xmath11 is steeply falling with @xmath1 and a depletion of the structure function ratios is naturally explained . recently their has been renewed interest in the emc effect and its possible origin by jlab measurements of the cross section ratios in inclusive electron scattering @xmath169 in the kinematic region @xmath57 where the cross section vanishes for scattering from free nucleons . measured by slac - e139 at @xmath170 ( left panel ) and by jlab - e02019 in the region @xmath171 ( right panel ) . the middle panel shows the correlation between the slope @xmath172 in the region @xmath173 and the height of the plateau of @xmath174 in the region @xmath175 ( after @xcite ) . ] the clas experiment @xcite has measured @xmath176 for @xmath48he , @xmath67c and @xmath71fe and jlab - e02019 @xcite measured @xmath47 for @xmath80he , @xmath48he , be , c , cu and au . in the right panel of fig . [ ab : src ] the cross section ratios @xmath48he / d , c / d and cu / d are shown as a function of @xmath1 . they rise with @xmath1 until they reach a plateau at @xmath177 . the height of this plateau increases with @xmath104 . such a behavior has already been observed with less accuracy by an experiment at slac @xcite . a similar pattern is seen in the clas comparison of the nuclear cross sections to @xmath80he . here the measurements extend up to @xmath178 and an indication of a second plateau is observed for @xmath179 . such a behavior is being interpreted as the manifestation of short - range nucleon - nucleon ( mostly p n ) correlations ( three - nucleon correlations for @xmath1 beyond 2 ) or , in another approach , as the ratio of the probabilities to find 6-quark or 9-quark clusters in nuclei compared to the reference nucleus . fig . [ ab : src ] shows a very interesting observation @xcite . in the left panel the e139 cross section ratios @xmath48he / d , c / d and fe / d are shown for @xmath170 . the lines correspond to the slopes @xmath180 in the region @xmath181 , that characterize the strength of the emc effect in this region and are unaffected by overall normalization uncertainties . in the middle panel the slopes of the e139 data ( tabulated in @xcite and corrected by @xmath182 @xcite ) are plotted against the height of the plateaus @xcite . obviously there is a strong correlation between these quantities as indicated by the straight line . it is rather unlikely that this correlation is purely accidental and one can therefore rather safely assume that a large fraction of the strength of the emc effect in the valence quark region is due to short - range nucleon - nucleon correlations . the emc effect is with us now for 30 years . it has stimulated huge experimental and theoretical efforts , but its origin is still not fully understood . recent data shed some new light on its possible origin , i.e. , short - range nucleon - nucleon correlations may play an important role for the observed nuclear modifications . still more precise data are needed , especially on the nuclear gluon and antiquark distributions at very low @xmath1 to constrain the initial state for the @xmath2 program at rhic and lhc . these hopefully will come from future measurements at jlab12 , rhic and lhc and eventually also from the proposed projects eic and lhec . a. m. cooper et al . , phys . lett . * b 141 * ( 1984 ) 133 ; m. a. parker et al . , nucl . phys . * b 232 * ( 1984 ) 1 ; + v. v. ammosov et al . , jetp lett . * 39 * ( 1984 ) 393 ; j. hanlon et al . , phys . rev . * d 32 * ( 1985 ) 2441 ; + a. e. asratyan et al . , sov . j. nucl * 41 * ( 1985 ) 763 ; sov . * 43 * ( 1986 ) 380 ; + m. aderholz et al . , phys . lett . * b 173 * ( 1986 ) 211 ; j. guy et al . , z. phys . * c 36 * ( 1987 ) 337 ; + j. guy et al . , phys . lett . * b 229 * ( 1989 ) 421 ; t. kitagaki et al . , phys . lett . * b 214 * ( 1988 ) 281 .

the present status of the emc effect , the modification of the per nucleon cross section in deep - inelastic lepton nucleus scattering by the nuclear environment , is reviewed . + friedrich - alexander - universitt erlangen - nrnberg + physics department + erwin - rommel - str . 1 + d-91058 erlangen , germany + e - mail : klaus.rith@physik.uni-erlangen.de

superallowed @xmath1 nuclear @xmath2 decay currently provides the most precise value for @xmath9 , the up - down element of the cabibbo - kobayashi - maskawa ( ckm ) matrix @xcite . this element is the key ingredient of the most demanding available test of ckm - matrix unitarity , a fundamental requirement of the electroweak standard model . to extract @xmath9 from the experimental data , small theoretical corrections of order @xmath101% must be applied to take account of unobserved radiative effects as well as the isospin symmetry - breaking that occurs between the analog parent and daughter states of each superallowed transition @xcite . even though these corrections are very small , experimental measurements have by now reached such high precision that the uncertainty on @xmath9 ( @xmath110.03% ) is currently dominated not by experiment but by the uncertainty on these theoretical corrections . in the determination of @xmath9 , an important strength of the nuclear measurements is that there are many @xmath1 transitions available for study , and currently there are thirteen of them , ranging from @xmath12c to @xmath13rb , that have been measured with high precision . with so many , it becomes possible to validate the analysis procedure by checking that all transitions individually yield statistically consistent results for @xmath9 . since the isospin - symmetry - breaking corrections depend on nuclear structure , they differ from transition to transition and are particularly sensitive to this consistency test . thus the appearance of an anomalous result from any transition could signal a problem with the structure - dependent correction for that case , a problem which might have implications for other cases as well . in the most recent survey of superallowed @xmath1 transitions , which appeared in 2005 @xcite , the results for all precisely measured cases there were twelve at that time were statistically consistent with one another . today , there are thirteen such cases and they still form a statistically consistent ensemble overall . however , recent precise penning - trap measurements @xcite of the @xmath14 value for the superallowed decay of @xmath7v have left the result for that transition more than two standard deviations away from the average of all other well - known transitions . this possible anomaly led us initially to reexamine the isospin - symmetry - breaking corrections for the @xmath7v transition , but what we learned from that reexamination prompted us to a more general reevaluation of the corrections for other transitions as well . our previous shell - model calculations for @xmath7v considered six valence nucleons occupying the @xmath15-shell orbitals outside a @xmath16ca closed shell . this model space generated reasonable energies and spins for the known states in @xmath7ti , the daughter of @xmath7v . however , an important part of the charge - dependent correction depends on the radial mismatch between the decaying proton in the parent nucleus and the resulting neutron in the daughter nucleus ; but both these nucleons are bound to @xmath17ti , so the structure of that nucleus turns out to be important too . what is most striking about @xmath17ti is that it has a @xmath18 state at an excitation energy of only 330 kev , which is strongly populated in single - nucleon pick - up reactions like @xmath19 and ( @xmath20he,@xmath21 ) . such low - lying @xmath22-shell states can contribute to the structural parentage of the initial and final states of the superallowed transition and consequently must affect the radial mismatch between them . this indicated to us that a complete calculation of the isospin - symmetry - breaking correction for the decay of @xmath7v should include contributions from shells deeper than the @xmath15 shell . two questions then arose . how many deeper shells need to be included and , if this effect is important for @xmath7v decay , how many other transitions will be similarly affected ? in section [ s : isbc ] of this paper , we address these questions and settle on criteria for including deeper shells . using these criteria and incorporating more recent effective interactions that have become available since our last work we then re - evaluate the isospin - symmetry - breaking corrections for all transitions of relevance to the study of superallowed @xmath23 decay . for the cases with @xmath24 the changes in the corrections are very small typically 0.03% but for the heavier nuclei the changes can be as large as 0.2% . most significantly , with the new calculated corrections , the result for @xmath7v is no longer anomalous . in section [ s : radc ] , we incorporate recent improvements made by marciano and sirlin @xcite to the calculation of the radiative corrections for superallowed decays and then in section [ s : ft ] we apply both types of corrections isospin - symmetry - breaking and radiative to the current experimental data for superallowed decays . the result for @xmath9 is changed appreciably , although it is still within quoted uncertainties of its old value , and the ckm - unitarity sum is improved . superallowed fermi beta decay between @xmath25 states depends uniquely on the vector part of the hadronic weak interaction . when it occurs between isospin @xmath26 analog states , the conserved vector current ( cvc ) hypothesis indicates that the @xmath27 values should be the same irrespective of the nucleus , _ viz . _ ft = = const , [ ftconst ] where @xmath28 gev@xmath29s ; @xmath30 is the vector coupling constant for semi - leptonic weak interactions ; and @xmath31 is the fermi matrix element . the cvc hypothesis asserts that the vector coupling constant , @xmath32 , is a true constant and not renormalised to another value in the nuclear medium . in practice , eq.([ftconst ] ) has to be amended slightly . firstly , there are radiative corrections because , for example , the emitted electron may emit a bremsstrahlung photon that goes undetected in the experiment . secondly , isospin is not an exact symmetry in nuclei so the nuclear matrix element , @xmath31 , is slightly reduced from its ideal value , leading us to write : |m_f|^2 = |m_0|^2 ( 1 - _ c ) , [ mf2 ] where @xmath33 is the exact - symmetry value , which for @xmath34 states is @xmath35 . thus , we define a corrected " @xmath6 value as t ft ( 1 + _ r ) ( 1 - _ c ) = = const , [ ftconst ] where @xmath0 is the isospin - symmetry - breaking correction , @xmath36 is the transition - dependent part of the radiative correction , and @xmath4 is the transition - independent part . fortunately these corrections are all of order 1% but , even so , to maintain an accuracy criterion of 0.1% they must be calculated with an accuracy of 10% of their central value . this is a demanding request , especially for the nuclear - structure - dependent corrections . to separate out those terms that are dependent on nuclear structure from those that are not , we split the transition - dependent radiative correction into two terms , _ r = _ r^ + _ ns , [ dr ] of which the first , @xmath5 , is a function only of the electron s energy and the charge of the daughter nucleus @xmath37 ; it therefore depends on the particular nuclear decay , but is _ independent _ of nuclear structure . the second term , @xmath3 , like @xmath0 , depends in its evaluation on the details of nuclear structure . to emphasize the different sensitivities of the correction terms , we rewrite the expression for @xmath6 as t ft ( 1 + _ r^ ) ( 1 + _ ns - _ c ) = , [ ftfactor ] where the first correction in brackets is independent of nuclear structure , while the second incorporates the structure - dependent terms . from eq.([ftfactor ] ) it can be seen that a measurement of any one superallowed transition establishes a single value for @xmath32 ; moreover , measurements of many transitions provides an excellent test of the validity of the whole analysis . since cvc requires a unique value of @xmath32 , all the extracted @xmath6-values should be identical within experimental uncertainties . the @xmath27-value that characterizes any @xmath2-transition depends on three measured quantities : the total transition energy , @xmath14 ; the half - life , @xmath38 , of the parent state ; and the branching ratio , @xmath39 , for the particular transition of interest . the @xmath14-value is required to determine the statistical rate function , @xmath40 , while the half - life and branching ratio combine to yield the partial half - life , @xmath41 . in 2005 we published a new survey of world data on superallowed @xmath42 beta decays @xcite . all previously published measurements were included , even those that were based on outdated calibrations if enough information was provided that they could be corrected to modern standards . in all , more than 125 independent measurements of comparable precision , spanning four decades , made the cut . in the two years since the survey was closed another ten relevant publications have appeared @xcite and we have now incorporated these results into our data base . based on these data for the thirteen most precisely known transitions , we obtain the @xmath27 values shown on the left side of figure [ fig1 ] ; then , by incorporating the corrections calculated by us in 2002 @xcite and used in our 2005 survey @xcite , we obtain the corrected @xmath6 values plotted on the right side of the figure . obviously the calculated corrections do a remarkable job eliminating the considerable scatter that is evident in the @xmath27-value plot on the left but is absent in the corrected @xmath6 values shown on the right . overall , the statistical agreement among the @xmath6 values is quite satisfactory , the normalized @xmath43 being 0.8 . thus , considering that the correction terms were evaluated completely independently of these data , the consistency among the @xmath6 values can be taken as strong evidence that the correction terms are , in general , soundly based . however , there is a small but noticeable deviation from the average at @xmath7v ( and possibly @xmath44sc ) , which has only been revealed by the recent penning - trap measurements @xcite of the transition @xmath14 values . though its statistical significance appears rather marginal in the figure , it must be remarked that the uncertainties quoted on these @xmath6 values have been very conservatively determined . the measured data for each input parameter @xmath14-value , half - life and branching ratio were separately evaluated @xcite and , if the measurements were inconsistent with one another , the weighted - average uncertainty for that parameter was increased to account for that inconsistency . in effect , for such cases , the original uncertainties quoted with the published measurements were all increased by a common scale factor " that was large enough to restore statistical consistency among the measurements . ( these scale factors are tabulated for each parameter in ref.@xcite ; they range from 1 to 3.6 . ) this method , which is also used by the particle data group @xcite , leads to final average values that have a high confidence level but it does so at the cost of producing uncertainties that are in many cases larger than would result from a strict statistical average . with this method of analysis in mind , the excursion of the @xmath7v @xmath6 value can not be entirely ignored as a possible signal that the nuclear - structure - dependent corrections in this mass region are deficient . it certainly proved to be sufficiently provocative that we were led to the reevaluation of correction terms that is reported here . for weak vector interactions in hadron states , the cvc hypothesis protects the decay amplitudes from strong - interaction corrections . however , there is a caveat . the cvc hypothesis also requires the hadron state to be an exact eigenstate of @xmath45 symmetry ( isospin ) . in nuclei , @xmath45 is always broken , albeit weakly , by coulomb interactions between protons . there may be other charge - dependent effects as well . these influences shift the value of the hadron matrix element from its exact symmetry limit to a new value and this shift has to be evaluated before weak - interaction physics can be probed with hadrons . in the case of superallowed @xmath2 decay , the hadron matrix element , @xmath31 , is given by eq.([mf2 ] ) and it is @xmath0 that we seek to evaluate . in the shell model for the cases of interest here , the @xmath46-particle wave functions representing the initial and final states for superallowed @xmath2 decay , @xmath47 and @xmath48 , are states of angular momentum zero and isospin one . in a second quantisation formulation , the fermi matrix element is written m_f = f | _ + | i = _ , f @xmath2 decay is the isospin ladder operator , @xmath49 creates a neutron in quantum state @xmath21 and @xmath50 annihilates a proton in quantum state @xmath2 . the single - particle matrix element , @xmath51 , is just a radial integral | _ + | = _ , _ 0^ r_^n(r ) r_^p(r ) r^2 dr _ , r_. [ radi ] if the proton and neutron radial functions @xmath52 and @xmath53 are identical , then the radial integral reduces to the normalization integral and has the value @xmath54 . now we introduce into eq.([mf2q ] ) a complete set of states for the @xmath55-particle system , @xmath56 , by writing m_f = _ , f | a_^ | | a _ [ mfpar ] this is the essence of our model : we have allowed the radial integral to depend on the parentage expansion . thus , we have added an additional label to @xmath57 and now write @xmath58 . if isospin is an exact symmetry , then the matrix elements of the creation and annihilation operators are related by hermiticity , @xmath59 . with that requirement , and with the radial integrals set to unity , the symmetry - limit matrix element is m_0 = _ , | f | a_^ | |^2 . [ m0 ] thus we see that the breakdown of isospin symmetry can enter the evaluation of @xmath31 in one of two ways : either the matrix elements of @xmath60 and @xmath49 are not related by hermiticity , or the radial integrals are not unity . since each effect is small , we can , to first order , write the isospin - symmetry breaking correction as the sum of two terms _ c = _ c1 + _ c2 [ dc1and2 ] where in evaluating @xmath61 all radial integrals are set to unity but the matrix elements are not assumed to be related by hermiticity , while in evaluating @xmath62 it is assumed that @xmath59 but the radial integrals are allowed to differ from unity . past calculations @xcite have indicated the radial overlap correction , @xmath62 , is the larger of the two corrections so we will study this first . for the @xmath62 calculation , the fermi matrix element is m_f & = & _ , | f | a_^ | |^2 r_^ + & = & _ , |f | a_^ | |^2 - _ , |f | a_^ | |^2 ( 1 - r_^ ) + & = & m_0 ( 1 - _ , |f | a_^ | |^2 _ ^ ) [ mf1 ] where @xmath33 is the exact - symmetry value , eq.([m0 ] ) , and @xmath63 has been introduced as a radial - mismatch factor _ ^ = ( 1 - r_^ ) . [ rmf ] recalling that @xmath62 is defined as @xmath64 we obtain _ c2 _ , |f contribution to @xmath62 therefore requires a large spectroscopic amplitude and a significant departure of the radial integral from unity . there is an opportunity here to take guidance from experiment . the square of each spectroscopic amplitude , @xmath65 , is related to the spectroscopic factor measured in neutron pick - up direct reactions . the exact relation , after inserting the isospin angular momentum couplings , is _ c2 _ , s _ , t_f^t _ _ ^ [ dc2_2 ] where @xmath66 is the spectroscopic factor for pick up of a neutron in quantum state @xmath21 from an @xmath46-particle state of isospin @xmath67 to an @xmath55-particle state of isospin @xmath68 . on setting @xmath69 and separately identfying sums to the isospin - lesser states with @xmath70 , denoted @xmath71 , and the isospin - greater states with @xmath72 , denoted @xmath73 , we obtain a very revealing formula _ c2 _ ^ < , s_^ < _ ^ > , s_^ > [ dc2_3 ] ccccccccc & & & & & & & & + & & & & & & + + & & & & & & & & + & & & & & & & & + & & & & & & & & + 0 & @xmath74 & @xmath75 & 2.7(11 ) & 0.134 & 3.33 & 0.45 & 3.36 & 0.45 + 330 & @xmath76 & @xmath77 & 1.9(8 ) & 0.157 & 2.67 & 0.42 & 2.45 & 0.39 + 1566 & @xmath78 & @xmath79 & 0.7(3 ) & 0.318 & 1.33 & 0.42 & 1.22 & 0.39 + 4723 & @xmath80 & @xmath75 & 3.6(16 ) & 0.085 & 2.67 & @xmath81 & 2.74 & @xmath82 + 4810 & @xmath83 & @xmath77 & 3.6(16 ) & 0.100 & 5.33 & @xmath84 & 4.92 & @xmath85 + 5760 & @xmath86 & @xmath79 & 3.2(12 ) & 0.224 & 2.67 & @xmath87 & 2.47 & @xmath88 + this equation provides the key to the strategy we will use in calculating @xmath62 . it demonstrates that there is a cancellation between the contributions of the isospin - lesser states and the isospin - greater states . moreover , if the orbital @xmath21 were completely full in the initial @xmath46-particle wavefunction , then the macfarlane and french sum rules @xcite for spectroscopic factors would require @xmath89 = @xmath90 and the cancellation in eq.([dc2_3 ] ) would be very strong . in fact , the cancellation would be complete if @xmath91 . as we will discuss further in the next section , this cancellation is not in general complete because the radial - mismatch factors for isospin - lesser states are larger than those for isospin - greater states . even so , cancellation is always significant , and it becomes most complete when closed - shell orbitals are involved . furthermore , the more deeply bound the closed - shell orbital , the greater the energy spread in the spectroscopic strength and the more complete the cancellation . thus , although the dominant contributions to @xmath62 come from unfilled orbitals , we conclude that closed - shell orbitals must play a role , albeit one that decreases in importance as the orbitals become more deeply bound . based on these observations , our strategy is to use experiment to guide us in determining which closed - shell orbitals are important enough to include . ideally , of course , one would take the spectroscopic factors determined from experiment and insert them into eq.([dc2_3 ] ) but , especially where delicate cancellations are involved , the reliability of ( forty - year - old ) experimental spectroscopic factors is certainly not up to the task . our strategy then is to use the shell model to calculate the spectroscopic amplitudes in eq.([dc2_1 ] ) but to limit the sum over orbitals @xmath21 just to those for which large spectroscopic factors have been observed in neutron pick - up reactions . we illustrate the strategy for the case of @xmath7v . the spectroscopic factors for neutron pick up from @xmath7ti have been measured in the @xmath92he@xmath93 reaction by borlin @xcite . he identified sixteen states in @xmath17ti , and in table [ t : sfactor ] we record the six states with the largest spectroscopic factors , _ i.e. _ @xmath94 . we note that the errors on the experimental spectroscopic factors are quite large , and in two cases the quoted @xmath95 values ( column 4 ) exceed the macfarlane - french sum rule @xcite for pure configurations ( column 6 ) . thus we do not use the experimental spectroscopic factor explicitly , but take them as a guide for which orbitals should be included in the shell - model calculation . in the case of @xmath7v decay , they tell us that orbitals @xmath75 , @xmath77 and @xmath79 should be included . in column five of table [ t : sfactor ] we give a typical value for the radial mismatch factor , @xmath63 , for the given orbital @xmath21 and isospin @xmath68 . column seven gives the contribution to @xmath62 from this @xmath21 and isospin @xmath68 if the macfarlane - french sum rule is used for the spectroscopic factor , while in columns eight and nine are shown the results of a detailed shell - model calculation . the results from the macfarlane - french sum rules and the shell - model calculation are remarkably similar . the summed @xmath62 for the shell - model calculation ( the sum of all entries in column 9 ) is @xmath96 , nearly a factor of two larger than our previous calculated value , which was published in 2002 @xcite . the difference between our calculations arises as follows : in 2002 our shell - model calculations for @xmath7v were based on the model space @xmath97 , with six valence nucleons occupying the @xmath15-shell orbitals . in fact , only the @xmath75 orbital contributed importantly to the @xmath62 calculation so the result was @xmath98 ( see the two rows for the @xmath75 orbital in table [ t : sfactor ] ) . absent from this 2002 calculation was any contribution from the core orbitals , @xmath77 and @xmath79 . in our present calculations , these orbitals are included , with the @xmath77 orbital contributing 0.14 % to @xmath62 and the @xmath79 contributing 0.11 % . but why stop there ? why not include the @xmath99 and possibly the @xmath100-shell orbitals in the computation ? our answer is that the neutron pick - up measurement saw little or no evidence for such core states , which implies that their spectroscopic strength is distributed widely over many states . in this case , the cancellation between isospin - lesser and isospin - greater states becomes more complete and their contribution to @xmath62 is reduced to a level that we believe can be neglected . with this approach , we are now in a position to revise our earlier results @xcite to include the effects of previously ignored core orbitals . again using measured spectroscopic factors from neutron pick - up reactions , we determined that changes were required for the @xmath101 and @xmath102 cases , in which @xmath100-shell holes must contribute in addition to the original @xmath22-shell configurations ; similarly , @xmath22-shell holes were required in addition to the @xmath15-shell particles for @xmath103 and @xmath104 . for @xmath105 and @xmath106 in the upper @xmath15-shell there are no experimental neutron pick - up reaction measurements to guide us . our previously published calculations for these nuclei were based on @xmath107 model spaces using @xmath108ni as a closed - shell core . it seemed prudent now for these cases at least to include the @xmath75 orbital in the calculation of @xmath62 , and we have made this change . in the cases with @xmath46 = 18 and 42 , we had previously included some contribution from deeper shells ; we did not need to make any changes in the former but did add the @xmath79 and @xmath99 shells to the latter . no additional orbitals were required for the cases with @xmath109 , 34 and 38 . in considering the radial integrals , we benefit from a very strong constraint : the asymptotic forms of all radial functions must match the measured separation energies , @xmath110 and @xmath111 , where @xmath110 is the proton separation energy in the decaying nucleus and @xmath111 the neutron separation energy in the daughter nucleus . the basic ingredients of these separation energies are well known and can be found in any atomic mass tables . it is the size of the difference between @xmath110 and @xmath111 and the presence or absence of nodes in the radial wave functions that are the principal factors in determining the magnitude of @xmath63 . our calculations of this mismatch factor follows the same path as that described in our earlier works @xcite . we use a saxon - woods potential defined for a nucleus of mass @xmath46 and charge @xmath112 as : v(r ) = - v_0 f(r ) - v_s g(r ) * l*. + v_c(r ) - v_g g(r ) - v_h h(r ) , [ vsw ] where f(r ) & = & \ { 1 + ( ( r - r)/a ) } ^-1 , + g(r ) & = & ( ) ^2 ( ) + & & \ { 1 + ( ) } ^-2 , + h(r ) & = & a^2 ( ) ^2 , + v_c(r ) & = & z e^2 / r , for r r_c + & = & ( 3 - ) , for r < r_c , [ pot ] with @xmath113 and @xmath114 . the first three terms in eq.([vsw ] ) are the central , spin - orbit and coulomb terms respectively . the fourth and fifth terms are additional surface terms whose role we discuss shortly . most of the parameters were fixed at standard values , @xmath115 mev , @xmath116 fm and @xmath117 fm . the radius of the coulomb potential was determined from the charge mean square radius , @xmath118 , of the decaying nucleus as determined from elastic electron scattering ; see eqs.(21 ) and ( 22 ) in ref.@xcite . the well radius , @xmath119 , was similarly fixed , by requiring that the charge density constructed from the square of the proton wave functions bound in the well should also match the charge mean square radius . initially , with @xmath120 and @xmath121 set to zero , the well depth , @xmath122 , was adjusted so that the binding energy of the least - bound orbital matched the experimental separation energy . from the shell model calculation , we obtained the @xmath46-particle wave functions , @xmath47 and @xmath48 , expanded into products of @xmath55-particle wave functions @xmath56 and single - particle functions @xmath123 . in eq.([mfpar ] ) and the discussion that followed it , we noted that the radial integral should depend on the separation energies relative to the @xmath55 state , @xmath124 . we ultimately allowed this to happen but initially we calculated the value of @xmath62 under the assumption that the proton and neutron radial functions , @xmath125 and @xmath126 , have asymptotic forms for all @xmath127 that are fixed at the separation energies , @xmath110 and @xmath111 , to the ground state of the @xmath55 nucleus . in this case , the sums over @xmath128 can be done analytically and the computed value of @xmath62 becomes independent of the shell - model effective interaction . this result , which we label @xmath129 , can be simply expressed with the help of eqs.([m0 ] ) and ( [ dc2_1 ] ) : _ c2^i 2 _ _ g . [ dc2i ] here @xmath130 is the shell - model orbital of the transferred neutron in the pick - up reaction from the @xmath46-particle state @xmath131 to the ground state of the @xmath55-particle nucleus . we next removed our simplifying assumption and evaluated the radial integrals with eigenfunctions of the saxon - woods potential whose well depth was adjusted so that each eigenfunction matched the separation energy of the @xmath55 state to which it corresponds , @xmath124 . for an @xmath55 state at excitation energy @xmath132 the corresponding separation energies are @xmath133 and @xmath134 . we label these results @xmath135 and note that the values now depend on the spectroscopic amplitudes , and hence on the shell - model effective interaction , but not strongly . so far , we have ignored the two surface terms in eq.([pot ] ) by setting @xmath136 and @xmath137 . it can be argued , however , that the central part of the potential , which in principle should be determined from some hartree - fock procedure , should not be continually adjusted . instead , any adjustments made to match separation energies should be to the surface part of the potential rather than to the depth of the well . thus , we also calculated @xmath62 by fixing @xmath122 separately for protons and neutrons to match the ground - state parent separation energies , @xmath110 and @xmath111 , and then adjusting the strength of the surface term , @xmath120 ( keeping @xmath137 ) so that the asymptotic forms matched the separation energies @xmath133 and @xmath134 . these results are labelled @xmath138 . finally , our fourth method of calculation was the same as the third , except that it was the second surface term , @xmath121 , that was adjusted to match separation energies , keeping @xmath136 . this second term , @xmath139 , is even more strongly peaked in the surface than @xmath140 . these results are labelled @xmath141 . on average , the method iii values of @xmath62 are about 2% lower than the method ii values ; and method iv values are about 7% lower than the method ii values for orbitals without any radial nodes . for orbitals with one or more nodes , there is more of the radial wave function in the surface region and methods iii and iv produce greater reductions . we now present our results for @xmath62 based on the extensions of the shell - model spaces mentioned at the end of sect . [ sss : st ] . in addition to adding the core orbitals mentioned there , however , in some cases we have also been able to make use of more recent effective interactions that have become available since our last work . specifically , we have used the following interactions in the various mass regions of interest : in the @xmath100-shell , we use the cohen - kurath interactions @xcite and the more recent pwbt interaction of warburton and brown @xcite . in the @xmath142-shell , besides the universal interaction of wildenthal @xcite , we employ two new versions , usd - a and usd - b , of brown and richter @xcite . in the @xmath15-shell we use the kb3 interaction of kuo - brown @xcite as modified by poves and zuker @xcite , the fpmi3 interaction of richter and brown @xcite , and the more recent gxpf1 interaction of honma @xcite . for cross - shell interactions between the major shells , we have used the interaction of millener and kurath @xcite . it should be noted that in many cases we found it necessary to introduce some truncations in the original model space in order to keep the calculations tractible . lrrrrrrr & & & & & & + & & + + & & & & & & + & & & & & & + & & & & & & + & & & & & + @xmath12c & 0.170(15 ) & 0.132 & 0.163 & 0.165 & 0.163 & 0.165(15 ) + @xmath143o & 0.270(15 ) & 0.217 & 0.274 & 0.271 & 0.271 & 0.275(15 ) + @xmath144ne & 0.390(10 ) & 0.251 & 0.386 & 0.387 & 0.382 & 0.410(25 ) + @xmath145 mg & 0.255(10 ) & 0.207 & 0.366 & 0.382 & 0.375 & 0.370(20 ) + @xmath146si & 0.330(10 ) & 0.223 & 0.421 & 0.407 & 0.392 & 0.405(25 ) + @xmath147s & 0.740(20 ) & 0.812 & 0.714 & 0.710 & 0.713 & 0.700(20 ) + @xmath148ar & 0.610(40 ) & 0.351 & 0.680 & 0.639 & 0.579 & 0.635(55 ) + @xmath149ca & 0.710(50 ) & 0.402 & 0.840 & 0.784 & 0.702 & 0.745(70 ) + @xmath44ti & 0.555(40 ) & 0.359 & 0.881 & 0.849 & 0.780 & 0.835(75 ) + & & & & & + @xmath146al@xmath150 & 0.230(10 ) & 0.156 & 0.292 & 0.280 & 0.271 & 0.280(15 ) + @xmath148cl & 0.530(30 ) & 0.312 & 0.583 & 0.561 & 0.498 & 0.550(45 ) + @xmath149k@xmath150 & 0.520(40 ) & 0.299 & 0.623 & 0.575 & 0.522 & 0.550(55 ) + @xmath44sc & 0.430(30 ) & 0.278 & 0.681 & 0.648 & 0.606 & 0.645(55 ) + @xmath7v & 0.330(25 ) & 0.273 & 0.587 & 0.543 & 0.506 & 0.545(55 ) + @xmath151mn & 0.450(30 ) & 0.315 & 0.638 & 0.598 & 0.594 & 0.610(50 ) + @xmath152co & 0.570(40 ) & 0.376 & 0.760 & 0.688 & 0.706 & 0.720(60 ) + @xmath153ga & 1.05(15 ) & 1.31 & 1.22 & 1.19 & 1.14 & 1.20(20 ) + @xmath154as & 1.15(15 ) & 1.32 & 1.41 & 1.34 & 1.24 & 1.35(40 ) + @xmath155br & 1.00(20 ) & 1.43 & 1.41 & 1.31 & 1.10 & 1.25(25 ) + @xmath13rb & 1.30(40 ) & 1.68 & 1.60 & 1.47 & 1.12 & 1.50(30 ) + we made calculations for all twenty superallowed transitions considered in our earlier work @xcite , and for each we calculated @xmath62 in the four methods , i - iv , described in sect.[sss : rmf ] and with the several interactions listed in the previous paragraph . in table [ t : dc2 ] we record only _ one sample result _ for @xmath129 , @xmath135 , @xmath138 and @xmath141 for each nucleus listed . however , our adopted @xmath62 " values result from our assessment of _ all _ multiple - parentage calculations made for each decay , not just those shown in the previous three columns . the uncertainty assigned to each adopted value reflects the uncertainty in the radius of the saxon - woods potential ( resulting from an uncertainty in the nuclear rms radius to which it is adjusted ) , the spread of results obtained with different shell - model interactions , and the spread of results obtained with the different procedures labelled ii , iii and iv in the table . the second ( and smaller ) contribution to @xmath0 is the isospin - mixing correction , @xmath61 . for its evaluation , the radial integrals are all set to unity , but the spectroscopic amplitudes in eq.([mfpar ] ) are not required to satisfy hermiticity . calculations of this correction turn out to be very sensitive to the details of the shell - model computation . this would be a very unfortunate property if we were not able to adopt certain strategies that act to reduce the model dependence considerably . lcccccccc + & & & & & & & & + & & 2002 & + + parent & & & & & & & + + nucleus & & & & & & & & + & & & & & & & & + @xmath156 : & & & & & & & & + @xmath12c & @xmath157 & @xmath158 & @xmath159 & @xmath160 & 9.24 & @xmath161 & @xmath162 & @xmath159 + @xmath143o & @xmath163 & @xmath164 & @xmath165 & @xmath166 & 6.64 & @xmath167 & @xmath168 & @xmath169 + @xmath144ne & @xmath170 & @xmath171 & @xmath172 & @xmath173 & 4.07 & @xmath174 & @xmath175 & @xmath176 + @xmath145 mg & @xmath177 & @xmath178 & @xmath159 & @xmath179 & 6.21 & @xmath180 & @xmath180 & @xmath159 + @xmath146si & @xmath181 & @xmath182 & @xmath183 & @xmath184 & 3.86 & @xmath185 & @xmath186 & @xmath187 + @xmath147s & @xmath188 & @xmath189 & @xmath190 & @xmath191 & 3.80 & @xmath192 & @xmath193 & @xmath194 + @xmath148ar & @xmath195 & @xmath196 & @xmath187 & @xmath197 & 3.97 & @xmath198 & @xmath198 & @xmath187 + @xmath149ca & @xmath199 & @xmath200 & @xmath201 & @xmath202 & 3.21 & @xmath203 & @xmath198 & @xmath201 + @xmath44ti & @xmath204 & @xmath205 & @xmath206 & @xmath207 & 3.16 & @xmath208 & @xmath209 & @xmath210 + @xmath211 : & & & & & & & + @xmath146al@xmath150 & @xmath181 & @xmath182 & @xmath183 & @xmath212 & 3.86 & @xmath213 & @xmath214 & @xmath187 + @xmath148cl & @xmath215 & @xmath196 & @xmath216 & @xmath197 & 3.97 & @xmath217 & @xmath218 & @xmath219 + @xmath149k@xmath150 & @xmath220 & @xmath200 & @xmath210 & @xmath202 & 3.21 & @xmath221 & @xmath222 & @xmath223 + @xmath44sc & @xmath204 & @xmath205 & @xmath224 & @xmath225 & 5.05 & @xmath226 & @xmath227 & @xmath201 + @xmath7v & @xmath228 & @xmath229 & @xmath230 & @xmath231 & 4.86 & @xmath232 & @xmath233 & @xmath234 + @xmath151mn & @xmath235 & @xmath236 & @xmath237 & @xmath238 & 3.62 & @xmath239 & @xmath240 & @xmath241 + @xmath152co & @xmath242 & @xmath243 & @xmath244 & @xmath245 & 2.26 & @xmath246 & @xmath247 & @xmath248 + @xmath153ga & @xmath249 & @xmath250 & @xmath251 & @xmath252 & 2.32 & @xmath253 & @xmath254 & @xmath255 + @xmath154as & @xmath256 & @xmath257 & @xmath258 & @xmath259 & 1.89 & @xmath260 & @xmath261 & @xmath262 + @xmath155br & @xmath263 & @xmath264 & @xmath265 & @xmath266 & 2.05 & @xmath267 & @xmath268 & @xmath265 + @xmath13rb & @xmath269 & @xmath270 & @xmath271 & @xmath272 & 0.523 & @xmath273 & @xmath274 & @xmath275 + + + + + there are three ways in which we incorporated charge dependence in our shell - model calculation . first , the single - particle energies of the proton orbits were shifted relative to those of the neutrons . the amount of shift was determined from the spectrum of single - particle states in the closed - shell - plus - proton versus the closed - shell - plus - neutron nucleus , where the closed shell was taken to be the nucleus used as a closed - shell core in the shell - model calculation . we took these single - particle shifts from experiment and did not adjust them . second , we added a two - body coulomb interaction among the valence protons and adjusted its strength so that the measured @xmath276-coefficient of the isobaric multiplet mass equation ( imme ) was exactly reproduced . third , we introduced a charge - dependent nuclear interaction by increasing all the @xmath34 proton - neutron matrix elements by about @xmath277 relative to the neutron - neutron matrix elements . the precise amount of this increment was determined by requiring agreement with the measured @xmath278-coefficient of the imme . this strategy of constraining the charge - dependence in the effective interaction by requiring it to reproduce the coefficients of the imme was adopted from the work of ormand and brown@xcite . experimental data were used in one more way to constrain our calculations . if isospin were an exact symmetry , then the parent @xmath25 ( @xmath26 ) state would decay exclusively to its analog state in the daughter nucleus . beta transitions to all other @xmath25 states in the daughter would be strictly forbidden . but , with isospin symmetry broken , weak transitions ( with branching ratios measured in parts per million ) can occur to these other @xmath25 states . in this case , we write the fermi matrix element squared to the @xmath279 non - analog @xmath25 state as |m_f^n|^2 = 2 _ c1^n [ mfn ] and the reduction in the analog transition fermi matrix element squared as |m_f|^2 = 2 ( 1 - _ c1 ) , [ mf0 ] neglecting , in this context , the contribution of @xmath62 . if all the @xmath25 states of a given model space had the same @xmath34 isospin designation , then the effect of isospin - symmetry breaking terms in the hamiltonian would be to deplete the analog - transition strength by an amount that is exactly matched by the sum of the strengths to the non - analog states : _ i.e. _ _ c1 _ n _ c1^n . [ dc1sum ] in practice , with large shell - model calculations the @xmath25 states in the model space will include some states whose isospin designation is not @xmath34 ; and eq.([dc1sum ] ) is not then exactly correct . nevertheless , it remains approximately true . significantly , in many cases the bulk of the analog state depletion shows up in a single excited @xmath25 state , usually ( but not always ) the first excited one . this allows us once again to use experiment to constrain and refine our calculation . in the limit of only two - state mixing , perturbation theory would indicate that _ c1 [ propde2 ] where @xmath280 is the energy separation of the analog and non - analog @xmath25 states . again , this is not an exact result , but it does highlight the importance of the shell - model hamiltonian producing a good quality spectrum of @xmath25 states with , in particular , the first excited non - analog @xmath25 state calculated to have an excitation energy close to its experimental value state . in these cases we optimized the agreement between theory and experiment for the excitation energy of that state ] . this is not always possible to achieve in the shell model , especially near closed shells where excited @xmath25 states tend to exhibit strong deformations . we used two strategies to bring the calculation into line with experimental information . our first was to adjust the centroids of the shell - model hamiltonian matrix elements specifically to get the excited @xmath25 state at about the right energy . our second was to scale our calculated @xmath61 value by a factor @xmath281 , the ratio of the square of the excitation energy of the first excited @xmath25 state in the model calculation to that known experimentally . we list in table [ t : dc1new ] the experimental values @xcite of the imme coefficients , @xmath276 and @xmath278 , and the known excitation energy @xmath282 of the first ( or second ) excited @xmath25 state in the daughter nuclei . as explained , all our shell - model calculations were adjusted to reproduce exactly the values of @xmath276 and @xmath278 , and to match , as closely as possible the excitation energy of the excited @xmath25 state . we compensated for any remaining discrepancies between the calculated and experimental values of @xmath282 by scaling the results for @xmath61 . as in table [ t : dc2 ] , we give ( in columns 68 ) the results from _ one sample calculation _ for each nucleus . then in column nine we present adopted @xmath61 values that result from our assessment of the results of _ all _ calculations made for each decay , not just the ones shown in columns 68 ; the uncertainties were chosen to encompass the spread in the results from those calculations and to include the uncertainty in the imme @xmath276 and @xmath278 coefficients . for comparison , in column 4 we list the values we adopted for @xmath61 in 2002@xcite . our strategies have remained unchanged , but here we have additionally used some more recent shell - model effective interactions as listed in sect . [ sss : smc ] . in nearly all cases , the new values of @xmath61 agree with the old values within their stated uncertainties . lccccc & & & & & + parent & 2002 & & + + nucleus & & & & & + & & & & & + & & & & + @xmath149k@xmath150 & @xmath283 & @xmath284 & @xmath285 & @xmath286 & @xmath287 $ ] + @xmath44sc & @xmath288 & @xmath226 & @xmath289 & @xmath290 & @xmath291 $ ] + @xmath7v & @xmath292 & @xmath293 & @xmath294 & @xmath295 & @xmath296 $ ] + @xmath151mn & @xmath241 & @xmath297 & @xmath298 & @xmath299 & @xmath300 $ ] + @xmath152co & @xmath299 & @xmath301 & @xmath302 & @xmath165 & @xmath303 $ ] + @xmath153ga & @xmath304 & @xmath305 & @xmath306 & @xmath307 & @xmath308 $ ] + @xmath154as & @xmath309 & @xmath310 & @xmath311 & @xmath248 & + @xmath155br & @xmath312 & @xmath313 & @xmath314 & @xmath315 & + @xmath13rb & @xmath316 & @xmath317 & @xmath318 & 0.050(30 ) & @xmath319 $ ] + for the heavier nuclei there are experimental data on fermi transitions to the non - analog excited @xmath25 states . the measured branching ratios @xcite have been converted to @xmath320 values , _ via _ eq.([mfn ] ) , and listed in table [ t : dc1exnew ] . again , for each nucleus , we list just one representative calculation and our adopted value . the assigned error reflects both the spread among the different calculations and the uncertainties in the imme coefficients . our 2002 adopted values @xcite are also listed . for nuclei @xmath321 , with the possible exception of @xmath151mn , the agreement between theory and experiment is entirely satisfactory . but in the upper @xmath15-shell , the calculated value for @xmath153ga is three times larger than measured in recent experiments @xcite . shell - model calculations in this region are complicated by the massive size of the hamiltonian matrices . to keep our calculations tractible , we kept the @xmath75 shell closed in these cases , but there is considerable evidence @xcite that this could be a poor assumption . conventionally , the radiative correction has been separated into two parts , one that contains the nucleus - dependent terms , called the ` outer ' radiative correction , and one that is independent of the nucleus , the ` inner ' radiative correction . principally due to the work of marciano and sirlin ( for example , refs.@xcite ) , the radiative correction applied to the uncorrected @xmath2-decay rate @xmath322 was expressed as follows : _ & = & _ ^0 ( 1 + _ r^ ) ( 1 + ) [ rate ] + _ r^ & = & + & & [ dr1 ] + & = & [ dr1 ] + & = & , [ dr2 ] where @xmath323 is the maximum electron energy in @xmath2-decay , and @xmath324 , @xmath325 , @xmath326 are the masses of the @xmath327-boson , proton and @xmath37-boson . the separation into outer and inner terms is accommodated in @xmath5 and @xmath4 respectively . in the outer correction , @xmath5 , the order-@xmath21 term contains the function @xmath328 : it is the average over the beta energy spectrum of the function @xmath329 , which was defined by sirlin ( see eq.(20b ) of ref.@xcite ) and is not reproduced here . its large-@xmath323 limit is shown in eq.([dr1 ] ) , indicating that the expression is dominated by the logarithm , @xmath330 . the last two terms in the outer correction , @xmath331 and @xmath332 , represent corrections to order @xmath333 and @xmath334 respectively . the origin of the @xmath328 term together with that of the leading term in the inner radiative correction , @xmath335 is the @xmath336-box and bremsstrahlung diagrams , which are taken together to remove the divergence as the photon energy goes to zero . both @xmath331 and @xmath332 also come from a standard qed calculation of the @xmath336-box and bremsstrahlung graphs @xcite , but in their case the electron was allowed to interact with the coulomb field of the nucleus . care was taken not to double count with the fermi function . the calculation was complete to order @xmath337 but only estimated in order @xmath338 . in the inner correction , @xmath4 , the second and third terms , @xmath339 , like the first term , also represent a @xmath336 box graph , but this time it involves an axial - vector weak interaction . the evaluation of this graph can be divided into two energy regimes : the high - energy ( or short - distance ) part given by the logarithm , and the low - energy ( or long - distance ) part denoted by @xmath340 . the parameter @xmath341 , referred to as the low - energy cut - off , divides these two energy regimes . marciano and sirlin @xcite allowed it to take on a range of values , 400 mev @xmath342 1600 mev ( revised slightly by sirlin @xcite to be @xmath343 , with @xmath344 being the @xmath345-vector - meson mass ) . the low - energy component , @xmath340 , was approximated by its born contribution c c_born = 3 ( 0.266 ) ( _ p + _ n ) = 0.885 , [ cborn1 ] where @xmath346 is the axial vector coupling constant accepted at the time and @xmath347 is the nucleon isoscalar magnetic moment . the factor @xmath348 is the value of the loop integral that was rendered finite by the use of dipole form factors for the nucleon electromagnetic , @xmath349 , and axial - vector , @xmath350 , vertices . the fourth term in eq.([dr1 ] ) , with the logarithm @xmath351 , arises from @xmath352-box graphs ; while the last term , @xmath353 , represents a small perturbative qcd correction that was evaluated by marciano and sirlin @xcite to be @xmath354 . the value of the outer radiative correction as defined in eq.([dr1 ] ) , ranges from 1.39 - 1.65% for the known superallowed emitters ( see ref.@xcite ) . following sirlin @xcite , the assigned uncertainties are set equal to @xmath355 as an estimate of the error made in stopping the calculation at that order . the value of the inner radiative correction as obtained from eq.([dr2 ] ) with @xmath356 from eq.([cborn1 ] ) is @xcite ( old ) = 2.40(8 ) % . [ drvvalu1 ] these results provide the essential foundation of the radiation corrections still used today . however a number of improvements have been introduced in the intervening 17 years . the low - energy part of the @xmath336-box diagram for an axial - vector weak interaction , denoted @xmath340 , was approximated by its born contribution in eq.([cborn1 ] ) , and was evaluated on a single nucleon . however , in a finite nucleus with many nucleons present , jaus and rasche @xcite observed that the two hadronic - interaction vertices , @xmath349 and @xmath350 , do not have to be with the same nucleon . thus , in finite nuclei there can be two types of contributions : those in which @xmath349 and @xmath350 vertices are with the same nucleon and those in which they are not . the evaluation of the former terms yields expressions @xcite that are proportional to @xmath357 , the isospin ladder operator , and so are also proportional to the fermi @xmath2-decay operator . therefore , they produce a universal correction the same in all nuclei with the value @xmath358 , which is given in eq.([cborn1 ] ) . the remaining terms , those in which the interactions are with different nucleons , must be evaluated with two - body operators that depend on the nuclear structure of the states involved . thus , the expression for @xmath356 given in eq.([cborn1 ] ) must be replaced by the following equation : c = c_born + c_ns , [ cns ] where @xmath359 comprises the nuclear - structure dependent terms . calculations of @xmath359 were first made in 1992 @xcite . a further modification was introduced in 1994 @xcite . in calculations of @xmath358 that had been made up to that time , the axial - vector and electromagnetic coupling constants , @xmath360 and @xmath361 see eq.([cborn1 ] ) had been given their free - nucleon values . yet there is ample evidence in nuclear physics that coupling constants for spin - flip processes are quenched in the nuclear medium , with the amount of quenching varying from nucleus to nucleus . thus , one should really be replacing @xmath362 , the value obtained with free - nucleon coupling constants , with @xmath363 . however , to separate the nucleus - dependent and nucleus - independent parts of the latter , we write c_born^quenched & = & q c_born^free + & = & c_born^free + ( q-1 ) c_born^free [ cbornq ] where @xmath364 is the factor by which the product of the weak and electromagnetic coupling constants is reduced in the medium relative to its free - nucleon value . the first term in eq.([cbornq ] ) , which remains universal , is retained in the inner radiative correction , replacing @xmath356 in eq.([dr1 ] ) . the second term becomes part of a separate nuclear - structure - dependent radiative correction , @xmath3 , which also includes @xmath365 , the value of @xmath366 recalculated with quenched operators . this correction is written as _ ns = , [ dns ] and is incorporated with the other nuclear - structure - dependent correction term , @xmath0 see eq.([ftfactor ] ) . calculated values of @xmath3 @xcite range from -0.360% to + 0.030% , each generally being smaller in magnitude than the corresponding value of @xmath0 . we return to @xmath3 in sect . [ ss : col ] . in 2005 , czarnecki , marciano and sirlin @xcite revisited the @xmath367 correction for neutron beta decay . they began by trivially updating the value of @xmath362 to reflect the current value of the axial - vector coupling constant , @xmath368 , to get c_born^free = 0.891 , [ cborn3 ] which replaces the value given in eq.([cborn1 ] ) . they then went on to re - evaluate @xmath4 , focusing particularly on the leading log corrections . using an established renormalization group summation @xcite for the leading short - distance logs , @xmath369 , they extended the method to the lower energy region between @xmath370 and @xmath325 to obtain @xmath371 . this resulted in the replacements 1 + & & s(m_p , ) = 1.02248 + 1 + & & l(2 e_m , m_p ) , [ sl ] where l(2 e_m , m_p ) = 1.026725 ^9/4 . [ lem ] the complete radiative correction , rc , including order @xmath337 and @xmath338 terms , could then be written @xcite 1 + rc & = & \ { 1 + } + & & \ { l(2 e_m , m_p ) + } + & & \ { s(m_p , ) + + nll } , + & & [ radc4 ] where @xmath372 is a next - to - leading log correction that czarnecki _ et al . _ estimate to be @xmath373 . the coefficient @xmath374 is a running qed coupling constant whose value at @xmath375 is @xmath376 and at @xmath377 is @xmath378 @xcite . this new result can still be organized to preserve the separation of nucleus - dependent and nucleus - independent components . the separation we hereby adopt is 1 + _ r^ & = & \ { 1 + } + & & \ { l(2 e_m , m_p ) + } [ dr3 ] 1 + & = & s(m_p , ) + c_born^free + & & + + nll . [ dr4 ] we will use this separation here and in our future work on superallowed @xmath2 decay . it results in a small change to the values of @xmath5 and @xmath379 that we used in our recent review @xcite . in sect.[ss : pre90 ] we explained that the terms @xmath380 in eq.([dr1 ] ) arose from the @xmath336-box graph for an axial - vector weak interaction . these two terms came from splitting the evaluation of this graph into two energy regimes . the division between the two regimes was chosen to be @xmath381 gev @xcite , roughly the mass of the @xmath345 resonance , and its range of uncertainty was taken to be from @xmath382 to @xmath383 . this @xmath384 range determination actually produced the largest single contributor to the uncertainty in the ckm matrix element , @xmath9 . to reduce the hadronic uncertainty in the radiative correction , marciano and sirlin @xcite have looked again at the @xmath336-box graph for an axial - vector weak interaction . this time they split it into three energy regimes , rather than two and , where possible , they drew on independent information to control their results : _ short distances , @xmath385 _ : this is a domain where qcd corrections remain perturbative . marciano and sirlin added higher - order terms , noting that these terms are identical ( in the chiral limit ) to qcd corrections to the bjorken sum rule for polarized electroproduction and can therefore be obtained from well - studied calculations for that process . _ intermediate distances , @xmath386 _ : in this region , they used an interpolation function between low and high energies , motivated by vector - meson and axial - vector - meson dominance . by limiting the number of terms to three , they had sufficient matching conditions to determine the coefficients uniquely . [ t ] .calculated transition - dependent radiative correction , @xmath5 , in percent units , and the component contributions . in our previous works ( @xmath387 ref.@xcite ) @xmath5 was defined as the sum of the contents of columns 2 - 4 ; this result is given in column 5 and labeled former @xmath5 . " as explained in the text , we have now redefined @xmath5 to include the additional term in column 6 ; the new values for @xmath5 are given in the last column . [ t : tab1 ] [ cols="<,^,^,^,^,^,^ " , ] the new inner correction is defined by eq.([dr4 ] ) , with @xmath362 taken from eq.([cborn4 ] ) . with its uncertainty obtained from marciano and sirlin @xcite , the result is = ( 2.361 0.038 ) % . [ drvvalu3 ] it is important to note that with the re - evaluation of @xmath362 , there is a consequent change in the nuclear - structure dependent correction @xmath3 given in eq.([dns ] ) . fortunately , the change is very small , being ( q-1 ) ( c_born^new - c_born^old ) ( ) & & -0.3 ( -0.062 ) 2.3 10 ^ -3 + & & 0.004 % . [ changedns ] in addition to making this change , we have also taken the opportunity to re - evaluate @xmath359 using the more recently available shell - model effective interactions described in sect . [ sss : smc ] . our revised @xmath3 values are listed in table [ t : tab2 ] . as in tables[t : dc2 ] and [ t : dc1new ] , we give ( in columns 3 and 4 ) the results from _ one sample calculation _ for each nucleus . then in column 5 we present adopted @xmath3 values that result from our assesment of _ all _ calculations made for each decay , not just the ones shown in columns 3 and 4 ; the uncertainties were chosen to encompass the spread in the results from those calculations . for comparison , in column 2 we list the values we adopted for @xmath3 in 2002 @xcite . in all cases the new values agree with the old ones within the quoted uncertainties . we have calculated improved results for the correction terms @xmath61 ( see table[t : dc1new ] ) , @xmath62 ( table[t : dc2 ] ) and @xmath3 ( table[t : tab2 ] ) ; and , based on the work of marciano and sirlin , we have presented revised values for @xmath5 ( table[t : tab1 ] ) and @xmath4 ( eq.([drvvalu3 ] ) ) . we are now in a position to extract corrected @xmath6 values from the current world data for superallowed @xmath1 transitions . we use the same data set as that described in sect.[s : sbd ] : it represents an interim update of our 2005 complete survey @xcite and includes ten additional published measurements @xcite . results are given in table[t : ft ] for the thirteen superallowed transitions whose @xmath27 values are known to a precision of 0.3% or better . the @xmath6 values given in column 6 were obtained from the data in the preceeding columns through the application of eq.([ftfactor ] ) . the corrected @xmath6 values are also plotted in figure [ fig2 ] . it is clear from the normalized @xmath43 given on the bottom line of the table that the statistical agreement among the @xmath6 values remains excellent . furthermore , it is evident from the figure that @xmath7v no longer shows any deviation from the overall average as it did in fig.[fig1 ] . however , it is equally evident that instead the @xmath151mn and @xmath152co @xmath6 values are now low , and by amounts that are no less statistically significant than the amount by which the @xmath7v value was previously high . rather than being a negative result , however , this possible discrepancy offers us the opportunity to use the cases of @xmath151mn and @xmath152co as a valuable test of our improved calculations . the @xmath14 value for each of them has been measured only twice with ( claimed ) high precision @xcite , and one of these references @xcite also included a measurement of the @xmath14 value for @xmath7v , which penning - trap measurements have recently shown @xcite to be low by 2 kev more than three times its originally quoted standard deviation . if , as seems likely , the problem with the @xmath7v measurement in ref.@xcite is not limited to that measurement alone , then doubt is certainly cast on the @xmath151mn and @xmath152co @xmath14-value results quoted in that reference as well . new penning - trap measurements of both @xmath14 values are currently in progress @xcite , and the question should be settled shortly . if the @xmath14 values in ref.@xcite prove to have been too low again , then the new penning - trap measurements will serve to increase the @xmath6 values for @xmath151mn and @xmath152co and could well bring them into close agreement with the average @xmath8 value . if so , this would add strong support to our new calculations . the average corrected @xmath8 value obtained from our new analysis , 3071.4(8 ) s , is lower by more than one standard deviation , compared to the comparable result obtained in our 2005 survey , 3072.7(8 ) s. if the new measurements do prove to increase the @xmath14 values for @xmath151mn and @xmath152co , then this discrepancy will decrease slightly , but there is no avoiding the fact that the inclusion of some core orbitals in the nuclear - structure - dependent correction terms has increased the correction in a number of cases , which in turn leads to a reduction in their @xmath6 values . a significant change in the nuclear model has led to a significant change but not a revolutionary one in the average @xmath8 value . the new average @xmath8 value yields a new value for @xmath9 via the equation v_ud^2 = , [ vudeq ] where @xmath388 is the well known weak - interaction constant for the purely leptonic muon decay @xcite . it has been our practice when using the @xmath8 value in this context to add 0.85(85)s to its value to account for possible systematic errors in the treatment of the radial wave function in the calculation of @xmath0 . ( this point is discussed in detail in section iii c of ref.@xcite . ) continuing this practice , we obtain the following result for the up - down element of the ckm matrix : |v_ud| = 0.97418(26 ) . [ vudvalu ] this result can be compared with the value 0.97380(40 ) , which was obtained in 2005 @xcite . the new value is ( just ) within the uncertainty of the previous value , and carries an uncertainty that is one third smaller . the final step is to combine this new value of @xmath389 with the other top - row elements of the ckm matrix , @xmath390 and @xmath391 , to test the unitarity of the matrix . taking the values of the latter two elements from the 2006 particle data group review @xcite we obtain the stunning result fully satisfied with a precision of 0.1% . we have presented new calculations of the nuclear - structure - dependent corrections to superallowed @xmath1 nuclear @xmath2 decay . the calculations incorporate core orbitals in the shell model in cases where independent experimental information indicates that they are required . where possible , they also make use of effective interactions that have been published since our previous calculation of these correction terms @xcite . as in that work , we have included twenty transitions in our calculations , thirteen that are by now rather well measured and seven more that are likely to be accessible to precise measurements in the future . the agreement among the corrected @xmath6 values for the thirteen well measured cases is very good , although there is a possible small discrepancy for the cases of @xmath151mn and @xmath152co . a new penning - trap measurement of the @xmath14 values for these two transitions is expected in the near future , and its effect on this discrepancy could serve to test the validity of our calculations . with our new corrections , the value of @xmath389 is increased by 0.04% , or by one standard deviation of the previous result @xcite . with the new value , the sum of squares of the top - row elements of the ckm matrix is in perfect agreement with unitarity . the improved calculations presented in this work were inspired by the remarkable recent improvements in experimental precision , particularly in the measurement of the @xmath7v @xmath14 value . the only way that the calculated corrections can be tested and improved is by such precise measurements , both on the currently well - known transitions and on other as - yet - unstudied superallowed transitions that have larger calculated corrections . if the calculated correction terms replace the significant scatter in the measured @xmath27 values ( see the left panel in fig.[fig1 ] ) with a set of self - consistent corrected @xmath6 values , then they can surely be relied upon to produce a secure value for @xmath389 . the present calculations testify to the value of increased experimental precision . the work of jch was supported by the u. s. dept . of energy under grant de - fg03 - 93er40773 and by the robert a. welch foundation under grant a-1397 . ist would like to thank the cyclotron institute of texas a & m university for its hospitality during annual two - month summer visits . g. savard , f. buchinger , j.a . clark , j.e . crawford , s. gulick , j.c . hardy , a.a . hecht , j.k.p . lee , a.f . levand , n.d . scielzo , h.p . sharma , k.s . sharma , i. tanihata , a.c . villari , and y. wang , , 102501 ( 2005 ) . t. eronen , v. elomaa , u. hager , j. hakala , a. jokinen , a. kankainen , i. moore , h. penttili , s. rahaman , j. rissanen , a. saastamoinen , t. sonoda , j. yst , j.c . hardy , and v.s . kolhinen , , 232501 ( 2006 ) . b. hyland , d. melconian , g.c . ball , j.r . leslie , c.e . svensson , p. bricault , e. cunningham , m. dombsky , g.f . grinyer , g. hackman , k. koopmans , f. sarazin , m.a . schumaker , h.c . scraggs , m.b . smith and p.m. walker , j. phys . g : nucl . part . phys . * 31 * , s1885 ( 2005 ) . t. eronen , v. elomaa , u. hager , j. hakala , a. jokinen , a. kankainen , i. moore , h. penttili , s. rahaman , s. rinta - antilla , a. saastamoinen , t. sonoda , j. yst , a. bey , b. blank , g. canchel , c dossat , j. giovinazzo , i matea , n. adimi , b * 636 * , 191 ( 2006 ) . b. hyland , c.e . svensson , g.c . ball , j.r . leslie , t. achtzehn , d. albers , c. andreoiu , p. bricault , r. churchman , d. cross , m. dombsky , p. finlay , p.e . garrett , c. geppert , g.f . grinyer , g. hackman , v. hanemaayer , j. lassen , j.p . lavoie , d. melconian , a.c . morton , c.j . pearson , m.r . pearson , a.a . phillips , m.a . schumaker , m.b . smith , i.s . towner , j.j . valiente - dobn , k.wendt , and e.f . zganjar , , 102501 ( 2006 ) . a. piechaczek , e.f . zganjar , g.c . ball , p. bricault , j.m . dauria , j.c . hardy , d.f . hodgson , v. iacob , p. klages , w.d . kulp , j.r . leslie , m. lipoglavsek , j.a . macdonald , h .- b . mak , d.m . moltz , g. savard , j. von schwarzenberg , c.e . svensson , i.s . towner , and j.l . wood , c * 67 * , 051305(r ) ( 2003 ) .

we report new shell - model calculations of the isospin - symmetry - breaking correction , @xmath0 , to superallowed @xmath1 nuclear @xmath2 decay . the most important improvement is the inclusion of core orbitals , which are demonstrated to have a significant impact on the mismatch in the radial wave functions of the parent and daughter states . we determine which core orbitals are important to include from an examination of measured spectroscopic factors in single - nucleon pick - up reactions . in addition , where new sets of effective interactions have become available since our last calculation , we now include them ; this leads to small changes in @xmath3 as well . we also examine the new radiative - correction calculation by marciano and sirlin and , by a simple reorganization , show that it is possible to preserve the conventional separation into a nucleus - independent inner " radiative term , @xmath4 , and a nucleus - dependent outer " term , @xmath5 we tabulate the new values for @xmath0 , @xmath3 and @xmath5 for twenty superallowed transitions , including the thirteen currently well - studied cases . with these new correction terms the corrected @xmath6 values for the thirteen cases are statistically consistent with one another and the anomalousness of the @xmath7v result disappears . these new calculations lead to a lower average @xmath8 value and a higher value for @xmath9 . the sum of squares of the top - row elements of the ckm matrix now agrees exactly with unitarity .

, where @xmath1 is the underlying genotype . @xmath2 with @xmath3 ; @xmath4 or @xmath5 indicates the state of the mutable site ( e.g. , amino acid position ) . the effect of a single , double , triple mutation is given by the red arrows . pairwise ( or second - order ) epistasis is defined as the differential effect of a mutation depending on the background in which it occurs , for example in ( b ) it is the degree to which the effect of one mutation ( e.g. @xmath6 ) deviates in the background of the second mutation ( @xmath7 ) . thus , the expression for second order epistasis is @xmath8 . the third order and higher cases are considered in the main text , [ fig : explain ] ] we begin with a formal definition of genotype , phenotype , and the representation of mutational effects . consider a specific sequence comprised of @xmath9 positions as a binary string @xmath2 with @xmath3 , where @xmath4 and @xmath5 represent the `` wild - type '' and mutant state of each position , respectively . this defines a total space of @xmath10 genotypes . the analysis could be expanded to the case of multiple substitutions per position , but we consider just the binary case for clarity here . each genotype @xmath1 has an associated phenotype @xmath11 , which is of the form that the independent action of two mutations means additivity in @xmath12 . for notational simplicity , we will simply write the genotype in a @xmath13-bit binary form , where @xmath13 is the order of the mutations that are considered . for example , the effect of a single mutation is simply @xmath14 , the difference in the phenotype between the mutant and wild - type states ( fig . [ fig : explain]a ) . the effect of a double mutant is given by @xmath15 ( red arrow , fig . [ fig : explain]b ) , and its linkage through paths of single mutations is defined by a two - dimensional graph ( a square network ) with four total genotypes . similarly , a triple mutant effect is @xmath16 ( red arrow , fig . [ fig : explain]c ) , and its linkage through paths of single mutations are enumerated on a three - dimensional graph ( a cube ) with eight total genotypes . more generally , and as described by horowitz and fersht @xcite , the phenotypic effect of any arbitrary @xmath17-dimensional mutation can be represented by an @xmath17-dimensional graph , with @xmath18 total genotypes . understanding the relationship of the phenotypes of multiple mutants to that of the underlying lower - order mutant states is the essence of epistasis , and is described below . a well - known approach in biochemistry for analyzing the cooperativity of amino acids in specifying protein structure and function is to use the formalism of thermodynamic mutant - cycles @xcite , one manifestation of the general principle of epistasis . in this approach , the `` phenotype '' is typically an equilibrium free energy @xmath19 ( e.g. of thermodynamic stability or biochemical activity ) , and the goal is to obtain information about the structural basis of this phenotype through mutations that represent subtle perturbations of the wild - type state . for pairs of mutations , the analysis involves measurements of four variants : wild - type ( @xmath20 ) , each single mutant ( @xmath21 and @xmath22 ) , and the double mutant ( @xmath23 ) , where the subscripts designate the mutated positions , and the superscript o indicates free energy relative to a standard state ( fig . 1b ) . from this , we can compute a coupling free energy between the two mutations ( @xmath24 ) as the degree to which the effect of one mutation ( @xmath25 ) is different when tried in the background of the other mutation ( @xmath26 ) : @xmath27 whereas the @xmath28 terms are individual measurements and @xmath29 terms are the effects of single mutations relative to wild - type , @xmath30 is a second order epistatic term describing the cooperativity ( or non - independence ) of two mutations with respect to the wild - type state . this analysis can be expanded to higher order . for example , the third order epistatic term describing the cooperative action of three mutations 1 , 2 , and 3 ( @xmath31 ) is defined as the degree to which the second order epistasis of any two mutations is different in the background of the third mutation : @xmath32 note that @xmath33 requires measurement of eight individual genotypes ( fig . more generally , we can define an @xmath17-th order epistatic term ( @xmath34 ) , describing the cooperativity of @xmath17 mutations , @xmath35 it is possible to write this expansion in a compact matrix form : @xmath36 where @xmath37 is the vector of @xmath18 epistasis terms of all orders , and @xmath38 is the vector of @xmath18 free energies corresponding to phenotypes of all the individual variants listed in binary order . to illustrate , for three mutations @xmath39 , and we obtain @xmath40 1 & 0 & 0 & 0 & 0 & 0 & 0 & \ \ 0\ \\ -1 & 1 & 0 & 0 & 0 & 0 & 0 & \ \ 0\ \\ -1 & 0 & 1 & 0 & 0 & 0 & 0 & \ \ 0\ \\ 1 & -1 & -1 & 1 & 0 & 0 & 0 & \ \ 0\ \\ -1 & 0 & 0 & 0 & 1 & 0 & 0 & \ \ 0\ \\ 1 & -1 & 0 & 0 & -1 & 1 & 0 & \ \ 0\ \\ 1 & 0 & -1 & 0 & -1 & 0 & 1 & \ \ 0\ \\ -1 & 1 & 1 & -1 & 1 & -1 & -1 & \ \ 1\ \end{pmatrix * } \setlength{\arraycolsep}{6pt } * \begin{pmatrix * } y_{000}\\ y_{001}\\ y_{010}\\ y_{011}\\ y_{100}\\ y_{101}\\ y_{110}\\ y_{111 } \end{pmatrix*}\ ] ] in this representation , subscripts in @xmath38 represent combinations of mutations ( e.g. @xmath41 , a double mutant ) and subscripts in @xmath37 represent epistatic order ( e.g. @xmath42 , pairwise epistasis between mutations 1 and 2 ) . thus , equations and correspond to multiplying @xmath38 by the fourth or eighth row of @xmath43 , respectively , to specify @xmath44 and @xmath45 . note that @xmath38 and @xmath37 contain precisely the same information , re - written in a different form . the matrix @xmath43 represents an operator linking these two representations of the mutation data and we will return to the nature of the operation in a later section . we can write a recursive definition for @xmath43 that defines the mapping between @xmath38 and @xmath37 for all epistatic orders @xmath17 : @xmath46 \boldsymbol{g}_n & \ \ 0 \ \ \\[0.2em ] -\boldsymbol{g}_n & \boldsymbol{g}_n \end{pmatrix*}\ \ \mathrm{with}\ \ \ \setlength{\arraycolsep}{6pt } \boldsymbol{g}_0 = 1 \label{eq : grecursive}\ ] ] the inverse mapping is defined by @xmath47 . this relationship gives the effect of any combination of mutants ( in @xmath38 ) as a sum over epistatic terms ( in @xmath37 ) . for example , the energetic effect of three mutations 1,2 , and 3 ( @xmath48 ) is : @xmath49 thus , in the most general case , the free energy value of a multiple mutation requires knowledge of the effect of the single mutations and all associated epistatic terms . for the triple mutant , this means the wild - type phenotype , the three single mutant effects , the three two - way epistatic interactions , and the single three - way epistatic term . this analysis highlights two important properties of epistasis : ( 1 ) the lack of any epistatic interactions between mutations dramatically simplifies the description of multiple mutations to just the sum over the underlying single mutation effects , and ( 2 ) the absence of lower - order epistatic interactions ( e.g. @xmath50 ) does not imply absence of higher order epistatic terms . in contrast to the biochemical definition , the significance of a mutation ( and its epistatic interactions ) may also be defined not solely with regard to a single reference state as the `` wild - type '' , but as an average over many possible genotypes . as we show below , such averaging better represents the epistatic level at which mutations operate , and in principle , can separate mutant effects that are idiosyncratic to particular genotypes from those that are fundamentally important . the concept of averaging epistasis over genotypic backgrounds is analogous to the idea of the schema average fitness in the field of genetic algorithms ( ga ) @xcite , which was recently introduced in biology @xcite . in its complete form , background - averaged epistasis considers averages over all possible genotypes for the remaining positions in the ensemble . for example , if @xmath51 , the epistasis between two positions 1 and 2 is computed as an average over both states of the third position ( @xmath52 , with the averaging denoted by @xmath53 ) ( see . 1c ) : @xmath54\\ + [ ( y_{011 } - y_{010 } ) - ( y_{001 } - y_{000 } ) ] \bigg\}\end{gathered}\ ] ] thus for @xmath51 , we can write all epistatic terms : @xmath55 \ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\ \\ \ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\ \\ \ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1\ \\ \ 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\ \\ \ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1\ \\ \ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1\ \\ \ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1\ \\ \ 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1\ \end{pmatrix * } \setlength{\arraycolsep}{6pt } * \begin{pmatrix * } y_{000}\\ y_{001}\\ y_{010}\\ y_{011}\\ y_{100}\\ y_{101}\\ y_{110}\\ y_{111 } \end{pmatrix*}\ ] ] where @xmath56 is a diagonal weighting matrix to account for averaging over different number of terms as a function of the order of epistasis ; @xmath57 , where @xmath58 is the order of the epistatic contribution in row @xmath59 . more generally , for any number of mutations @xmath17 : @xmath60 where @xmath38 is the same vector of phenotypes of variants as defined above , @xmath61 is the vector of background averaged epistatic terms , and @xmath62 is the operator for background - averaged epistasis , defined recursively as @xmath63 \ \boldsymbol{h}_n & \boldsymbol{h}_n\\[0.2em ] \ \boldsymbol{h}_n & -\boldsymbol{h}_n \end{pmatrix*}\ \ \mathrm{with}\ \ \ \setlength{\arraycolsep}{6pt } \boldsymbol{h}_0 = 1\ ] ] the recursive definition for the weighting matrix @xmath56 is @xmath64 \ \frac{1}{2}\boldsymbol{v}_n & 0 \\[0.2em ] \ 0 & -\boldsymbol{v}_n\ \end{pmatrix*}\ \ \mathrm{with}\ \ \ \setlength{\arraycolsep}{6pt } \boldsymbol{v}_0 = 1\ ] ] the matrix @xmath62 has special significance ; its action mathematically corresponds to a generalized fourier analysis @xcite known as the walsh - hadamard transform . this converts the phenotypes of individual variants ( in @xmath38 ) into a vector of averaged epistasis ( in @xmath61 ) , an operation that can also be seen as a spectral analysis of the high - dimensional phenotypic landscape defined by the genotypes studied . in this transform , the phenotypic effects of combinations of mutations are represented as sums over averaged epistatic terms . in summary , the definition of epistasis proposed in evolutionary genetics is a global definition over sequence space , averaging the epistatic effects of mutations over the ensemble of all possible variants . in contrast , the biochemical definition given in the previous section is a local one , treating a particular variant as a reference for determining the epistatic effect of mutations . a third approach for analyzing epistasis is linear regression . for example , when we have a complete dataset of phenotypes of all @xmath18 genotypes , we can use regression to define the extent to which epistasis is captured by only considering terms to some order @xmath65 . that is , whether terms up to the @xmath66@xmath67 order are sufficient for effectively capturing the full complexity of a biological system . the standard form for a linear regression is a set of equations : @xmath68 for each genotype @xmath1 . the @xmath69 terms denote the regression coefficients corresponding to the ( epistatic ) effects between subscripted positions , and @xmath70 is the residual noise term . in matrix form this can be written as @xmath71 where @xmath72 tabulates which regression coefficients are summed over for genotypes @xmath1 . for @xmath39 , regressing to full order , we can write @xmath73 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0\\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{pmatrix * } * \begin{pmatrix } \beta_{000}\\ \beta_{001}\\ \beta_{010}\\ \beta_{011}\\ \beta_{100}\\ \beta_{101}\\ \beta_{110}\\ \beta_{111 } \end{pmatrix } + \boldsymbol{\bar{\epsilon}}\ ] ] following the same rule for subscripts as before . @xmath72 has the recursive definition : @xmath74 \ \boldsymbol{x}_n & \ \ 0 \ \ \ \\[0.2em ] \ \boldsymbol{x}_n & \boldsymbol{x}_n\ \end{pmatrix*}\ \ \mathrm{with}\ \ \ \setlength{\arraycolsep}{6pt } \boldsymbol{x}_0 = 1 \label{eq : recursivex}\ ] ] it is worth noting that the inverse of @xmath72 is @xmath75 , the operator for biochemical epistasis ( eq . ; see supplementary information ) . thus , the multi - dimensional mutant - cycle analysis is indistinguishable from regression to full order the case in which @xmath76 and @xmath77 . however , the usual aim of regression is to approximate the data with fewer coefficients than there are data points , i.e. , @xmath65 . to express this , we simply remove the columns from @xmath72 that refer to the epistatic orders excluded from the regression ( i.e. , @xmath78 ) : @xmath72 is multiplied by an @xmath18-by-@xmath79 matrix @xmath80 , the identity matrix with columns corresponding to epistatic orders higher than @xmath66 removed . @xmath79 is the number of epistatic terms up to @xmath66 and is given by @xmath81 . thus for regression to order @xmath66 , we can define @xmath82 , and write @xmath83 the linear regression is performed by solving the so - called normal equations @xmath84 where @xmath85 is necessarily square and invertible as long as @xmath86 is full column rank and hence @xmath85 is full rank . note that in this analysis we compute epistatic terms only up to the @xmath66@xmath67 order , but use phenotype / fitness data of all @xmath18 combinations of mutants . the more general case in which we estimate epistatic terms with less than @xmath18 data points is distinct and is discussed below . if the biochemical definition of epistasis is a local one , exploring the coupling of mutations of all order with regard to one `` wild - type '' reference , and the ensemble view of epistasis is a global one , assessing the coupling of mutations of all order averaged over all possible genotypes , then the regression view of epistasis is an attempt to project to a lower dimension - capturing epistasis as much as possible with low - order terms . the analysis presented above leads to a simple unifying concept underlying the calculations of epistasis . in general , all the calculations are a mapping from the space of phenotypic measurements of genotypes @xmath38 to epistatic coefficients @xmath87 , in a general form @xmath88 , where @xmath89 is the epistasis operator . we give the bottom line of the different operators below ; their formal mathematical derivations can be found in the supplementary information . the most general situation is that of the background - averaged epistasis with averaging over the complete space of possible genotypes . in this case @xmath90 where @xmath62 is a @xmath91 matrix corresponding to the walsh - hadamard transform ( @xmath17 is the number of mutated sites ) and @xmath56 is a matrix of weights to normalize for the different numbers of terms for epistasis of different orders . the biochemical definition of epistasis using one `` wild - type '' sequence as a reference is a sub - sampling of terms in the hadamard transform . in this case @xmath92 where @xmath72 is , as defined in eq . . in essence , @xmath93 picks out the terms in @xmath62 that concern the wild - type background . note that both these mapping are one - to - one , such that the number of epistatic terms ( in @xmath87 ) is equal to the number of phenotypic measurements ( in @xmath38 ) and no information is lost . in contrast , regression to lower orders necessarily implies fewer epistatic terms than data points , which means the mapping is compressive and information is lost . in this case @xmath94 where @xmath95 ( @xmath96 ) is the identity matrix but with zeros on the diagonal at the orders that are higher than which we regress over . the fundamental point is that all three formalisms for computing epistasis are just versions of the walsh - hadamard transform , with terms selected as appropriate for the choice of a single reference sequence or limitations on the order of epistatic terms considered . from a computational point of view , it is interesting to note that regression using the hadamard transform makes matrix inversion unnecessary ( compare with eq . ) . to illustrate the different analyses of epistasis , we consider a small case study of three spatially proximal mutations that define a switch in ligand specificity in psd95-pdz3 , a member of the pdz family of protein interaction modules ( fig . [ fig : pdz]a ) . two mutations are in psd95-pdz3 ( g330 t and h372a ) , and one mutation in its cognate ligand peptide ( t-2f ) . the phenotype is the binding affinity , @xmath97 , and the absence of epistasis implies additivity in the corresponding free energy , expressed as @xmath98 in kcal mol@xmath99 . binding affinities for this system are from ref . @xcite , and given in figure [ fig : pdz]b . these quantitative phenotypes are then transformed to epistatic terms using eq.[eq : omegahadamard]-[eq : regresshadamard ] ( table 1 ) . a number of simple mathematical relationships are evident in the data . first , regression is carried out only to the second - order and therefore the third - order epistatic term for this analysis does not exist ( or , equivalently , is set to zero if the epistatic vector @xmath100 is defined to be of full length @xmath18 ) . second , there are some equalities . the regression terms at the highest order ( second , in this case ) are equal to the corresponding terms for the averaged epistasis . this is because @xmath101 sets columns corresponding to orders higher than the regression order to zero , leaving rows corresponding to the highest regression order with only one non - zero element , on the diagonal . for these rows the entries in the epistasis operators @xmath102 and @xmath103 are equal . another more trivial equality is the highest - order term for the mutant - cycle and averaged epistasis formalisms ; there is only one contribution for the highest order , and therefore no backgrounds to average over . values in @xmath104 m for all eight combinations of two amino acids at the three mutable positions.[fig : pdz ] ] @height 9.5pt depth4pt width0ptrr|rrrrr & & & & & + & & & & & + & & & & & + & & & & & + & & & & & + & & & & & + & & & & & + & & & & & + & & & & & + & & & & & + & & & & & + & & & & & & + + [ table : pdz ] the data also illustrate the key properties of the different formalisms . the g330 t , h372a , and t-2f mutations represent a collectively cooperative set of perturbations , as indicated by a significant third - order epistatic term by both mutant cycle and background averaged definitions ( @xmath105 kcal mol@xmath99 ) . but the three formalisms differ in the energetic value of the lower order epistatic terms . for example , g330 t is essentially neutral for wild - type ligand binding but shows a dramatic gain in affinity in the context of the t-2f ligand ; thus , a large second - order epistatic term by the biochemical definition ( @xmath106 kcal mol@xmath99 ) . however , the coupling between g330 t and t-2f is nearly negligible in the background of h372a ; as a consequence , the background averaged second - order epistasis term @xmath107 is smaller ( @xmath108 kcal mol@xmath99 ) . similarly , both biochemical and regression formalisms assign a large first - order effect to the t-2f ( 1 * * ) and h372a ( * 1 * ) single mutations , while the corresponding background - averaged terms are nearly insignificant . for example , the free energy effect of mutating h372a ( @xmath109 ) is @xmath110 kcal mol@xmath99 in the wild - type background , but is @xmath111 kcal mol@xmath99 in the background of the t-2f ligand mutation - a nearly complete reversal of the effect of this mutation depending on context . thus with background averaging , the first order term for h372a ( @xmath112 ) is close to zero . this makes sense ; given the experiment described in figure 2 , the h372a mutation should not be thought of as a general determinant of ligand affinity . instead it is a conditional determinant , with an effect that depends on the identity of the amino acid at the @xmath113 position of the ligand . note that the degree of averaging depends on the number of mutated sites , and thus the interpretation of mutational effects will depend on the scale of the experimental study . these examples show that background averaging has the effect of `` correcting '' mutational effects for the existence of higher - order epistatic interactions . without background averaging , the effect of a mutation ( at any order ) idiosyncratically depends on a particular reference genotype and will fail to account for higher order epistasis which modulates the observed mutational effect . thus , background averaging provides a measure of the effects of mutation that represents its general value over many related systems , and more appropriately represents the cooperative unit within which the mutation operates . the analytical expressions in eq.[eq : omegahadamard]-[eq : regresshadamard ] involves the measurement of phenotypes ( @xmath38 ) for all @xmath18 combinatorial mutants , a fact that exposes two fundamental problems . first , it is only practical when @xmath17 is small . in such cases ( e.g figure 2 , @xmath51 ) , the data can be combinatorially complete permitting a full analysis - the local and global structure of epistasis , possible evolutionary trajectories , and adaptive trade - offs @xcite . but for the typical size of protein domains ( @xmath114 ) , the combinatorial complexity of mutations precludes the collection of complete datasets . second , even if it were possible , the sampling of all genotypes is not desired ; indeed , the majority of systems in such an ensemble are unlikely to be functional and and averages over them are not meaningful with regard to learning the epistatic structure of native systems . how then can we apply these epistasis formalisms in practice , especially with regard to background averaging ? to develop general principles , we begin with two obvious approaches that lead to well - defined alternative expressions for averaged epistasis . first , consider the case in which the data are only `` locally complete '' ; that is , we have all possible mutants up to a certain order @xmath115 . we can then define a measure that is intermediate between epistasis with a single reference genotype and epistasis with full background - averaging , which we will refer to as the _ partial _ background - averaged epistasis . for example , for three positions ( @xmath51 ) with data complete only up to order ( @xmath116 ) , the partial background - averaged effect of the first position ( rightmost subscript ) , is calculated as @xmath117 . compared to the full background - averaged epistasis , the partial averages just leaves out the last term , @xmath118 , which represents the unavailable phenotype of the triple mutant @xmath119 . more generally , we can define this measure of epistasis as another special case of the hadamard transform : @xmath120 where @xmath121 designates the element - wise product . @xmath122 is again a diagonal weighting vector , now given by @xmath123 where @xmath58 is the epistatic order associated with row @xmath59 as defined earlier , and @xmath124 . note that @xmath125 because mutants of order higher than @xmath126 are considered absent in the dataset . the matrix @xmath127 simply serves to multiply by zero the terms in the hadamard matrix that include orders higher than @xmath126 . interestingly , the @xmath127 matrices display a self - similar hierarchical pattern ( fig . [ fig : sierpinski ] ) and are related to so - called sierpinski triangles ( see ref @xcite ) . this permits a recursive definition in both @xmath17 and @xmath126 for the product @xmath128 , which we will designate as @xmath129 : @xmath130 \ \boldsymbol{f}_{\!_{n-1,p } } \ \ \ & \boldsymbol{f}_{\!_{n-1,p-1}}\ \\[0.2em ] \ \boldsymbol{f}_{\!_{n-1,p-1 } } & -\boldsymbol{f}_{\!_{n-1,p-1}}\ \end{pmatrix * } \label{eq : partbackgiter}\ ] ] + with @xmath131 for @xmath132 , and @xmath133 is a @xmath91 matrix of zeros , except for a 1 in the upper left corner . this analysis assumes that data are complete up to the order @xmath126 . if not , analytical schemes for background - averaged epistasis such as eqs[eq : partbackg]-[eq : partbackgiter ] are not obvious . a second analytically tractable case for incomplete data arises in regression , where the idea is to estimate epistatic terms up to a specified order from available data . this involves solving a set of equations similar to the normal equations : @xmath134 where @xmath135 is an @xmath136 matrix constructed from the @xmath18 by @xmath18 identity matrix by deleting the @xmath137 rows corresponding to the unavailable phenotypic data , and @xmath138 , with @xmath80 defined as above . in order for this system of equations to be solvable , a necessary constraint is that @xmath139 ; that is , the number of data points available should be larger than or equal to the number of regression parameters . in addition , the data must be such that it is possible to uniquely solve for all epistatic terms in the regression . for example , if two mutations always co - occur in the data , it is obviously impossible to calculate their independent effects . in such cases , the number of solutions to eq . is infinite ( @xmath140 is not invertible ) . introduced to calculate the partial background - averaged epistasis , for @xmath39 . ( a ) @xmath141 for when data for mutants up to second - order is available and ( b ) @xmath142 for when only first - order mutants are available . both matrices are self - similar , which allows their generation for arbitrary order , and are related to the so - called logical sierpiski triangle . for example @xmath143 , where @xmath144 is the anti - diagonal identity matrix and @xmath145 is the sierpinski matrix ( i.e. multigrade and in boolean logic ) for three inputs.[fig : sierpinski ] ] in practice , even with `` high - throughput '' assays , we can only hope to measure a tiny fraction of all combinatorial mutants due to the vast number of possibilities . in this situation , the problem of inferring epistasis by regression may be further constrained by imposing additional conditions , termed regularization . for example , kernel ridge regression @xcite and lasso @xcite include a weighted norm of the regression coefficients in the minimization procedure . regularization comes with its own set of caveats @xcite , but its application is , unlike the approaches in eq . and , not conditional on specific structure of the data or depth of coverage . however , none of these approaches directly addresses the problem of optimally defining appropriate ensembles of genotypes over which averages should be taken . in principle , the idea should be to perform background averaging over a representative ensemble of systems that show invariance of functional properties of interest . how can we generally find such ensembles without the impractical notion of exhaustive functional analysis of the space of possible genotypes ? one idea is motivated by the empirical finding of _ sparsity _ in the pattern of key epistatic interactions within biological systems . indeed , evidence suggests that in proteins , the architecture is to have a small subset of amino acids that shows strong and distributed epistatic couplings surrounded by a majority of amino acids that are more weakly and locally coupled @xcite . more generally , the notion of a sparse core of strong couplings surrounded by a milieu of weak couplings has been argued to be a signature of evolvable systems @xcite . if it can be more generally verified , the notion of sparsity might be exploited to define relevant strategies for optimally learning the epistatic structure of natural systems . one approach is to minimize the so - called @xmath146-norm ( the sum of absolute values of the epistatic coefficients ) in a constrained optimization , which has the effect of producing many epistatic coefficients with zero or very small values @xcite , while projecting onto background - averaged epistatic terms : @xmath147 this procedure is akin to the technique of compressed sensing @xcite , a powerful approach used in signal processing to recognize the low - dimensional space in which the relevant features of a high - dimensional dataset occur given the assumption of sparsity of these features . the application of this theory for mapping biological epistasis has to our knowledge not been reported before , but might be explored with focused high - order mutational analyses in specific well - chosen model systems . the necessary technologies for such experiments are now becoming available , and should help define practical data collection strategies for studying epistasis more generally . it is worth pointing out that other approaches that use ensemble - averaged information to understand biological systems have been developed and experimentally tested . for example , statistical methods that operate on multiple sequence alignments of proteins @xcite calculate quantities related to epistasis that are averaged over the space of homologous sequences . importantly , these approaches have been successful at revealing a hierarchy of cooperative interactions between amino acids that range from local structural contacts in protein tertiary structures @xcite to more global functional modes @xcite . for defining good experimental approaches to epistasis , a conceptual advance may come from an attempt to formally map the constrained optimization problem described in eq.[eq : cs ] to the kind of ensemble averaging that underlies the statistical coevolution approaches . a fundamental problem is to define the epistatic structure of biological systems , which holds the key to understanding how phenotype arises from genotype . here we provide a unified mathematical foundation for epistasis in which different approaches are found to be versions of a single mathematical formalism - the weighted walsh - hadamard transform . in the most general case , this transform corresponds to an averaging of mutant effects over all possible genetic backgrounds at every order order of epistasis . this approach corrects the effect of mutations at every level of epistasis for higher order terms . importantly , it represents the degree to which the effects of mutations are transferable from one model system to another , the usual purpose of most mutagenesis studies . in contrast , the thermodynamic mutant cycle @xcite ( commonly used in biochemistry ) constitutes a special case of taking a single reference genotype and thus no averaging @xcite . this analysis represents the effects of mutations that are specific to a particular model system . regression ( commonly used in evolutionary biology ) is an attempt to capture features of a system with epistatic terms up to a defined lower order , often to bound the extent of epistasis or to predict the effects of higher - order combinations of mutations @xcite . the similarity of the regression operator to that of the mutant cycle ( see eq . ) indicates that this approach is also focused around the local mutational environment of a chosen reference sequence . in general , background averaging would seem to provide the most informative representation of the effect of a mutation . however , with the exception of very small - 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jordan elimination : + + @xmath154 \boldsymbol{x}_n & \boldsymbol{x}_n\ & \ 0 & \ \ \mathbb{i}\\ \end{array } \right ) \rightarrow \left ( \begin{array}{rr|rr } \mathbb{i } & \ \ 0 \ \ & \boldsymbol{x}^{^{-1}}_{\ n } & 0 \ \ \ \\[0.2em ] \mathbb{i } & \mathbb{i } \ \ & 0 \ \ \ & \boldsymbol{x}^{^{-1}}_{\ n}\\ \end{array } \right)\rightarrow \left ( \begin{array}{rr|rr } \mathbb{i } & \ \ 0 \ \ & \boldsymbol{x}^{^{-1}}_{\ n } & 0 \ \ \ \\[0.2em ] 0 & \mathbb{i } \ \ & -\boldsymbol{x}^{^{-1}}_{\ n } & \boldsymbol{x}^{^{-1}}_{\ n}\\ \end{array } \right ) $ ] + + hence we have for the inverse of @xmath72 : + + @xmath155 \ \boldsymbol{x}^{^{-1}}_{\ n } & \ \ 0 \ \ \ \ \\[0.2em ] \ -\boldsymbol{x}^{^{-1}}_{\ n } & \boldsymbol{x}^{^{-1}}_{\ n } \ \end{pmatrix*}\ \ \mathrm{with}\ \ \ \boldsymbol{x}^{^{-1}}_0 = 1 $ ] + + + which is identical to the recursive form for @xmath43 : + + @xmath156 \ \boldsymbol{g}_n & \ \ 0 \ \ \ \\[0.2em ] \ -\boldsymbol{g}_n & \boldsymbol{g}_n\ \end{pmatrix*}$ ] + + + we further have : + + @xmath157 \ \ \boldsymbol{h}_n & \boldsymbol{h}_n\ \\[0.2em ] \ \ \boldsymbol{h}_n & -\boldsymbol{h}_n\ \end{pmatrix*}\ \ \mathrm{with}\ \ \ \boldsymbol{h}_0 = 1 $ ] + + + and @xmath158 \ \frac{1}{2 } \boldsymbol{v}_{\!n } & 0 \ \ \ \\[0.2em ] 0 \ \ \ & -\boldsymbol{v}_{\!n}\ \end{pmatrix*}\ \ \mathrm{with}\ \ \ \boldsymbol{v}_{\!0 } = 1 $ ] + + + + with the above relations we can derive the equality in the main text expressing @xmath43 as a hadamard transform : + + @xmath159 + + for @xmath160 the statement is trivial . we now show by induction that this relation holds for all @xmath17 . + + @xmath161 \ \boldsymbol{g}_n & \ \ 0 \ \ \ \\[0.2em ] \ -\boldsymbol{g}_n & \boldsymbol{g}_n \ \end{pmatrix * } = \begin{pmatrix*}[r ] \ \boldsymbol{v}_{_{\!n } } \boldsymbol{x}_{_{n}}^t \boldsymbol{h}_n & 0 \ \ \ \ \ \ \\[0.2em ] \ -\boldsymbol{v}_{_{\!n } } \boldsymbol{x}_{_{n}}^t \boldsymbol{h}_n & \boldsymbol{v}_{_{\!n } } \boldsymbol{x}_{_{n}}^t \boldsymbol{h}_n\ \end{pmatrix * } $ ] + + + @xmath162 \ \frac{1}{2 } \boldsymbol{v}_{\!n } & 0 \ \ \ \\[0.2em ] 0 \ \ \ & -\boldsymbol{v}_{\!n } \end{pmatrix * } \begin{pmatrix*}[r ] \ \ 2 \boldsymbol{x}_{_{n}}^t \boldsymbol{h}_n & 0 \ \ \ \ \ \\[0.2em ] \ \ \boldsymbol{x}_{_{n}}^t \boldsymbol{h}_n\ & -\boldsymbol{x}_{_{n}}^t \boldsymbol{h}_n\ \end{pmatrix * } = \begin{pmatrix*}[r ] \ \frac{1}{2 } \boldsymbol{v}_{\!n } & 0 \ \ \ \\[0.2em ] 0 \ \ \ & -\boldsymbol{v}_{\!n}\ \end{pmatrix * } \begin{pmatrix*}[r ] \ \boldsymbol{x}_{_{n}}^t & \boldsymbol{x}_{_{n}}^t\ \ \\[0.2em ] \ 0 \ \ & \boldsymbol{x}_{_{n}}^t\ \ \end{pmatrix * } \begin{pmatrix*}[r ] \ \ \boldsymbol{h}_n & \boldsymbol{h}_n\ \\[0.2em ] \ \ \boldsymbol{h}_n & -\boldsymbol{h}_n\ \end{pmatrix*}$ ] + + + @xmath163 + + + + + @xmath164 ( eq . ) + + + we will use @xmath82 and @xmath165 as defined in the main text . + + for the right - hand side we can write + + @xmath166 + + where we used @xmath167 , which can be proven straightforwardly by induction using the generative function for @xmath62 . + rearranging and using @xmath168 , we obtain + + @xmath169 + + we thus have to prove + + @xmath170 + + left - multiplying both sides by @xmath171 ( mind the hat is only on the first operator ) and right - multiplying by @xmath62 we are left to prove + + @xmath172 + + left - multiplication by @xmath80 yields + + @xmath173 + + which , again using the relation we proved in section a above , can be rewritten as + + @xmath174 + + or + + @xmath175 + + given the commutative properties of diagonal matrices @xmath95 and @xmath176 . + + this equality indicates that setting certain rows of @xmath177 to zero ( left - hand side ) is the same as setting both those rows and corresponding columns of @xmath177 to zero ( right - hand side ) . this is obviously not true for every set of rows and columns , and needs more discussion . + + we can prove this iteratively starting at regression to order @xmath178 and going down to lower order . if regression is done to order @xmath178 , this means that only the last row of @xmath177 is set to zero , and by construction of @xmath177 ( see above ) the last column only has a non - zero element in this row . this means that in this case the equality is correct . another way to see this is looking at matrix @xmath43 for @xmath51 in its explicit representation in the main text ( here @xmath43 being identified with @xmath177 ) and noting that the highest order epistatic term @xmath45 is the only one that receives a contribution from the highest order ( @xmath17 ) mutant term @xmath119 . + + next , if regression is performed instead to order @xmath179 , not only the last row of @xmath177 is set to zero , but also the rows corresponding to @xmath178 order mutants . analogously to above , the only terms in the vector @xmath37 that receive contributions from the @xmath178 order mutants are the ones in the rows corresponding to @xmath178 order of epistasis ( since the row corresponding to @xmath17@xmath67 order is already set to zero ) , meaning that their corresponding column again has only one non - zero element . hence setting these rows to zero will directly set their corresponding column to zero , and the equality holds . + + and so forth for regression to order @xmath180 , etc . + + qed + +

defining the extent of epistasis the non - independence of the effects of mutations is essential for understanding the relationship of genotype , phenotype , and fitness in biological systems . the applications cover many areas of biological research , including biochemistry , genomics , protein and systems engineering , medicine , and evolutionary biology . however , the quantitative definitions of epistasis vary among fields , and its analysis beyond just pairwise effects remains obscure in general . here , we show that different definitions of epistasis are versions of a single mathematical formalism - the weighted walsh - hadamard transform . we discuss that one of the definitions , the backgound - averaged epistasis , is the most informative when the goal is to uncover the general epistatic structure of a biological system , a description that can be rather different from the local epistatic structure of specific model systems . key issues are the choice of effective ensembles for averaging and to practically contend with the vast combinatorial complexity of mutations . in this regard , we discuss possible approaches for optimally learning the epistatic structure of biological systems . there has been much recent interest in the prevalence of epistasis in the relationships between genotype , phenotype , and fitness in biological systems @xcite . epistasis here is defined as the non - independence ( or context - dependence ) of the effect of a mutation , which is a generalization of bateson s original definition of epistasis as a genetic interaction in which a mutation masks the effect of variation at another locus @xcite . it is also in line with fisher s broader definition of epistacy @xcite . epistasis limits our ability to predict the function of a system that harbors several mutations given knowledge of the effects of those mutations taken independently @xcite , and makes these relationships increasingly more complex @xcite . from an evolutionary perspective , the presence of epistatic interactions may limit or entirely preclude trajectories of single - mutation steps towards peaks in the fitness landscape @xcite . with regard to human health , epistasis complicates our understanding of the origin and progression of disease @xcite . thus , interest in the extent of epistatic interactions in biological systems has originated from the fields of protein biochemistry , protein engineering , medicine , systems biology , and evolutionary biology alike . originally epistasis was considered in the context of two genes , but we can define it more broadly as the non - independence of mutational effects in the genome , whether the effects are within , between , or even outside protein coding regions ( e.g. in regulatory regions ) . the perturbations may go beyond point mutagenesis , but we limit the discussion here for clarity of presentation . importantly , the definition of epistasis can be extended beyond pairwise effects to comprise a hierarchy of 3-way , 4-way , and higher - order terms that represent the complete theoretical description of epistasis between the parts that make up a biological system . how can we quantitatively assign an epistatic interaction given experimentally determined effects of mutations ? since epistasis is deviation from independence , it is crucial to first explicitly state the null hypothesis : asserting what exactly it means to have _ independent _ contributions of mutations . this by itself can be non - trivial . in some cases the phenotype is directly related to a thermodynamic state variable , and the issue is then straightforward : independence implies additivity in the state variable . for example , for equilibrium binding reactions between two proteins , independence means additivity in the free energy of binding @xmath0 , such that the energetic effect of a double mutation is the sum of the energetic effects of each single mutation taken independently . however , in general , many phenotypes can not be so directly linked to a thermodynamic state variable , and quantification of epistasis needs to be accompanied by a proper rationale for the choice of null hypothesis . in what follows we will assume this step has already been carried out and we will equate independence with additivity of mutational effects . epistasis between two mutations is then defined as the degree to which the effect of both mutations together differs from the sum of the effects of the single mutations . in this paper , we describe three theoretical frameworks that have been proposed for characterizing the epistasis between components of biological systems ; these frameworks originate in different fields and use seemingly different calculations to describe the non - independence of mutations @xcite . we show that these formalisms are different manifestations of a common mathematical principle , a finding that explains their conceptual similarities and distinctions . each of these formalisms has its value depending on depth of coverage and nature of sampling in the experimental data , and the purpose of the analysis . in the end , the fundamental issue is to develop practical approaches for optimally learning the epistatic structure of biological systems in the face of explosive combinatorial complexity of possible epistatic interactions between mutations . demonstrating the mathematical relationships between the different frameworks for analyzing epistasis is a first key step in this process .

an intense research effort has been launched after the stabilization of graphene @xcite , a true two - dimensional system , due to the emergence of unconventional electronic properties with broad potential applications @xcite . a benchmark for a two dimensional system is the quantum hall effect , which indeed was observed in graphene@xcite , even at room temperatures@xcite . the research effort on graphene also includes its response to intense ac electric fields@xcite . the concurrence of an ac electric field in plane and a perpendicularly applied static magnetic field , acting simultaneously on a graphene system , will be the focus of our work here , motivated by another recently discovered emergent effect , namely the microwave induced zero resistance states in very high mobility , but still `` conventional '' , quantum hall systems@xcite . one characteristic feature of these states is that they are immune to the in plane ac field polarization@xcite , an aspect that continues to be addressed theoretically in `` conventional '' 2d systems@xcite . furthermore , the microwave intensities in these experiments could go beyond a perturbative regime , leading to the necessity of treating the electron - ac field interaction from a non - perturbative approach@xcite . these elements settle the scenario and context of our work : we investigate the evolution of the graphene quasi - energy spectrum in the presence of both , an ac field parallel and a static magnetic field perpendicular to the carbon atoms plane . on the contrary to electrons at the bottom of the gaas conduction band , graphene is intrinsically anisotropic and the response to ac fields should reveal fingerprints of this anisotropy . we address the problem from a tight - binding approach applying a floquet - fourier transformation and renormalization procedure that permits us to find the density of states of the quasi energy spectrum@xcite . this approach leads to the limitation of handling the problem only for finite systems . on the other hand , emerging effects for dot like structures may be identified , and indeed , small quantum dots out of a graphene layer is rapidly becoming a reality@xcite . the tight - binding approach for the graphene electronic structure coupled to an ac field is given by an hamiltonian divided in two parts , @xmath0 , where @xmath1 includes the interaction of the lattice with the ac field , while the interaction of the static magnetic field with the lattice is included in @xmath2 , @xmath3 @xmath4\bigr.+\ ] ] @xmath5\bigr\ } \label{eq.1}\ ] ] and @xmath6 where the index @xmath7 is for the ac electric field component parallel to the zigzag direction , and @xmath8 is for the ac field parallel to the armchair direction , as visualized in the bricklayer representation of a graphene finite lattice , fig . [ fig.1 ] . furthermore , @xmath9 are the index of the @xmath10 coordinates of the sites . the phase factor is defined as @xmath11 , where @xmath12 is the quantum magnetic flux and @xmath13 is the magnetic flux per a half unit cell of the graphene lattice , because @xmath14 , in mean value @xmath15 , and @xmath16 , the graphene lattice parameter . while the @xmath17 orbital energy will be taken constant , and we choose @xmath18 with no diagonal disorder . the hopping parameter are @xmath19 and @xmath20 as the nearest neighbors interaction and are taken constant and equal to 2.97 ev , along the continuous lines of the brickwall in fig . [ fig.1]@xcite . the ac field is described by @xmath21 and @xmath22 , the frequency and the field intensity , respectively . a floquet - fourier transformation , considering the so called floquet eigenstates @xmath23 with @xmath24 indicating the photon number states , redefines the hamiltonian @xmath25 as an infinite matrix with the elements given by@xcite : @xmath26 @xmath27\delta _ { m'm}\ ] ] @xmath28 this matrix is truncated at dimensions given by @xmath29 . @xmath30 and @xmath31 are the lateral sizes of the graphene finite lattice in number of atomic sites , while @xmath32 is the maximum photon index . since the ac field couples the _ photon replicas _ , for example , the floquet state indexed by @xmath24 to states labeled by @xmath33 or @xmath34 , multiple photon processes become relevant with increasing field intensity and @xmath32 settles how many replicas are taken into account . this floquet matrix is a tridiagonal block matrix with @xmath35 diagonal blocks , given by @xmath36 representing a photon replica with the matrix elements given by the left hand side of eq . the coupling of system to the intense ac electric field is represented by the off - diagonal blocks @xmath37 with the elements given by @xmath38 . the problem is numerically handled by means of a renormalization procedure@xcite , based on the associated green s function . the final result is the dressed green s function for one of the photon replicas , say @xmath39 , and a quasi - density of floquet states , @xmath40 can then be obtained as @xmath41 , $ ] in the atomic sites basis . the construction of our system is based on the topological equivalence between the brickwall - like and honeycomb lattices , as already pointed out by iye _ _ et al__@xcite , and as illustrated in fig . [ fig.1 ] for a rectangular dot of @xmath42 carbon - like sites . the actual system discussed in the following is a rectangular dot of @xmath43 sites@xcite . the edges of this rectangular graphene dot are zigzag along the @xmath7 direction with period @xmath44 and armchair along the @xmath8 with period @xmath45 ( see representation in fig . [ fig.1 ] ) . ] the starting point is the inspection of the density of states of the graphene quantum dot in the presence of a perpendicular magnetic field , but in absence of an ac field@xcite , shown in fig . [ fig.2 ] . in this figure both the electron ( @xmath46 ) and hole - like ( @xmath47 ) halfs of the spectrum are shown . along the horizontal axis at e=0 one can see the central landau level ( ll ) , @xmath48 . at higher(lower ) energies one clearly identifies the second , @xmath49 , and the third , @xmath50 , lls with the typical @xmath51 dependence . between the central and the second electron(hole)-like lls ( @xmath49 ) we see the edge states going down(up ) in energy with increasing magnetic field . at the upper(lower ) left corner , there is a wide region of strong mixing of the edge and bulk - like states , since there the magnetic length is comparable to the system size , ( this is the so called weak field limit , as elegantly discussed previously for a square lattice model@xcite ) . at higher magnetic fields , the right side of fig . [ fig.2 ] , the lattice effects start to develop giving rise to the self - similar hofstadter spectrum for a honeycomb lattice . our interest is focused on the energy versus magnetic field window in which the lls are already well defined but the lattice effects are not yet relevant . referring to fig . [ fig.2 ] , this window of interest spans from @xmath52 . hence we are focusing on the energy and magnetic field window between the weak field limit and the threshold for the lattice effects . indeed for the present small systems , bulk - like lls start to be defined here at very large magnetic fields . however , the present results can be scaled to lower magnetic fields in larger dots , since the proper length scale for defining a ll is simply a dot dimension larger than the magnetic length . the inclusion of an ac field with the electric field component in the graphene plane , parallel to either a zigzag or to an armchair direction is the main discussion stage in what follows . in fig . [ fig.3 ] , we show the electron - like part of the quasi density of states of the graphene dot electronic structure dressed by photons , defined by an energy of @xmath53 ev and a field intensity of @xmath54 v / cm applied parallel to the zigzag direction . initially these values have to be discussed in order to build up a clear perspective for the present work . this field intensity corresponds to @xmath55 mev , considering @xmath56 in eq . [ eq.3 ] as the lattice constant for graphene . these are again very high energies and field intensities when compared to actual experimental conditions ( the ac field are normally in the terahertz range@xcite for the effects under scrutiny ) . nevertheless , the high absolute values for the ac field intensity and photon energy also scale down for larger systems . in summary , the computational limitation to small systems lead the formation of lls beyond the weak field limit to high magnetic fields and , in order to observe interesting photon dressing effects , the ac field parameters are also increased . therefore the discussion of the results has to consider normalized units , @xmath57 and @xmath58 , leading to a view also valid for realistic and experimentally available parameter sets , as discussed below . reports on the microwave induced zero resistance states lead to estimates of the available field intensities in the 100 ghz range of @xmath59 . such a ratio already points toward the necessity of non - perturbative approaches@xcite . hence , in fig . [ fig.3 ] we limit ourselves to even lower intensities @xmath60 , in order to carefully follow the modification induced by dressing with photons the bare electronic structure of graphene . in the spectrum shown in fig . [ fig.3 ] we clearly see the @xmath61 replica of the @xmath48 ll ( the horizontal line at @xmath62 ev ) while only faint signatures of the @xmath63 replica of the @xmath64 ll can be perceived ( leading to a shadowy region below @xmath64 ll ) , as well as replicas of edge states . fig . [ fig.4 ] depicts the quasi density of states for an ac field with the same frequency and intensity , but now with the electric field polarized along the armchair direction . albeit the same field intensity , the replicas seem to be more intense , in particular the edge states ones , as can be particularly seen in an energy stripe around @xmath65 ev . this structure in the set of quasi edge states will be addressed in the following by means of further increasing the amplitude of the ac field . so far these results indicate a clear difference between the intensity of the photon replicas for fields applied either in the zig - zag and armchair directions , but a clear description of the effect is still missing . in this context , the quasi density of states for the graphene dot under an ac field along the zigzag direction is shown for higher frequency and field intensity parameters in fig . [ fig.5 ] : @xmath66 ev and @xmath67 . by comparing with fig . 3 and having in mind the photon doubled frequency , two replicas of bulk lls can be seen : now , besides the @xmath63 of ll @xmath64 , an analogous replica for ll @xmath68 can also be seen . a very strong replica of the central ll can be seen at @xmath69 ev , which now merges into the weak field limit states . besides the anisotropy in the intensity of the replicas that could be experimentally identified , no clear coupling between quasi states belonging to different replicas have been observed for this case of ac electric field parallel to the zigzag direction . indeed , anticrossings between quasi states are expected in several systems leading to interesting effects like collapsing superlattice minibands due to intense ac field induced dynamic localization@xcite or , more recently and in the context of the present work , new gaps openings in graphene systems @xcite . such couplings between different quasi states are revealed by increasing the intensity of an ac field polarized along the armchair direction , fig . [ fig.6 ] . now the hole - like edge states associated to the @xmath61 replica of the central ll ( rising in energy with increasing magnetic field ) anticross with the electron - like edge states associated to the bare ( @xmath39 ) ll ( lowering in energy with increasing magnetic field ) . therefore , these anticrossings occur at half the energy separation of both replicas , namely at @xmath70 ev for the chosen parameters . this picture can be clearly identified by the @xmath71 of superposing hole - like edge states shown in the lower half of the bare spectrum in fig . 2 with the electron - like edge states of the very same bare spectrum . such superposition , actually impossible in a bare system , becomes feasible with the building up of the ll replicas induced by the intense ac field . the appropriate anticrossing behavior leading to the formation of a nicely defined band is connected to the proper symmetry of an ac field along the armchair direction . a careful inspection of the miniband build up by the anticrossings of hole - like and electron - like states reveals a periodic modulation with increasing magnetic field . the period of this modulation , @xmath72 , represents a flux quantum through an area corresponding approximately 250 graphene unit cells , i.e. , of the order of the dot size . this suggests that this miniband shows the behavior of a quantum ring spectrum@xcite near the edges of the dot . the simplest model for a quantum ring is a one dimensional tight - binding ring of sites , enclosing a magnetic flux , which can actually be treated analytically@xcite . in experimental systems , like a nanostructured two dimensional electronic gas in gaas , quantum rings have finite thickness but bona fide quantum ring spectrum can be identified with these very miniband modulations with magnetic field@xcite . one should have in mind that such periodic modulation of the electronic structure is an outcome of the ring geometry , which is absent in our system . the actual potential felt by the electron is a photon dressed potential with a different effective symmetry@xcite , here the case of a ring . from another point of view , the formation of a quantum ring like electronic structure without a ring structure but in the presence of an ac field is a unique feature of a graphene quantum dot . an ac field induced quasi quantum ring is only possible due to the hole - like edge states associated to the central ll @xmath61 replica anticrossing with the @xmath39 electron - like edge states . it is worth to mention that it could be seen as an emergent property of the dirac - like electronic structure , since such an effect can not be observed for conventional two dimensional electronic systems for which below the lowest ll there is only a magnetic barrier@xcite and all lls show a linear dispersion with magnetic field . a remaining important question concerns the field polarization dependence . at a first glance , like mentioned in the introduction , the consequences of the intrinsic anisotropy of grapheme could be explored particularly in a low field quantum hall regime , due to the insensitivity of microwave induced zero resistance states to the field polarization in gaas based systems . here qualitatively different pictures emerge with different ac field orientations . the quantum ring like spectrum only appears for the ac electric field parallel to the armchair direction . nevertheless , the field induced ll replicas appear irrespective of the field polarization , albeit a quantitative difference in the intensity . since the quantum ring like miniband is build up by means of proper anticrossings , rather involved symmetry properties should be involved . a further hint refers to the interaction between bulk lls and their replicas . looking to the region of the spectrum in fig.6 where the @xmath61 replica of the @xmath48 ll crosses with the bulk ( @xmath39 ) @xmath64 ll , we see that this crossing occurs very near the weak field limit . therefore , if we do not consider high photon energies , the investigation of the coupling between these lls is hindered by the presence of edge states . it is also due to the small size of the dot considered here ( restricted by numerical costs ) that we have to consider very high photon energies to follow the interactions between bulk lls and their replicas . in spite of these features , the results suggest that the @xmath73 ll replica seems to preserve its identity throughout the entire range down to the zero magnetic field limit , a very different situation from what is expected for lls with either a @xmath51 or linear dispersion with the magnetic field@xcite . however , a closer observation reveals th at this preservation of the identity of the ll replicas occurs only for the armchair case . for ac fields parallel to the zigzag direction , fig . 5 , the bulk ll merges with the whispering gallery of edge states in the weak field limit and is completely smeared out . an analogous smearing out happens to replicas of lls with linear dispersion with the magnetic field , observed for square lattice models @xcite . therefore , electric fields parallel to the zigzag direction seem to have similar properties than electric fields parallel to one of the sides of a square lattice . it should be mentioned that real material edges should be a mixture of zigzag and armchair edges and the robustness of the effect has to be addressed in further work on such ac field induced quantum rings . in actual geometrically defined quantum rings , the characteristic magnetic field periodic spectrum is robust considering edge disorder @xcite . the several results depicted here may be summarized in two main findings . ( i ) the unique electronic structure of graphene near the dirac point , in the presence of a magnetic field , leads to anisotropic response to the ac field orientation , as revealed in the quasi density of states . ( ii ) the ac field induces a quasi - quantum ring ( only for the ac field along the armchair direction ) within the edge states spectrum , an emergent effect due to the flatness of the central ll and the special coupling among electronic - like and hole - like replicas of edge states . the authors would like to acknowledge the financial support from cnpq in the framework of a latin american research network . alcp also acknowledges support from fapesp , pas and phr , an additional partial support from cnpq and vracad - unmsm , respectively . 100 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 ) . a. k. geim and k. s. novoselov , nature mater . * 6 * , 183 ( 2007 ) . a. h. castro neto , f. guinea , n. m. r. peres , k. s. novoselov , and a. k. geim , rev . 81 * , 109 ( 2009 ) . k. s. novosolov , a. k. geim , s. v. morozov , d. jiang , m. i. katsnelson , i. v. grigorieva , s. v. dubonos , and a. a. firsov , nature * 438 * , 197 ( 2005 ) . y. zhang , y - w . tan , h. l. stormer , and p. kim , nature * 438 * , 201 ( 2005 ) . k. s. novoselov , z. jiang , y. zhang , s. v. morozov , h. l. stormer , u. zeitler , j. c. maan , g. s. boebinger , p. kim , and a. k. geim , science * 315 * , 1379 ( 2007 ) . v. p. gusynin , s. g. sharapov , and j. p. carbotte , phys . * 96 * , 256802 ( 2006 ) . s. a. mikhailov and k. ziegler , j. phys . : condens . matter * 20 * , 384204 ( 2008 ) and references therein . f. j. lopez - rodriguez and g. g. naumis , phys . rev . b * 78 * , 201406(r ) ( 2008 ) . t. oka , and h. aoki , phys . b * 79 * , 081406(r ) ( 2009 ) . r. g. mani , j. h. smet , k. von klitzing , v. narayanamurti , w. b. johnson , and v. umansky , nature * 420 * , 646 ( 2002 ) . m. a. zudov , r. r. du , l. n. pfeiffer , and k. w. west , phys . rev 90 * , 046807 ( 2003 ) . r. g. mani , physica e * 22 * , 1 ( 2004 ) . j. h. smet , b. gorshunov , c. jiang , l. pfeiffer , k. west , v. umansky , m. dressel , r. meisels , f. kuchar , and k. von klitzing , phys . lett . * 95 * , 116804 ( 2005 ) . s. wang and t - k . ng , phys . b * 77 * , 165324 ( 2008 ) . j. iarrea and g. platero , phys . b * 76 * , 073311 ( 2007 ) . p. h. rivera and p. a. schulz , phys . b * 70 * , 075314 ( 2004 ) . c. stampfer , j. gttinger , f. molitor , d. graf , t. ihn , and k. ensslin , appl . phys . lett . * 92 * , 012102 ( 2008 ) . l. a. ponomarenko , f. schedin , m. i. katsnelson , r. yang , e. w. hill , k. s. novoselov , and a. k. geim , science * 320 * , 356 ( 2008 ) . d. a. bahamon , a. l. c. pereira and p. a. schulz , phys . b * 79 * , 125414 ( 2009 ) . y. iye , e. kuramochi , m. hara , a. endo , and s. katsumoto , phys . rev b * 70 * , 144524 ( 2004 ) . the present system is rather small ( @xmath74 sites ) , while the smallest experimental graphene quantum dots are of the order of 20 000 sites . nevertheless , the features of the spectra discussed here can be scaled up to larger systems , as discussed in the text . u. sivan , y. imry , and c. hartzstein , phys . rev . b * 39 * , 1242 ( 1989 ) . s. s. gylfadottir , m. nitta , v. gudmundsson , a. manolescu , physica e * 27 * , 278 ( 2005 ) . m. holthaus , phys . lett . * 69 * , 351 ( 1992 ) . p. recher , b. trauzettel , a. rycerz , ya . m. blanter , c. w. j. beenakker , and a. f. morpurgo , phys . rev b * 76 * , 235404 ( 2007 ) . tan and j. c. inkson , semicond . technol . * 11 * , 1635 ( 1996 ) . r. kotlyar , c. a. stafford , and s. das sarma , phys . b * 58 * , 3989 ( 1998 ) .

a graphene quantum dot under intense ac field and static low magnetic field is investigated . from a tight - binding perspective , applying a fourier - floquet transformation and renormalization process , we observe that graphene -intrinsically anisotropic- reveals field polarization signatures in the quasi - density of states . for the ac field polarized along the armchair direction , the dressed electronic structure shows an emergent property : an ac field induced quantum ring . this is inferred by the orientation - dependent formation of a miniband of energy states periodically modulated with increasing magnetic field , exactly analogous to the behavior of a quantum ring spectrum .

nous nous intressons la frquence de contre - exemples au principe de hasse dans une famille de varits algbriques dfinies sur @xmath0 . les courbes de degr @xmath1 dans @xmath2 sont lobjet du travail de bhargava @xcite . le cas des surfaces de chtelet a t rcemment tudi par la bretche et browning @xcite . le but de cet article est de faire de mme pour les varits affines @xmath3 , dfinies par @xmath4 avec @xmath5 . larithmtique de @xmath6 a t tudie par colliot - thlne @xcite , qui a notamment montr que le choix de coefficients @xmath7 donne un contre - exemple au principe de hasse . notre investigation quantitative est fonde sur son travail . la varit @xmath6 est un espace principal homogne du tore coflasque @xmath8 daprs un rsultat de sansuc ( * ? ? ? 8.7 ) , lobstruction brauer manin est la seule obstruction au principe de hasse . une @xmath0-compactification lisse de @xmath6 et @xmath10 une caractristique intressante de @xmath6 est le fait quil existe un gnrateur universel explicite pour le groupe de brauer @xmath11 . en fait , suite ( * ? ? ? 4.1 ) , si @xmath12 on a @xmath13 avec lalgbre de quarternions @xmath14 comme gnrateur , tandis que , si lun des @xmath15 est dans @xmath16 , @xmath6 est @xmath0-rationnelle , et donc @xmath17 . nous utilisons cette description explicite pour dterminer la frquence laquelle il existe des contre - exemples au principe de hasse pour les varits . nous paramtrons les varits @xmath6 par lensemble @xmath18 il est vident que toute @xmath0-varit est @xmath0-isomorphe la varit dfinie par la mme quation avec @xmath19 . notre intrt principal est de dterminer la rpartition des lments de @xmath20 tels que @xmath21 est vide ( ou non - vide ) . la lumire de ( * ? ? ? 5.1(a ) ) , pour toute place @xmath22 de @xmath0 et chaque @xmath19 , on a @xmath23 . il ny a donc jamais dobstruction locale pour lexistence de @xmath0-points . soit @xmath24 , pour @xmath25 . nous estimons asymptotiquement , lorsque @xmath26 tend vers linfini , le cardinal @xmath27 notre rsultat principal est le suivant . [ main ] lorsque @xmath28 , on a @xmath29 o @xmath30 la diffrence @xmath31 est le nombre de varits @xmath6 paramtres par @xmath32 pour lesquelles @xmath33 . le cardinal @xmath34 tant facile estimer , nous obtenons le rsultat suivant . lorsque @xmath28 , on a @xmath35 en particulier , on a une proportion asymptotique de 100% des varits y qui ont des @xmath0-points . pendant llaboration de cet article , le premier auteur a t soutenu par un _ iuf junior _ et le _ projet anr ( pepr ) _ , tandis que le second auteur a t soutenu par la bourse _ nous rappelons quelques points cls du travail de colliot - thlne @xcite , sur les varits @xmath6 dfinies en , lorsque @xmath36 appartient lensemble @xmath20 dfini en . selon ( * ? ? ? 5.1(c ) ) , on a @xmath37 sil existe un nombre premier @xmath38 tel quaucun des @xmath39 ne soit pas un carr dans @xmath40 . supposons que pour chaque premier @xmath38 lun au moins des @xmath41 ou @xmath42 est un carr dans @xmath40 , alors il dcoule de ( * ? ? ? 5.1(d ) ) que @xmath43 si , et seulement si , @xmath44_p\equiv 1 { \,(\operatorname{mod}{2})}.\ ] ] ici @xmath45_p:{\mathbb{q}}_p^*\times { \mathbb{q}}_p^*\rightarrow { \mathbb{z}}/2{\mathbb{z}}$ ] est dfini par @xmath46_p}$ ] , o @xmath47 est le symbole de hilbert . lorsque @xmath48 , nous considrons @xmath49 et @xmath50 notre problme est donc dvaluer , lorsque @xmath26 tend vers linfini , la quantit @xmath51 o les dfinitions de @xmath52 et @xmath53 sont videntes . notre analyse de @xmath52 et @xmath53 est inspire du travail de friedlander et iwaniec @xcite . nous commenons avec lobservation @xmath54 o @xmath55 est la fonction de mbius . nous tendons la dfinition de la fonction @xmath55 de telle sorte que @xmath56 . le deuxime facteur est facile estimer . il vient @xmath57 il est clair que lun au moins des @xmath41 ou @xmath42 est un carr dans @xmath58 pour chaque premier @xmath59 . quand @xmath60 et @xmath42 est impair , la condition relative @xmath60 contenue dans @xmath61 est que lun au moins des @xmath41 or @xmath42 est congru @xmath62 modulo @xmath63 . rappelant que @xmath41 sont des entiers sans facteur carr , nous avons alors lgalit @xmath64 avec @xmath65 avec @xmath66 dfini sur les impairs par @xmath67 si @xmath68 et @xmath69 si @xmath70 , la loi de rciprocit quadratique snonce , lorsque @xmath71 et @xmath72 sont des nombres entiers impairs premiers entre eux , sous la forme @xmath73 nous terminons cette section par quelques mots sur les symboles de hilbert ( cf . soit @xmath38 un nombre premier et @xmath74 . supposant que @xmath75 et @xmath76 , o @xmath77 sont des @xmath38-units , alors pour tout @xmath78 nous avons @xmath79 tandis que lorsque @xmath60 , nous avons @xmath80 lorsque @xmath81 , nous considrons @xmath82 nous aurons besoin de ( * ? ? ? 2 ) , concernant @xmath83 o @xmath84 est un caractre modulo @xmath85 et @xmath86 dsigne le nombre de facteurs premiers distincts de @xmath87 . posons @xmath88 avec @xmath89 notons aussi @xmath90 la fonction caractristique des caractres principaux . [ l : ficor2 ] soient @xmath91 fix et @xmath92 lorsque @xmath93 @xmath94 @xmath95 , @xmath85 sont des entiers satisfaisant @xmath96 , @xmath97 et @xmath84 est un caractre modulo @xmath85 , on a @xmath98 dans ( * ? ? ? * cor . 2 ) , ce rsultat est dmontr pour @xmath99 , la condition supplmentaire @xmath100 tant inutile . les cas @xmath101 se dmontrent de la mme manire . soient @xmath102 , @xmath103 , @xmath104 , @xmath105 et @xmath106 . du lemme [ l : ficor2 ] , nous dduisons lestimation de la somme @xmath107 o les @xmath108 sont des caractres modulo @xmath109 . [ c : estq ] soit @xmath91 fix . lorsque @xmath110 , @xmath111 @xmath112 @xmath95 , @xmath113 , @xmath114 sont des entiers satisfaisant @xmath115 , @xmath116 et @xmath108 sont des caractres modulo @xmath109 , on a @xmath117 avec @xmath118 notons @xmath119 la quantit estimer . une interversion de mbius fournit @xmath120 nous pouvons restreindre la sommation aux entiers @xmath121 o @xmath122 la contribution complmentaire tant majore par @xmath123 . nous appliquons ensuite le lemme [ l : ficor2 ] avec @xmath99 . @xmath124 ici nous avons @xmath125 ce qui fournit donc le rsultat quitte modifier la valeur du paramtre @xmath126 . lorsque @xmath127 est grand , le corollaire [ c : estq ] est inutilisable . nous aurons besoin ainsi du rsultat suivant ( * ? ? ? * lemme 2 ) . [ l : filem2 ] soient @xmath128 @xmath129 des suites de nombres complexes telles que @xmath130 , do nt le support est inclus dans les nombres impairs . lorsque @xmath131 , on a @xmath132 pour estimer @xmath52 partir de , nous considrons @xmath133 lorsque @xmath134 et @xmath135 , nous notons @xmath136 la contribution dans @xmath137 des couples @xmath138 tels que @xmath139 et @xmath140 , @xmath141 @xmath142 . les couples @xmath143 appartiennent un ensemble @xmath144 modulo @xmath63 , avec @xmath145 lorsque @xmath146 et @xmath147 avec @xmath148 , @xmath141 et @xmath149 , la formule scrit aussi @xmath150 il vient @xmath151 la loi de rciprocit quadratique et la multiplicativit des caractres fournissent @xmath152 avec @xmath153 un calcul simple fournit le rsultat suivant . [ lem : calculu ] lorsque @xmath154 et @xmath155 , on a @xmath156 nous avons toujours @xmath157 en effet , le cas @xmath158 tant trivial , regardons le cas @xmath159 . il dcoule de que @xmath160 ce qui montre la formule dans ce cas . les raisonnements sont identiques pour @xmath161 ou @xmath162 . nous avons donc bien lexpression attendue . comme @xmath163 nous avons @xmath164 ce qui fournit le rsultat recherch . nous reprenons la dmarche dveloppe dans @xcite . pour cela , nous considrons @xmath165 o @xmath166 est un paramtre qui sera choisi suffisamment grand en fonction de la valeur de @xmath126 prise dans les applications du corollaire [ c : estq ] . la contribution @xmath167 des couples dentiers @xmath138 tels que @xmath168 est @xmath169 . dornavant , nous nous restreignons au cas @xmath170 . la contribution @xmath167 des couples dentiers @xmath138 tels que @xmath171 et @xmath172 est @xmath173 . de mme lorsque @xmath174 et @xmath175 . grce au lemme [ l : filem2 ] , du fait de la prsence du facteur @xmath176 , la contribution du cas @xmath177 et @xmath178 est @xmath179 ce qui suffit lorsque @xmath180 . nous avons la mme majoration lorsque @xmath181 et @xmath182 grce la prsence du facteur @xmath183 . il nous reste traiter le cas @xmath184 ou @xmath185 . dans le premier cas , nous sommes amens considrer lorsque @xmath186 la somme @xmath187 alors que , dans le deuxime cas , nous estimerons lorsque @xmath188 la somme @xmath189 en effet @xmath190 , lorsque @xmath154 , ne dpend pas de @xmath191 . nous avons @xmath192 o @xmath193 . cette somme peut donc tre estime grce au corollaire [ c : estq ] . nous obtenons @xmath194 o nous avons utilis la formule @xmath195 de mme , posant @xmath196 nous obtenons grce @xmath197 enfin , nous en dduisons @xmath198 avec @xmath199 o @xmath200 a t dfini en . grce , nous avons @xmath201 le lemme [ lem : calculu ] et les galits fournissent ainsi @xmath202 un simple calcul fournit @xmath203 en choisissant @xmath204 and @xmath205 , nous obtenons @xmath206 o @xmath207 partir de , il vient ainsi @xmath208 notre objectif dans cette section est destimer la somme @xmath209 les calculs sont plus compliqus que pour lestimation de @xmath52 mais relvent des mmes mthodes . lorsque @xmath210 , nous aurons besoin dune expression simple de la fonction @xmath211 dfinie en . comme @xmath212 pour tout nombre premier impair @xmath38 ne divisant pas @xmath213 , nous avons @xmath214 rappelons la dfinition de @xmath20 . nous paramtrons les @xmath19 par @xmath215 avec @xmath216 , @xmath217 , @xmath218 @xmath219 @xmath220 des nombres impairs , @xmath221 et les conditions de coprimalit @xmath222 lorsque @xmath228 sont sans facteur carr et @xmath229 , alors @xmath230 . de plus , lorsque @xmath231 et @xmath232 , le fait que @xmath210 implique que @xmath233 est un carr dans @xmath58 ce qui est exclu . lorsque @xmath231 , nous nous restreignons @xmath234 et donc @xmath235 et @xmath236 . ainsi daprs , nous avons @xmath237 . il vient @xmath238 en dveloppant le produit , nous obtenons @xmath239 de plus , nous avons @xmath240 en utilisant les notations , nous obtenons @xmath241 puisque la loi de rciprocit quadratique fournit @xmath242 si @xmath249 alors @xmath250 et ainsi @xmath251 . si @xmath252 et @xmath253 , alors @xmath210 implique @xmath254 . la formule implique encore @xmath255 si @xmath256 , alors @xmath210 implique @xmath257 . donc la formule implique le rsultat . si @xmath258 , alors @xmath210 implique que @xmath259 est impair et que @xmath259 ou @xmath42 est congru @xmath260 . lorsque @xmath261 , la formule implique le rsultat . avec les notations , nous avons @xmath266 dans la sommation , nous remplaons @xmath267 par @xmath268 tels que @xmath269 lorsque @xmath147 avec @xmath148 , @xmath141 @xmath149 o @xmath146 , la formule scrit aussi @xmath270 avec @xmath271 et @xmath272 satisfaisant . la formule fournit alors @xmath273 avec @xmath274 le paramtrage fournit aussi @xmath275 grce au lemme [ lem : h1 ] . lorsque @xmath276 avec @xmath277 , @xmath141 @xmath142 o @xmath146 , nous devons donc sommer le terme @xmath278 avec des sommations sur les entiers satisfaisant et @xmath279 , o @xmath280 ici , nous avons @xmath281 la sommation sur @xmath36 sopre de la mme manire que pour @xmath282 . quitte ngliger une contribution englobe dans le terme derreur , nous pouvons supposer que @xmath283 avec @xmath284 . puis en appliquant le lemme [ l : filem2 ] du fait du facteur @xmath285 , nous pouvons nous restreindre au cas @xmath286 ou @xmath287 . de mme , grce au lemme [ l : filem2 ] et le facteur @xmath288 , nous pouvons dsormais supposer que @xmath289 . le facteur @xmath290 ne dpend pas de @xmath291 mais seulement de leur valeur modulo @xmath63 . en fixant @xmath292 , la somme sur @xmath293 lorsque @xmath286 estimer est @xmath294 avec @xmath295 respectivement , en fixant @xmath296 , la somme sur @xmath191 estimer lorsque @xmath287 est @xmath297 avec @xmath298 la contribution principale provient du cas o les deux modules @xmath299 @xmath114 ( resp . @xmath300 @xmath301 ) des caractres somms sont gaux un . aprs cette sommation , il reste quatre variables sommer @xmath302 ( resp . @xmath303 ) . ensuite nous sommons les @xmath307 congrus @xmath308 avec @xmath309 impair , premiers @xmath310 en appliquant le lemme [ l : ficor2 ] avec @xmath311 . nous obtenons un terme principal @xmath312 avec @xmath313 nous aurons besoin du lemme suivant . nous utilisons lexpression pour les @xmath144 et le lemme [ lem : h2 ] pour le calcul de @xmath317 . la somme @xmath318 se dcompose sous la forme suivante @xmath319 pour la troisime ligne , on a not que @xmath320 ce qui achve la dmonstration . ensuite , avec la notation de [ s : lemmes ] , nous avons @xmath321 pour un impair @xmath95 . du lemme [ lem : last ] , il dcoule la formule @xmath322 puis enfin , nous sommons sur @xmath323 et @xmath324 . nous avons @xmath325 nous sommes donc amens une premire contribution de @xmath53 gale @xmath326 nous passons maintenant la deuxime contribution , lie . grce au corollaire [ c : estq ] , lorsque @xmath327 , le terme principal obtenu est @xmath328 avec @xmath329 ensuite , nous sommons les @xmath330 congrus @xmath308 avec @xmath309 impair , premiers @xmath331 en appliquant le lemme [ l : ficor2 ] avec @xmath311 . nous obtenons un terme principal @xmath332 nous avons @xmath333 o nous avons utilis la dfinition de @xmath200 . daprs les lemmes [ lem : calculu ] et [ lem : last ] , nous avons @xmath334 puis enfin une sommation sur @xmath323 et @xmath324 fournit une deuxime contribution @xmath53 gale la moiti de .

nous tudions le comportement asymptotique du nombre de varits dans une certaine classe ne satisfaisant pas le principe de hasse . cette tude repose sur des rsultats rcemment obtenus par colliot - thlne @xcite .

young , massive star clusters ( ymcs ) are the most notable and significant end products of violent star - forming episodes triggered by galaxy collisions , mergers , and close encounters . their contribution to the total luminosity induced by such extreme conditions dominates , by far , the overall energy output due to gravitationally - induced star formation ( e.g. , holtzman et al . 1992 , whitmore et al . 1993 , oconnell et al . 1994 , conti et al . 1996 , watson et al . 1996 , carlson et al . 1998 , de grijs et al . 2001 , 2003a , b , c , d , e ) . the question remains , however , whether or not at least a fraction of the compact ymcs seen in abundance in extragalactic starbursts , are potentially the progenitors of globular cluster ( gc)-type objects . if we could settle this issue convincingly , one way or the other , the implications of such a result would have profound and far - reaching implications for a wide range of astrophysical questions , including ( but not limited to ) our understanding of the process of galaxy formation and assembly , and the process and conditions required for star ( cluster ) formation . because of the lack of a statistically significant sample of similar nearby objects , however , we need to resort to either statistical arguments or to the painstaking approach of case by case studies of individual objects in more distant galaxies . the present state - of - the - art teaches us that the sizes , luminosities , and in several cases spectroscopic mass estimates of most ( young , massive ) extragalactic star cluster systems are fully consistent with the expected properties of young milky way - type gc progenitors ( e.g. , meurer 1995 , van den bergh 1995 , ho & filippenko 1996a , b , schweizer & seitzer 1998 , de grijs et al . 2001 , 2003d ) . however , the postulated evolutionary connection between the recently formed ymcs in regions of violent star formation and starburst galaxies , and old gcs similar to those in the galaxy , m31 , m87 , and other old elliptical galaxies is still a contentious issue . the evolution and survivability of ymcs depend crucially on the stellar initial mass function ( imf ) of their constituent stars ( cf . smith & gallagher 2001 ) : if the imf is too shallow , i.e. , if the clusters are significantly depleted in low - mass stars compared to ( for instance ) the solar neighbourhood , they will disperse within a few orbital periods around their host galaxy s centre , and most likely within about a billion years of their formation ( e.g. , gnedin & ostriker 1997 , goodwin 1997a , smith & gallagher 2001 , mengel et al . 2002 ) . ideally , one would need to obtain ( i ) high - resolution spectroscopy of all clusters in a given cluster sample in order to obtain dynamical mass estimates ( we will assume , for the purpose of the present discussion , that our ymcs are fully virialised based on their ages of @xmath7 yr , i.e. , many crossing times old ) and ( ii ) high - resolution imaging ( e.g. , with the _ hubble space telescope ; hst _ ) to measure their luminosities and sizes . in this paper , we explore the potential of a novel method to compare the properties of ymcs in the context of those of old gc systems , and predict their evolution over a hubble time . in section [ diagnostic.sec ] we outline the basic diagnostic tool we will use , based on the distribution of old gcs in @xmath8 space ( luminosity vs. central velocity dispersion ) . we extend this idea to younger clusters in section [ extension.sec ] , and discuss the uncertainties involved in our assumptions in section [ uncertainties.sec ] . section [ discussion.sec ] provides a detailed discussion of the implications of our results , and we conclude the paper in section [ summary.sec ] . it is well - known that the central velocity dispersion , @xmath2 , of old gcs in the galaxy and in m31 is tightly correlated with their @xmath0-band luminosity , @xmath9 ( e.g. , meylan & mayor 1986 , paturel & garnier 1992 , djorgovski 1991 , 1993 , djorgovski & meylan 1994 , djorgovski et al . mclaughlin ( 2000a ) suggests that this is a consequence of the tighter relationship between a cluster s binding energy , @xmath10 , and its luminosity , @xmath11 , which is one of the defining relationships of the gc fundamental plane . in fig . [ diagnostic1.fig ] we show this @xmath8 relationship for old gcs , represented by the filled symbols . we not only include the galactic and m31 gcs ( 56 and 21 objects , respectively ; pryor & meylan 1993 , djorgovski et al . 1997 , dubath & grillmair 1997 , dubath , meylan & mayor 1997 ; photometry from crampton et al . 1985 , bonoli et al . 1987 , reed , harris & harris 1994 ) , but have also added for the first time the data points for the ( @xmath12 gyr ) old compact magellanic cloud clusters ( 9 clusters ; dubath et al . 1993 , 1997 ; photometry from bica et al . 1996 , de freitas pacheco , barbuy & idiart 1998 ) , and the old gcs in m33 ( larsen et al . 2002 ) and the fornax dwarf spheroidal ( dsph ) galaxy ( dubath et al . 1992 , 1993 ) with available velocity dispersion measurements ( 4 and 3 gcs , respectively , for m33 and the fornax dsph ) . although uncertainty estimates are available for both the photometry and the central velocity dispersions , we decided not to include error bars for the individual objects for reasons of clarity . as an example , slightly larger than typical error bars are shown for ngc 2419 ; generally speaking , the uncertainties in the central velocity dispersion are @xmath1340 per cent ( or 0.100.15 dex ) , while the photometric uncertainties are mostly smaller than the symbol sizes . we find that the additional local group gcs follow , within the measurement uncertainties , the @xmath8 relationship for the galactic and m31 gcs . this is consistent with unpublished results for the fornax dsph and small magellanic cloud ( smc ) gcs referred to by djorgovski & meylan ( 1994 ) . the best - fitting relationship between the gc luminosities and their central velocity dispersion is represented by the long - dashed line in fig . [ diagnostic1.fig ] , which has the functional form @xmath14 or @xmath15 with correlation coefficient @xmath16 , when expressed in the logarithmic units used in fig . [ diagnostic1.fig ] . based on his identification of a gc fundamental plane , mclaughlin ( 2000a ) predicted a dependence of the form @xmath17 for the pre - core collapse gcs in the milky way . he found that the form of the correlations obtained by projecting the gc fundamental plane depends only weakly on cluster properties such as galactocentric distance and concentration in fact , these affect the normalisations of the relations rather than their slopes . our larger data set displays a relationship that is very similar to the predicted one . the most discrepant data point among the old gcs is that of the galactic gc ngc 2419 , as indicated in fig . [ diagnostic1.fig ] . it is one of the most luminous galactic gcs , and yet has one of the lowest measured central velocity dispersions ; both of these observational parameters are well determined and the uncertainties are too small to allow for the cluster to fall within the normal scatter around the best - fitting relationship ( cf . olszewski , pryor & schommer 1993 ) . the arrow extending from the gc s location to higher velocity dispersions indicates the expected value for its central velocity dispersion based on its structural parameters and calculated using single - mass isotropic king models with a constant mass - to - light ( m / l ) ratio of @xmath18 ( gnedin et al . ngc 2419 is a large ( half - light radius , @xmath19 pc ) , old ( @xmath20 gyr ) outer halo gc , located at a galactocentric distance of @xmath21 kpc ( harris 1996 ) . it is possibly not a normal gc , but has been speculated to be the stripped core of a former dsph galaxy ( e.g. , van den bergh & mackey 2004 ; but see section [ ymcdiagram.sec ] ) . its exclusion from the gc sample used to derive the best - fitting relationship between @xmath2 and @xmath1 does not alter this relationship significantly . the slope of the combined best - fitting relationship for all old local group gcs with measured velocity dispersions is , within the measurement uncertainties , consistent with the slopes most recently determined by djorgovski et al . ( 1997 ) for both the galactic and m31 gcs individually ( @xmath22 vs. @xmath23 ) and for the combined galactic / m31 gc sample ( @xmath24 ) . in fig . [ diagnostic1.fig ] we have also indicated the 2 , 3 and @xmath25 envelopes toward fainter luminosities of the scatter of the gc data points about the best - fitting relationship ( short - dashed lines ; we have adopted a gaussian distribution of the scatter , for reasons of simplicity ) . we will return to these envelopes in section [ extension.sec ] , where we will discuss the distribution and evolution of the younger clusters also included in this figure , and shown as the open circles and open squares . encouraged by the tightness of the @xmath8 relationship for old local group gcs , we added the data points for the ymcs for which velocity dispersion measurements are available in the literature . these are indicated by the numbered open circles ; table [ clusids.tab ] provides an overview of the ymc identifications and their age and metallicity measurements , and photometry . the ymcs are ranked in order of decreasing ( central ) velocity dispersion . since most velocity dispersion measurements in the literature are given as the `` observed '' velocity dispersion , corresponding to the one - dimensional line - of - sight component , and denoted by @xmath26 or @xmath27 , where relevant we corrected these measurements to reflect the _ central _ value of the velocity dispersion profile . in practice , this corresponds to applying an aperture correction to the measurements from the effective size of the apertures used ( typically corresponding to @xmath28 , for a given ymc ) . we adopted djorgovski et al.s ( 1997 ) correction for m31 gcs of @xmath29 ( see also mclaughlin 2000a ) . although the exact value of the clusters concentration , @xmath30 , is unknown in most cases , this correction is applicable where @xmath31 ( so that @xmath32 ) . this condition is met for all of the ymcs in our sample . djorgovski et al . ( 1997 ) estimated the uncertainty of this correction to be a few per cent , i.e. comparable to the measurement errors . we note that while this procedure possibly introduces uncertainties that are hard to quantify , our subsequent analysis is based on these values in _ logarithmic _ parameter space , where the impact of these uncertainties is minimised , @xmath33 dex ( see mclaughlin 2000a ) . yet , since the central velocity dispersions defining the @xmath34 relationship span more than an order of magnitude , our analysis of the relationship in logarithmic space does not penalise us in terms of the resultant accuracy . it is less straightforward to understand the effects of conversions of the original photometric data to the @xmath0 band used to construct fig . [ diagnostic1.fig ] . yet , because of the relatively small number of ymcs with measured ( central ) velocity dispersions , we endeavoured to include as large a data set as possible in order to increase the statistical relevance of the comparison done in this paper . the penultimate column in table [ clusids.tab ] indicates whether a given photometric entry was taken from the original reference , or derived from the original data . in the following sections , we will discuss our approach to these derivations on an object - by - object basis . our photometric conversion procedures are based on the following general principles : * where we needed to adopt a distance modulus to a given ymc s host galaxy , we used the most up - to - date values contained in the hyperleda database , except for m82 , where we adopted @xmath35 based on freedman et al.s ( 1994 ) cepheid - based distance to the m81/m82/ngc 3077 group . * conversions from a given passband to the @xmath0 band are age and metallicity sensitive ; we used the best available age and metallicity estimates , together with the most up - to - date galev simple stellar population ( ssp ) models ( schulz et al . 2002 , anders & fritze v . alvensleben 2003 ) , and assuming a kroupa ( 2001 ; hereafter kroupa01 ) imf , covering the mass range from 0.1 to 100 m@xmath36 ( see section [ mv.sec ] for details ) . the kroupa01 imf is one of the current best descriptions of the mass distribution of the stellar populations in the solar neighbourhood . below , we will also discuss the impact of adopting this imf on the uncertainties in our resulting , converted @xmath0-band magnitudes . puxley & brand ( 1999 ) obtained high - resolution mid - infrared spectroscopy of the two nuclear star clusters in ngc 1614 , using the gemini 8 m telescope . they calculated the objects individual bolometric luminosities to be @xmath37 and ( @xmath38 l@xmath39 , respectively . using the appropriate bolometric correction for the sun , we derive @xmath40 and @xmath41 mag , respectively . the uncertainties here are dominated by the uncertainties in the original conversion from mid - infrared flux to bolometric luminosity . the combination of using the bolometric correction for the sun and a metallicity of 2 z@xmath36 contributes an uncertainty of up to @xmath42 mag . this is of a similar order as the uncertainties in the original photometry , where given in table [ clusids.tab ] . photometry of the nuclear cluster in ngc 1042 was published by bker et al . ( 2004 ) and walcher et al . ( 2004 ) as @xmath43 mag . for the best age estimate of @xmath44 yr , our galev models for the appropriate metallicity indicate @xmath45 mag , thus leading to @xmath46 mag . the uncertainties in this conversion owing to the imf parameterisation adopted are minimal ; comparing the @xmath47 values for all of the imfs discussed in section [ mv.sec ] below , and assuming solar metallicity ( see table [ clusids.tab ] ) , we find a maximum difference among the @xmath47 colours predicted of @xmath48 mag , ranging from @xmath49 mag for the salpeter ( 1955 ) imf truncated at 1 m@xmath36 , to @xmath50 mag for a non - truncated salpeter imf . by having adopted the kroupa01 imf , we have essentially halved this uncertainty . a more important contribution to the photometric uncertainty arises from the fact that we have assumed the ngc 1042 nc to behave as a clean ssp . however , we note that this is perhaps a questionable assumption : nuclear clusters are more likely to be contaminated by secondary and tertiary star - formation episodes than more isolated star clusters in the outer regions of their host galaxies ( e.g. , cid fernandes et al . 2004 ) , so that in essence we are measuring the properties of a luminosity - weighted mean stellar population in this case . we will return to this discussion below . of the ymcs in the antennae galaxies , only clusters [ ws95]355 and [ m03 ] required photometric conversions to the @xmath0 band ; for the other ymcs we adopted the original photometry . because of their young ages , of @xmath51 and @xmath52 myr , the photometric uncertainties in the conversions to the @xmath0 band are more significant for these clusters than for the older nuclear cluster in ngc 1042 . for [ ws95]355 , mengel et al . ( 2002 ) reported only an upper limit in the @xmath0 band , but a well - determined flux in @xmath53 . it is in this age range where uncertainties in the treatment of the more massive component of any ssp , and in particular that of the red supergiants , render colour transformations significantly uncertain . adopting the same set of imfs as above , we find that @xmath54 mag , ranging from @xmath55 mag for the kroupa01 and kroupa , tout & gilmore ( 1993 , ktg93 ) imfs to @xmath56 mag for the truncated salpeter imf . as we will see in section [ mv.sec ] below , when evolved to an age of 12 gyr , this ymc does stand out , by @xmath57 mag , from the majority of the other ymcs in our sample . therefore , we believe that we can confidently include this object in our statistical analysis of the @xmath58 diagnostic diagram , despite this large photometric uncertainty , and despite the considerable uncertainty introduced by the poorly bracketed effects of internal extinction in the antennae system ( see sect . [ mv.sec ] below ) . unfortunately , we can not be as confident for cluster [ m03 ] . for this object , our only photometric data consists of the combination of a dynamical mass estimate ( @xmath59 m@xmath36 ) and a @xmath60-band m / l ratio of @xmath61 ( mengel 2003 ) . using @xmath62 , we then obtain @xmath63 . similar analysis as presented in the previous paragraph shows that the inherent photometric uncertainties at its young age caused by imf variations amount to @xmath64 mag , ranging from @xmath65 mag for the ktg93 imf to @xmath66 mag for the truncated salpeter imf . by adopting the kroupa01 imf as our imf parameterisation , we reduce this uncertainty to @xmath67 mag . contrary to [ ws95]355 , [ m03 ] does not stand out from the sample objects in any specific way , and in view of the large photometric uncertainty , we can only conclude that this cluster appears to follow the trend set by the bulk of the sample ( see section [ mv.sec ] ) . our @xmath0-band magnitudes for the three ymcs in ngc 1487 , also observed by mengel ( 2003 ) , were obtained using exactly the same procedure as used for antennae ymc [ m03 ] . once again , because of the ymcs ages clustering around 8 myr , the photometric uncertainty owing to the @xmath60-to-@xmath0 conversion is significant and highly imf dependent , with the most likely uncertainty on the order of @xmath68 mag , as discussed above . as we will see in section [ mv.sec ] , although these three objects show tentative differences with respect to the majority of our cluster sample , when evolved to a common age of 12 gyr , the large photometric uncertainty does not allow us to draw firm conclusions on these perceived differences . of the three sample ymcs drawn from the large cluster sample in m82 , we used the original photometry of smith & gallagher ( 2001 ) for m82-f , which the authors attempted to correct for the effects of a few saturated pixels . nevertheless , we are more confident using the corrected @xmath0 magnitude ( the quoted uncertainty in which already includes the effects caused by the saturated pixels ) than mccrady et al.s ( 2003 ) near - infrared _ hst _ photometry , in view of the much larger uncertainties introduced by filter conversions using a given imf ( see above ) . mccrady et al . ( 2005 ) report new acs observations of m82-f in the _ hst _ f555w band , but do not give the cluster s integrated magnitude in this filter . in view of the uncertainties involved in converting their f814w luminosity to a @xmath0-band flux , we are hesitant to take this approach . for objects mgg-9 and 11 , we have to resort to a similar technique as applied to the ngc 1487 clusters and to ymc [ m03 ] in the antennae galaxies . mccrady et al . ( 2003 ) provide _ hst_-equivalent @xmath69 ( f160w ) and @xmath70-band ( f222 m ) photometry for these two objects . given their age of @xmath71 myr , the uncertainty due to the passband conversion amounts to @xmath72 mag for the same range of imf parameterisations as used above . in addition , as we will show below ( section [ mv.sec ] ) , the additional photometric uncertainties owing to the intrinsic uncertainties in the f160w - band extinction estimates of mccrady et al . ( 2003 ) are considerable . based on the analysis of the effects of passband conversions on the quality of the input photometry for the diagnostic @xmath8 diagram , we conclude that the resulting uncertainties are most significant for the youngest objects . these converted @xmath0-band magnitudes should therefore be treated with caution . in our sample of 27 ymcs , this affects six objects , for which @xmath73 mag . for the remainder of the sample , the photometric uncertainties in the input data are significantly smaller , and mostly on the order of up to several tenths of a magnitude . : @xmath74 `` nc '' refers to nuclear clusters ; the original antennae cluster data is from whitmore & schweizer ( 1995 ; [ ws95 ] ) , whitmore et al . ( 1999 ; [ w99 ] ) and mengel ( 2003 ; [ m03 ] ) ; @xmath75 we adopted an age of 8 myr for this cluster ; @xmath76 based on broad - band photometry ; @xmath77 12 + log(o / h ) @xmath78 at a radius of 4 kpc and rising inward ; @xmath79 although no metallicity estimates are available , we adopted solar metallicity on the basis that the cluster was likely formed from pre - enriched material ; @xmath80 they adopted @xmath81 ; @xmath82 12 + log(o / h ) @xmath83 in the galactic centre ; @xmath84 @xmath85 and @xmath86 all give similar results ; we adopted solar metallicity ; @xmath87 these absolute magnitudes were corrected for extinction by the original authors , so that they represent @xmath88 ; @xmath89 based on the absolute magnitude in the _ hst _ f555w filter . * references * : 1 , agero & paolantonio ( 1997 ) ; 2 , aitken et al . ( 1981 ) ; 3 , aloisi et al . ( 2001 ) ; 4 , anders et al . ( 2004 ) ; 5 , bker et al . ( 1999 ) ; 6 , bker et al . ( 2005 ) ; 7 , de marchi et al . ( 1997 ) ; 8 , devost et al . ( 1997 ) ; 9 , efremov et al . ( 2002 ) ; 10 , greggio et al . ( 1998 ) ; 11 , ho & filippenko ( 1996b ) ; 12 , hunter et al . ( 2000 ) ; 13 , kobulnicky & skillman ( 1997 ) ; 14 , larsen et al . ( 2001 ) ; 15 , larsen et al . ( 2004 ) ; 16 , larsen & richtler ( 2004 ) ; 17 , maraston et al . ( 2001 ) ; 18 , maraston et al . ( 2004 ) ; 19 , mccrady et al . ( 2003 ) ; 20 , mccrady et al . ( 2005 ) ; 21 , mengel et al . ( 2002 ) ; 22 , mengel ( 2003 ) ; 23 , oconnell et al . ( 1994 ) ; 24 , origlia et al . ( 2001 ) ; 25 , puxley & brand ( 1999 ) ; 26 , schweizer & seitzer ( 1998 ) ; 27 , smith & gallagher ( 2001 ) ; 28 , tosi & daz ( 1985 ) ; 29 , vzquez et al . ( 2004 ) ; 30 , verma et al . ( 2003 ) ; 31 , walcher et al . ( 2004 ) . in order to compare the ymc loci with those of the gcs , we evolved the ymc luminosities to a common age of 12 gyr ( see the dotted arrows toward fainter luminosities in fig . [ diagnostic1.fig ] ) , using the most recent galev ssp models , and assuming a `` standard '' salpeter imf , covering the mass range from 0.1 to 100 m@xmath36 . we took special care to adopt the most appropriate ssp models , based on their current age and metallicity ( see table [ clusids.tab ] ) . in the remainder of this paper , wherever we refer to the evolution of our ymc sample to an age of 12 gyr , we implicitly assume this standard salpeter imf , and stellar evolution following the galev ssps , unless indicated otherwise . at first sight , we identify three main results based on this photometric evolution : 1 . almost all ymcs appear to evolve to loci on the fainter side of the old gc relationship . this may give us a handle on the functional form of the realistic imf , if we assume that these ymcs will evolve to obey the gc @xmath8 relationship at old age . in addition , it may help us to determine whether the ymc formation process itself is ( close to ) universal ; 2 . for most ymcs , luminosity evolution governed by a salpeter - type imf results in these objects ending up very close to the best - fitting gc relationship by the time they reach an age of 12 gyr ; 3 . a small fraction ( @xmath13 per cent ) of the ymcs appear to form a distinct group at significantly fainter luminosities than expected for old gc - type objects , if we evolve their luminosities assuming a salpeter - type imf . this implies that if their _ initial _ mass function ( mf ) was similar to the salpeter law , their _ present - day _ mf must be significantly depleted in low - mass stars if they are assumed to evolve to the gc relationship , as we will see below . alternatively , if the imf was unlike a salpeter - type imf , then comparison with the clusters discussed in point ( ii ) would suggest that imf variations exist in the highest - density regions in active starbursts , the birth places of these ymcs . in this context , it is worth noting that the tightness of the @xmath8 relationship for the local group gcs , and the lack of any significant dependence of gc properties on metallicity ( see also sect . [ uncertainties.sec ] and mclaughlin 2000b ) points to a universal imf in at least the local group . of the 20 ymcs with projected central velocity dispersions smaller than those of the most massive gc candidates in the local group ( @xmath90cen in the galaxy , and g1mayall ii in m31 ) 13 objects have the potential to evolve to a position in the @xmath8 diagnostic diagram within @xmath91 of the best - fitting gc relationship . since _ all _ of the gcs in our local group gc sample fall well within this @xmath91 envelope , we adopt this envelope as the stability boundary for a cluster to survive for a hubble time ( we realise that this is , of course , a relatively arbitrary assumption , but we will use it simply to guide the discussion ) . of the remaining 7 ymcs with projected central velocity dispersions smaller than those of @xmath90cen and g1 , 5 objects overshoot even the @xmath92 envelope if we adopt a standard salpeter imf for their stellar content . if this imf assumption is valid , then these objects would appear to be too dynamically hot , given their luminosities , to become old gc counterparts . if they are to evolve to loci close to the well - established gc relationship , their imf ( or their present - day mf ) must be significantly different from salpeter ; we will return to this issue in section [ mv.sec ] . the five objects with the largest projected central velocity dispersions are suspected to be either nuclear star clusters , or perhaps stripped dsph or dwarf elliptical ( de ) nuclei ( cf . ngc 7252-w3 = object 3 ; maraston et al . their range of central velocity dispersions overlaps that of the recently discovered `` ultracompact dwarf galaxies '' ( ucds ) in the fornax cluster ( e.g. , hilker et al . 1999 , drinkwater et al . 2000 , 2003 ) . the nature of these latter objects is as yet unclear : they may be very large star clusters ( perhaps stripped nuclear clusters ) , or instead extremely compact de galaxies , such as m32 . on the assumption that these objects constitute a new class of galaxies , drinkwater et al . ( 2003 ) argued that they follow the faber - jackson ( fj ) relation for elliptical galaxies , which has a slope that is markedly different from that of the gc relationship . the fj relation for elliptical galaxies , and the loci of the fornax ucds are also indicated in fig . [ diagnostic1.fig ] . intriguingly , the crossing point between the fj and gc relationships is very close to the locations of @xmath90cen and m31-g1 in the diagnostic diagram of fig . [ diagnostic1.fig ] ; both objects have been suggested to be the stripped nuclei of dwarf galaxies captured by their host galaxies . unfortunately , however , neither the location by itself of the fornax ucds on the fj relationship , nor of any of the other ( nuclear ) star clusters , provides conclusive evidence as to the nature of these extremely massive objects , unless their dominant stellar populations are older than @xmath93 gyr . for the fornax ucds to evolve to the gc relationship , their dominant stellar populations need only be as young as ( or younger than ) @xmath94 gyr , somewhat depending on metallicity , again assuming that they are governed by a standard salpeter - type imf and stellar evolution as described by the galev ssp models . hilker et al . ( 1999 ) analysed two of the five fornax ucds in more detail spectroscopically , and concluded that while object cgf 5 - 4 is most likely older than @xmath95 gyr ( ages as young as 3 gyr can be excluded with confidence ) , the location of object cgf 1 - 4 in the mg@xmath96 vs. @xmath97fe@xmath98 diagram suggests an age as young as @xmath99 gyr ( @xmath100 uncertainty ) , based on its h@xmath101 line strength . in addition , drinkwater et al . ( 2000 ) point out that the spectra of these objects are best fit by k - type stellar templates , consistent with an old ( metal - rich ) stellar population . this suggests that they might be related to gcs , since de galaxies observed with the same set up are best fit by younger f and early g - type templates . thus , the nature of these intriguing objects is still an open issue . if we now consider our sample objects with the largest central velocity dispersions in this context , and evolve their dominant stellar populations to a common age of 12 gyr , we find that they tend toward the best - fitting gc line , although within the uncertainties ( see section [ uncertainties.sec ] ) they are also consistent with objects following the fj relationship . we also note that while we have used ssp models to evolve the luminosities of these nuclear clusters to old age , this is strictly speaking not correct . nuclear clusters are not well described by `` simple '' stellar populations , but exhibit ( sometimes significant ) age ranges ( e.g. , cid fernandes et al . the implication of this is that , in fact , we may have _ overestimated _ the lengths of the luminosity evolution arrows in fig . [ diagnostic1.fig ] for these objects , depending on how much their stellar contents deviate from the ssp approximation , and from a salpeter - type imf ( see section [ mv.sec ] ) . the main consequence of this is that these nuclear clusters may indeed follow the fj relationship if they are able to survive to old age . thus , by placing the recently discovered ucds in this context , we believe that they may be closely related to nuclear star clusters , and perhaps are the stripped nuclei of de galaxies , akin to @xmath90cen , m31-g1 , and ngc 7252-w3 ( maraston et al . 2004 ; see also drinkwater et al . 2003 ) . let us now briefly return to the suggestion by van den bergh & mackey ( 2004 ) that the unusual gc ngc 2419 may also be a similar type of object . if this were the case , we would expect the cluster to be located close to either the fj relation in fig . [ diagnostic1.fig ] or if it were a genuine gc to the fundamental plane correlation for galactic gcs ( e.g. , dubath et al . 1997 , their fig . 16 , mclaughlin 2000a ) . in either case , the location of ngc 2419 is , respectively , @xmath102 and @xmath103 ( where @xmath104 represents the measurement uncertainty ) removed from the fiducial relationship . therefore , we conclude that it is unlikely that ngc 2419 is the stripped core of a dsph galaxy . we note that , thus far , we have only considered the evolution of the ymcs in terms of their luminosity , and have ignored the possibility of significant evolution of the central velocity dispersion over a hubble time . following an initial phase of mass loss caused by stellar evolution , the long - term dynamical evolution of star clusters is dominated by evaporation due to internal relaxation and stripping due to external , tidal shocks . the latter process removes mass ( and luminosity ) , but should not significantly affect the central velocity dispersion ( e.g. , djorgovski 1991 , 1993 , djorgovski & meylan 1994 ) . it is unclear , however , how the central velocity dispersion evolves over time as a result of internal evolution in the presence of external tidal fields , significant binary fractions , the effects of mass segregation and core collapse . @xmath5-body simulations present an ideal way to investigate this problem . however , despite the vast literature on @xmath5-body simulations of star clusters , we are not aware of any paper which presents the evolution of the central , projected velocity dispersion of the simulated clusters . therefore , in section [ sigma.sec ] we investigate the evolution of the observable properties of a set of simulated @xmath5-body clusters in order to constrain the expected evolution of the observed @xmath2 . in the previous sections we have constructed a diagnostic tool that could potentially tell us whether a given ymc might evolve into a gc - type object over a hubble time , based on only two observables : the cluster s ( central ) velocity dispersion and its @xmath0-band luminosity ( or absolute magnitude ) . this provides a simpler and potentially more reliable method to predict , to first order , the evolutionary fate of ymcs than existing methods . in particular , the most common method to assess this issue is based on the comparison of dynamical cluster mass estimates with a variety of imf descriptions in the ( age vs. m / l ratio ) plane . this method introduces two complications that we can in principle avoid using the @xmath8 approach : in order to estimate an object s dynamical mass , one needs to ( i ) assume that the virial theorem applies ( which is generally assumed to hold for clusters older than @xmath105 myr ) , and ( ii ) obtain a reliable measurement of the cluster radius . while the complication introduced by the assumption of virialisation is minimal ( although it may play a significant role for the youngest objects in our sample ! ) , measuring reliable cluster radii is problematic for all but the nearest objects . in addition , using the half - light radius as an estimate of the volume occupied by the cluster implicitly assumes that the m / l ratio is constant across the cluster an assumption that may be unjustified in the presence of significant mass segregation , as shown observationally ( see , e.g. , de grijs et al . 2002b , and references therein ; see also section [ m82clus.sec ] below and the discussion in mccrady et al . thus , here we have presented a simpler and potentially more reliable method to predict the approximate evolution for a given ymc than currently available . we will now compare the predictions from this new method to those obtained from the dynamical mass estimates , in order to assess the robustness of the @xmath8 approach , on a case by case basis , for those of our sample clusters for which this information is available . where appropriate , we will also point out those cases where discrepancies between our new results and previous predictions occur ; these provide a useful insight into the uncertainties inherent to the use of any of the methods currently employed in this field . for the purposes of this discussion , we will consider whether the observational data are consistent with the assumption that all surviving old star clusters will obey the local group gc correlation between @xmath1 and @xmath2 , within the uncertainties . mengel et al . ( 2002 ) concluded , aided by ground - based @xmath60-band luminosities , that clusters [ w99]1 and [ w99]2 appeared to have a deficit of low - mass stars ( see their fig . 7 ) , either because of a shallower - than - salpeter imf slope down to stellar masses of @xmath106 m@xmath36 , or because of a low - mass imf cut - off . their results for ymcs [ w99]15 , 16 and [ ws95]355 are more consistent with a steeper imf slope , similar to or steeper than the standard salpeter slope ( or , alternatively , an overabundance of low - mass stars compared to the standard salpeter imf ) , down to low masses . these results are supported by their _ hst_-based @xmath0-band observations for [ w99]1 , 15 and 16 ( although the uncertainties for cluster [ w99]1 make it a potential object with a salpeter - type slope ; see their fig . 6 ) , although the opposite trend is found for object [ w99]2 , at a level of 23 times the uncertainty in the measurements . this object appears to be characterised by a decidedly larger proportion of low - mass stars based on its @xmath0-band photometry than seemed to be the case based on the @xmath60-band data ( see below for a discussion ) . it is striking that they seem to find systematically steeper imf slopes ( or , equivalently , imfs richer in low - mass stars ) in the higher - density overlap region between the two merging galaxies ( containing clusters [ w99]15 , 16 and [ ws95]355 ; although [ w99]16 may not be located in the densest region , we believe its ambient density to be much higher than that in the outer regions of the system ; see also mengel et al . [ 2002 ] ) , while the low - mass deficient imfs are found in the outer spiral arms ( containing objects [ w99]1 and 2 ) . mengel ( 2003 ) obtained similar quality measurements for the additional ymcs [ m03 ] and [ w99]331 , both of which appear to be characterised by a `` normal '' imf with a salpeter - type slope down to 0.1 m@xmath36 in their diagnostic ( age vs. @xmath107 ) diagram . if we adopt the assumption that these ymcs will eventually evolve to loci close to the @xmath8 relation for old gcs at least , if they survive sufficiently long then our diagnostic @xmath8 diagram suggests that clusters [ ws95]331 , [ ws95]355 , [ w99]15 , and [ w99]16 ( objects 19 , 6 , 7 and 11 in table [ clusids.tab ] , respectively ) are characterised by a present - day mf that differs significantly from a standard salpeter - type ( i)mf ; evolved to an age of 12 gyr using a salpeter imf , their luminosities will fade to well beyond the @xmath92 envelope . this conclusion remains valid even in view of the large photometric uncertainty associated with [ ws95]355 ( see section [ antunc.sec ] ) . antennae ymcs [ w99]1 and [ m03 ] ( objects 20 and 23 , respectively ; note the large photometric uncertainty associated with [ m03 ] ) , on the other hand , appear to have an ( i)mf that is closer to the salpeter function down to low stellar masses , if we assume that when the current generation of ymcs in the local universe evolves to gc - type ages , they should also occupy the gc relationship . depending on the uncertainties in the luminosity evolution ( see section [ uncertainties.sec ] ) , cluster [ w99]2 s ( object 15 ) evolved location in the @xmath8 plane is also consistent with such a salpeter - type ( i)mf . we note , however , that all of these objects may well have non - salpeter - type mfs , considering that our simple modelling lets them evolve to significantly fainter magnitudes than expected if they were to obey the well - defined local group gc relationship at similar age . in order for a ymc to survive to old age , it needs to have sufficient low - mass stars to remain bound for a hubble time . this condition is met for salpeter - type imfs extending down to masses on the order of 0.1 m@xmath36 , but not for objects with much shallower slopes , or ( obviously ) a low - mass cut - off . thus , from a detailed comparison between our results and those presented in figures 6 and 7 of mengel et al . ( 2002 ) and in mengel ( 2003 ) , we conclude that , on average , we obtain similar predictions for the future evolution of the antennae ymcs , although our detailed conclusions may differ for some of the individual objects . for instance , while mengel et al . ( 2002 ) suggest that [ ws95]355 and [ w99]15 may be better represented by a slightly steeper than salpeter slope , @xmath108 for the full mass range from 0.1 to 100 m@xmath36 , we do not believe that the uncertainties inherent to the data warrant such a fine distinction . while for objects [ w99]1 and 2 they obtain somewhat conflicting results from their @xmath0 and @xmath60-band data , our conclusions ( based on the @xmath0-band data ) agree for [ w99]1 , but differ for [ w99]2 . these discrepant results may in part be explained by the difficulty of obtaining clean cluster photometry from ground - based ( @xmath60 ) versus _ hst_-based ( @xmath0 ) data ; the difference in m / l ratios in mengel et al.s ( 2002 ) between the @xmath0 and the @xmath60 band is as expected if source confusion played a more important role in the ground - based images . in addition , in the presence of significant mass segregation , one would also expect to obtain different results between the @xmath0 and @xmath60-band m / l ratios ( e.g. , mccrady et al . 2003 , 2005 ) , in a similar sense as seen here . however , the data of mengel et al . ( 2002 ) show a general offset between the @xmath0 and the @xmath60 band for all of their objects , so that this can not be the only explanation . in essence , this shows the extent to which one can rely on any individual approach ; it shows , in particular , that conclusions on the evolution of the objects that are predicted to evolve to the area close to the 2@xmath92 transition region in fig . [ diagnostic1.fig ] should be treated with caution . finally , most of the objects that we predict to overshoot the @xmath92 boundary by a significant amount by the time they reach an age of 12 gyr are located in the higher - density regions of the system . it is likely that the ambient pressure in the interaction region is significantly higher , and externally driven dynamical evolution proceeds faster than in the more quiescent spiral arm regions ( section [ implications.sec ] ) ; this may render invalid the assumption that these clusters are in virial equilibrium , in particular in view of their very young ages , of @xmath109 myr ( mengel et al . 2002 , mengel 2003 ; see table [ clusids.tab ] ) . based on the @xmath107 determinations in mengel ( 2003 ) and their location in the ( age vs. @xmath107 ) diagram , the luminosities of ymcs ngc 1487 - 1 and 2 are consistent with salpeter - type imf slopes down to masses of @xmath106 m@xmath36 . cluster ngc 1487 - 3 , on the other hand , has a much lower @xmath60-band m / l ratio for approximately the same age ( see mengel 2003 ) , which is indicative of a steeper imf slope . evolved to a common age of 12 gyr in fig . [ diagnostic1.fig ] , clusters ngc 1487 - 1 and 2 are found in the boundary region between gc stability and gc dissolution , i.e. , between the 2 and 3 @xmath110 envelopes . the uncertainties in the @xmath0-band photometry that we obtained from our @xmath60-to-@xmath0 conversions , and also the luminosity evolution may reduce the lengths of their luminosity evolution arrows ( see section [ uncertainties.sec ] ) , so that these objects may potentially evolve into gc - type objects over a hubble time ( but see section [ implications.sec ] ) . compared to ngc 1487 - 1 and 2 , object ngc 1487 - 3 , appears to be an outlier , which may evolve to well beyond the @xmath92 envelope if its present - day mf is salpeter - like . however , we note that the large photometric uncertainty introduced by our passband conversion only allows us to conclude this tentatively . if we compare the loci of the ngc 1487 ymcs in the ( age vs. @xmath107 ) diagram of mengel ( 2003 ) with their expected evolution in the @xmath8 diagram of fig . [ diagnostic1.fig ] , we conclude that our results are consistent with those of mengel ( 2003 ) . clusters 1 and 2 are ( perhaps marginally ) consistent with salpeter - type mfs , while ymc 3 is characterised by an overabundance of low - mass stars compared to clusters 1 and 2 ( and compared to the standard salpeter imf ) , and is better represented by an imf with a steeper - than - salpeter slope ( @xmath111 ) for a stellar mass range from 0.1 to 100 m@xmath36 . once again , these objects are among the youngest in our sample , and as such they may not yet be entirely virialised . when we evolve the luminosities of clusters f , mgg-9 and mgg-11 to a common age of 12 gyr , they are all found within @xmath112 about the gc relationship . this implies , again adopting the assumption that all old gcs are confined to a narrow distribution in @xmath8 space and characterised by a salpeter imf , that these three m82 clusters may potentially evolve into gc - type objects . mccrady et al . ( 2003 , 2005 ) suggest that all three clusters are affected by significant mass segregation , whether primordial or dynamical : every single ymc studied in sufficient ( spatially resolved ) detail to date is known to show significant mass segregation , from the youngest ages ( see de grijs et al . 2002a , b for a discussion ) . in the presence of significant mass segregation , the estimated ymc masses are lower limits . mccrady et al . ( 2003 ) concluded that mgg-9 and mgg-11 are consistent with salpeter - like imfs , _ in the presence of significant ( primordial ) mass segregation . _ neglecting the effects of mass segregation , mgg-11 appears to be high - mass dominated . this scenario seems to be confirmed by our results based on fig . [ diagnostic1.fig ] . smith & gallagher ( 2001 ) , on the other hand , concluded that m82-f will likely dissolve within the next @xmath113 gyr . they concluded that its imf was likely truncated at a lower mass of 23 m@xmath36 , thus retaining too few low - mass stars to produce a bound cluster over time - scales longer than a gyr . however , mccrady et al . ( 2003 , 2005 ) provide evidence for mass segregation in cluster f ( resulting in more compact profiles at redder wavelengths ) , while they also redetermine the age to be toward the lower limit of the uncertainty range quoted by smith & gallagher ( 2001 ) . the latter authors result is also affected by a somewhat uncertain correction for the saturated cluster centre in the _ hst _ @xmath0-band image . taking all of these effects together , mccrady et al . ( 2003 , 2005 ) conclude that m82-f may be deficient in low - mass stars ( i.e. , a simple application of ssp models to the observed m / l ratio suggests a low - mass cut - off at @xmath114 m@xmath36 ) , although in view of the significant mass segregation present , it is equally likely characterised by a `` standard '' imf . these results support our conclusion . of the two m83 clusters in our sample , object ngc 5236 - 502 appears to be characterised by a standard salpeter imf , based on the fact that adopting this imf will let the ymc evolve to a location close to the old gc relationship . this is fully consistent with the conclusion reached by larsen & richtler ( 2004 ) , based on their more complex analysis of the cluster s dynamical mass and its corresponding m / l ratio . cluster ngc 5236 - 805 , however , appears to overshoot the @xmath91 envelope somewhat , if it were governed by a similar initial and/or present - day mf , although the uncertainties inherent in the luminosity evolution ( see section [ uncertainties.sec ] ) still allow for this object to have a close - to - salpeter mf . thus , we conclude that our results for this object are also consistent with larsen & richtler s ( 2004 ) independent assessment . the measurements for ngc 1569-a1 are affected by significant uncertainties . the original high - dispersion spectra of ho & filippenko ( 1996a ) are contaminated by flux from the its binary companion cluster , a2 , which was first realised by de marchi et al . however , since a1 is almost twice as bright as a2 , de marchi et al . ( 1997 ) argued that the basic velocity dispersion measurement of ho & filippenko ( 1996a ) still reflects that of the main component , a1 . in addition , because of the contamination by a2 , the age determination of component a1 is affected by significant uncertainties ( see table [ clusids.tab ] ) . for the purpose of the present paper , we have used the most up - to - date photometry of de marchi et al . ( 1997 ) and the best age determination of @xmath115 myr ( hunter et al . 2000 , origlia et al . when we evolve the cluster s luminosity to an age of 12 gyr , it is found on the @xmath91 envelope of the gc relation . the uncertainties inherent in the luminosity evolution are such that any correction will result in this evolution being reduced and thus the cluster would end up closer to the gc relation . therefore , we predict that ngc 1569-a1 will likely become an old gc ( in the absence of external disruptive forces ; see section [ implications.sec ] ) . as a consequence , we also suggest that the cluster s imf may be close to the standard salpeter imf . our conclusions are consistent with those of de marchi et al . ( 1997 ) , based on their analysis of the evolution of the m / l ratio , assuming a salpeter imf down to the hydrogen - burning limit , and with origlia et al . ( 2001 ) , based on ssp fits governed by variety of imfs . our results are also consistent with ho & filippenko ( 1996a ) , despite different assumptions used for the mass determinations ; these authors also concluded that to a first approximation the ngc 1569-a imf appeared to be similar to that of typical galactic gcs . ho & filippenko ( 1996b ) concluded , using a similar approach as for ngc 1569-a ( i.e. , a1 and a2 combined ) , that ngc 1705-i has all the properties ( m / l ratio , radius , mass ) of a young , metal - rich gc ( but note the caveat mentioned above regarding their mass determinations ) . in the most recent detailed study of the stellar content of ngc 1705-i , vzquez et al . ( 2004 ) conclude based on _ hst_/stis spectroscopy and an analysis of the cluster s m / l ratio that there is no significant evidence for an anomalous imf at the low - mass end , contrary to previous suggestions ( see references in vzquez et al . this is fully consistent with the location of the ymc in our diagnostic @xmath8 diagram when evolved to an age of 12 gyr . larsen et al . ( 2004 ) obtained high - dispersion spectra for four ymcs in the dwarf irregular galaxies ngc 4214 and ngc 4449 . for all clusters , they find m / l ratios that are similar to or slightly higher than for a salpeter or kroupa01-type imf . they thus rule out any present - day mf that is deficient in low - mass stars compared to these imfs . they conclude that these objects might therefore evolve to become old gcs over a hubble time . this conclusion is fully supported by the location of the evolved ymcs in our diagnostic diagram of fig . [ diagnostic1.fig ] . just as for the ymcs in ngc 4214 and ngc 4449 , larsen et al . ( 2004 ) also conclude that the present - day mf of ngc 6946 - 1447 resembles a salpeter or kroupa - type mf quite closely . they essentially confirmed their earlier result for this cluster ( larsen et al . 2001 ) where they concluded that the estimates for its dynamical mass and its photometric mass based on ssps governed by a salpeter imf were similar within the model uncertainties . thus , this object also has the potential of evolving into an old gc if not disrupted prematurely by external factors . this is again fully consistent with the cluster s evolved location in our diagnostic @xmath8 diagram . finally , in a detailed spectroscopic and photometric study , maraston et al . ( 2004 ) conclude that the dynamical virial mass for ngc 7252-w3 , based on their newly obtained high - dispersion spectroscopy , is in excellent agreement with photometric values previously estimated ( schweizer & seitzer 1998 , maraston et al . 2001 ) from the cluster luminosity by means of stellar m / l ratios predicted by ssp models with a salpeter imf down to stellar masses of @xmath106 m@xmath36 . while this conclusion is consistent , within the uncertainties , with the object s evolved location in our diagnostic diagram of fig . [ diagnostic1.fig ] , its velocity dispersion places it in the realm of the nuclear clusters and ucds , so that caution needs to be exercised when comparing results in this context . thus , it appears that the simple diagnostic @xmath8 diagram results in consistent predictions regarding the evolution of ymcs in the local universe , without the need to convert the observed velocity dispersions into dynamical masses and thus introducing additional assumptions and their associated uncertainties . discrepancies between predictions on the ymcs evolutionary fate resulting from the application of different methods serve as a useful diagnostic providing insight into the likely range of uncertainties involved in any of these predictions . we note that our predictions should be treated as first - order predictions ( as should those resulting from using dynamical mass estimates ) . they do not include external factors that might speed up the dissolution of otherwise firmly bound star clusters ; we will address this issue in section [ implications.sec ] . nevertheless , to first order , the fact that most clusters , when evolved using a standard solar - neighbourhood salpeter - type imf , appear to end up close to the gc relationship ( although systematically somewhat to fainter magnitudes ) instills some confidence in the universality of this imf for extragalactic ymcs , leaving little leeway for significant imf variations , _ assuming that they may potentially survive for a hubble time_. we note in passing that dynamical evolution of @xmath2 will tend to move our sample clusters even closer to the old gc relation , adding weight to this conclusion ( see section [ sigma.sec ] ) . we will discuss those objects that still appear to overshoot the gc relation in more detail in section [ implications.sec ] . finally , in fig . [ diagnostic1.fig ] we have also included the relevant data points for the compact lmc and smc clusters younger than 10 gyr at the present time ( open squares ; dubath et al . 1993 , 1997 ; photometry from bica et al . 1996 , de freitas pacheco , barbuy & idiart 1998 ) . if these objects are characterised by a salpeter - type present - day mf and imf , as is supported by observational evidence ( see , e.g. , de grijs et al . 2002a , b for a representative sample of compact lmc clusters ) , they will fade by up to @xmath116 mag ( and in most cases by more than @xmath117 mag ) before they reach an age of 12 gyr . however , very few of the compact lmc and smc clusters extend to fainter absolute magnitudes than contained within the @xmath118 envelope of the best - fitting gc relation . this implies either that cluster disruption , at least in the magellanic clouds , must occur before a cluster fades to this limit , or that the old gc relation for the lower - density lmc environment is significantly different from ( and much broader than ) that in the galaxy and m31 . if we assume that the gc relation is independent of environment , as seems to be suggested by the good agreement of the old gcs in the local group , we predict that at least half of the lmc and smc clusters younger than 10 gyr will dissolve before reaching gc - type ages . the small number of lmc and smc clusters currently beyond the @xmath119 boundary may either be caused by statistical sampling effects or perhaps we have caught objects in the process of dissolution . once again , the presence of these objects gives a good indication of the uncertainties involved in using the @xmath8 diagnostic diagram : there is most likely a transition region in the diagram where clusters may or may not evolve to , depending on the details of their internal and environmental properties . in this context , we note that the lmc provides a fairly low - density stellar environment , particularly outside the central , barred region . the two magellanic cloud objects toward brighter magnitudes than the best - fitting gc relationship are the youngest lmc cluster for which we have velocity dispersion information , ngc 1818 ( 25 myr ; de grijs et al . 2002a ) and ngc 419 in the smc . if they are characterised by salpeter - type imfs down to @xmath106 m@xmath36 ( cf . de grijs et al . 2002b ) , these objects are likely to fade by @xmath120 and @xmath114 mag , respectively . judging from their location in fig . [ diagnostic1.fig ] , we predict that while ngc 419 may possibly become an object equivalent to ngc 121 ( the only gc - equivalent object in the smc ) , ngc 1818 will likely disperse long before . we emphasise that in this case we have independent measurements of the cluster s present - day mf ( de grijs et al . 2002a , b ) , so that this is a firm conclusion . in this context , it is interesting to compare these results for the massive , compact star clusters in the local group to the galactic open clusters . the galactic cluster population exhibits a clear dichotomy , in the sense that all galactic gcs are older than @xmath105 gyr , while few galactic open clusters are older than a few gyr . if we include the roughly 40 galactic open clusters with relevant observational data ( lohmann 1972 , sagar & bhatt 1989 ) in our diagnostic diagram , they occupy a well - delineated region centred at @xmath121 , and lying on the extrapolation of the gc relationship . considering that , if they were governed by a salpeter - type imf down to the hydrogen - burning limit , they would fade by at least another 2 mag , their location in the @xmath8 diagram is consistent with the observational fact that there are no known open clusters of typical gc age in the galaxy . having established that , to first order , the @xmath8 diagram provides us with a diagnostic tool to assess the similarities ( and differences ) of ymcs compared to old gcs , we will now assess the uncertainties inherent to this approach . in section [ mv.sec ] we will first address the uncertainties related to the evolution in luminosity of a given cluster . subsequently , in section [ sigma.sec ] we will present the results of detailed _ n_-body simulations to obtain a feeling for the uncertainties associated with the evolution of the central velocity dispersion over a hubble time . the main issue we need to address regarding the luminosity evolution of our sample ymcs , as represented by the `` luminosity evolution arrows '' in fig . [ diagnostic1.fig ] , is the accuracy of the arrow lengths . in addition , we will address a number of issues related to the accuracy of the photometric measurements of the objects themselves . regarding the former , the key issues to be discussed are the dependence of the luminosity evolution on ( i ) metallicity and ( ii ) the adopted imf ( and , therefore , on the adopted ssp models ) . in fig . [ uncertainties.fig]a , we show the expected length of the luminosity evolution arrow as a function of cluster age , ( @xmath122 ) , for the five different metallicities included in the galev ssps . for the purposes of this discussion , we have adopted a salpeter imf , covering stellar masses from 0.1 to 100 m@xmath36 . it is clear that the effect of adopting an incorrect metallicity is roughly constant as a function of age , and amounts to an error of @xmath123 mag over the entire age range spanned by our ymc sample if solar metallicity were incorrectly assumed . the effect decreases slightly for cluster ages @xmath124 yr . we note that we have taken great care to adopt the most appropriate metallicity for our sample ymcs ( see table [ clusids.tab ] ) , so that we are confident that we have minimised the uncertainties associated with the choice of cluster metallicity . secondly , we explore the effects of varying the imf , @xmath125 . we consider the effects of varying both the slope , @xmath126 , and the low - mass cut - off of the imf . in order to do so , we calculated the age dependence of the length of the `` evolution arrows '' in fig . [ uncertainties.fig]b for five different imf representations , and solar metallicity . except for imf ( ii ) below , where we use the starburst99 ssps ( leitherer et al . 1999 ) , we use the galev ssps in all cases , and assume the imf to cover the mass range from 0.1 to 100 m@xmath36 . we consider the following imfs , the effects of which on the luminosity evolution are shown in fig . [ uncertainties.fig]b : 1 . the `` standard '' salpeter imf , for masses between 0.1 and 100 m@xmath36 , and @xmath127 for the entire mass range ; 2 . the @xmath127 salpeter imf , but for the mass range @xmath128 m@xmath36 ; 3 . the scalo ( 1986 ) imf , for masses @xmath129 , characterised by @xmath130 4 . the ktg93 imf , with @xmath131 and 5 . the kroupa01 imf : @xmath132 figure [ imfs.fig ] displays the functional forms of these imfs , normalised to a standard salpeter imf at 1 m@xmath36 , which contains a total mass of 1 m@xmath36 . this standard salpeter imf is shown as the solid line , and is used as reference in the following . except for the truncated salpeter imf ( short - dashed line ) , the other , more realistic imfs are characterised by a turnover at or below 1 m@xmath36 , and an enhanced contribution of intermediate - mass stars ( @xmath133 ) compared to the full salpeter imf . in all cases , however , they are dominated by the lower - mass stars and provide , therefore , a solid basis for any compact virialised system to survive for up to a hubble time ( e.g. , gnedin & ostriker 1997 , goodwin 1997a , smith & gallagher 2001 , mengel et al . 2002 ) . it is clear that the effects on the luminosity evolution arrow of varying the imf are significant for all ages below several @xmath134 yr . any correction to the length of the luminosity evolution arrow caused by a significant change in the imf ( for the imfs discussed in this paper ) is in the sense that the length of the arrow will be _ reduced _ ; for the kroupa01 and truncated salpeter imfs the effect is expected to be negligible . thus , by adopting a more realistic imf than the standard salpeter representation ( such as the ktg93 imf , which accurately describes the solar neighbourhood imf ) , those clusters that in our current diagnostic diagram of fig . [ diagnostic1.fig ] would evolve to locations well beyond the @xmath92 envelope of the gc relationship if they were characterised by a salpeter - type imf down to the hydrogen - burning limit might well evolve to a location within @xmath135 . in addition , if we had assumed any more realistic imf description for the luminosity evolution of our sample ymcs , the evolved loci of most of these objects might have scattered more symmetrically around the best - fitting gc relation , instead of systematically ending up on the faint side of the correlation ( we have confirmed this for the case of the ktg93 imf ) . based on the currently available data , we can not draw firm conclusions on the actual ( i)mfs of our sample clusters . detailed follow - up _ n_-body simulations , including the effects of primordial and dynamical mass segregation , and of varying binary fractions , are required to address this issue more robustly . this is , however , beyond the scope of the present work . on the other hand , the fact that most clusters , when evolved using a standard solar - neighbourhood salpeter - type imf , appear to end up close to the gc relationship is suggestive of the near - universality of an imf for extragalactic ymcs of any of the currently fashionable forms discussed in this paper . based on the available evidence , it is therefore more likely that the six ymcs that appear to have a central velocity dispersion that is significantly too large for their mass ( luminosity ) will disperse before reaching gc - type ages , than that they were characterised by significantly different _ initial _ mfs ( and possibly very different _ present - day _ mfs ; see also section [ implications.sec ] ) . thirdly , there are a number of observational uncertainties that affect the accuracy of the location of the data points at the present epoch . some of the sample ymcs are affected by significant extinction in their host galaxies , so that any extinction correction introduces uncertainties in the clusters location at the present time . the objects most affected by these uncertainties are * * ngc 6946 - 1447 * : @xmath136 mag ( schlegel et al . 1998 ) ; * * ngc 1042-nc * : based on @xmath53-band photometry , only corrected for galactic extinction . we believe that the main uncertainty in the photometry of this cluster is related to our assumption of it being a clean ssp , as discussed above ; * * ic 342-nc * : @xmath137 mag ( mccall 1989 , madore & freedman 1992 ) , but patchy and variable . bker et al . ( 1999 ) measured @xmath138 mag toward the ymc , equivalent to @xmath139 mag , with an uncertainy of @xmath140 mag due to the patchiness of the extinction ; * * ngc 1614-nc1,2 * : based on bolometric luminosities , derived from mid - infrared observations , so that the accuracy of the conversion depends on the accuracy of the bolometric correction adopted . in addition , @xmath141 mag , in a clumpy distribution ; * * m82-f * : @xmath142 mag ( smith & gallagher 2001 , but see mccrady et al . mccrady et al . ( 2005 ) conclude that their @xmath69-band spectra are negligibly affected by extinction , while @xmath143 mag ; * * m82 mgg-9 * and * mgg-11 * : photometry based on near - infrared _ hst _ observations ; @xmath144 and @xmath145 mag , respectively ( mccrady et al . 2003 ) . translated to the @xmath0 band , the extinction becomes considerable , at @xmath146 and @xmath147 mag , respectively ; * * ngc 5236 clusters * : @xmath148 mag ( schlegel et al . 1998 ) . the internal extinction @xmath149 mag , and @xmath150 mag for ngc 5236 - 502 and ngc 5236 - 805 , respectively ( larsen & richtler 2004 ) . cluster 502 is located close to both a conspicuous dust lane , and to a fainter , bluer companion cluster ; both objects are unresolved at ground - based spatial resolution . * * ngc 4214 - 13 * : @xmath151 mag ( larsen et al . 2004 ) ; * the * antennae clusters [ ws95]355 * ( photometry based on @xmath53 band data , since only an upper limit could be obtained in @xmath0 ; significant extinction ) , * [ ws95]331 * and * [ m03 ] * ( both based on @xmath60-band photometry ; significant extinction ) . based on a comparison of mengel et al . ( 2001 , 2002 ) , antennae ymcs * [ w99]1 , 10 * , and * 16 * are affected by @xmath152 , @xmath153 and @xmath154 mag of extinction ; the other antennae objects are more highly extincted , although the details are lacking in the original papers . * the * ngc 1487 clusters * : based on ground - based @xmath60-band photometry ; no extinction estimates available . [ cols=">,<,<,>,^,>,^,^,^ " , ] notes : @xmath74 based on @xmath69 and @xmath53-band spectroscopy ( first and second line , respectively ) ; @xmath75 based on barycentric motions ; @xmath155 if virialised ; @xmath76 the differences among the existing photometric mass estimates are mostly caused by varying distance estimates to the galaxy ( ho & filippenko 1996b ) ; @xmath77 the photometric mass estimates are for a kroupa01 and a salpeter imf , covering masses down to 0.1 m@xmath36 ( first and second line , respectively ) . * references : * 1 , this work , based on data from bker et al . ( 2004 , 2005 ) ; 2 , anders et al . ( 2004 ) ; 3 , bker et al . ( 1999 ) ; 4 , de marchi et al . ( 1997 ) ; 5 , gilbert & graham ( 2001 ) ; 6 , ho & filippenko ( 1996a ) ; 7 , ho & filippenko ( 1996b ) ; 8 , larsen et al . ( 2001 ) ; 9 , larsen et al . ( 2004 ) ; 10 , larsen & richtler ( 2004 ) ; 11 , maraston et al . ( 2004 ) ; 12 , mccrady et al . ( 2003 ) ; 13 , mccrady et al . ( 2005 ) ; 14 , melnick et al . ( 1985 ) ; 15 , mengel et al . ( 2002 ) ; 16 , mengel ( 2003 ) ; 17 , meurer et al . ( 1992 ) ; 18 , meurer et al . ( 1995 ) ; 19 , puxley & brand ( 1999 ) ; 20 , smith & gallagher ( 2001 ) . figure [ masscf.fig ] provides a projection of the `` ymc fundamental plane '' defined in the space of the ymcs luminosities , velocity dispersions and sizes . we show the distribution of our sample ymcs in the plane defined by the photometric vs. the dynamical mass estimates ; the photometric mass estimates are based on converting the cluster luminosities to masses using the galev ssps under the assumption of a salpeter imf from 0.1 to 100 m@xmath36 . the solid line of equality represents the loci where our sample clusters would be found if they were characterised by this salpeter imf , and a constant m / l ratio throughout . the other lines , offset from the solid line , are calculated for the alternative imfs considered for the photometric mass estimates listed in table [ masses.tab ] . we can conclude that most of our sample ymcs are scattered closely around the line of equality , which provides additional evidence that they are characterised by imfs ( or present - day mfs ) similar to the standard salpeter imf . only few objects , including the m82 clusters f and mgg-11 , and ngc 1705-i , are found in the region where we expect to see the effects of either a low - mass cut - off or significant mass segregation . this lends support to mccrady et al.s ( 2003 , 2005 ) suggestion that these m82 clusters are affected by significant primordial mass segregation , and suggests a similar effect for ngc 1705-i . in this context , we note that the straightforward application of the virial theorem , eq . ( [ virial.eq ] ) , which is based on a single - mass model for all stars contained in the system , tends to underestimate a system s dynamical mass by a factor of @xmath114 compared to more realistic multi - mass models ( e.g. , mandushev et al . [ 1991 ] , based on an analysis of the observational uncertainties ) . this effect potentially reduces the number of clusters in fig . [ masscf.fig ] scattered toward mfs defined by low - mass cut - offs or ymcs dominated by significant mass segregation even further . the origin of the tight relationship between the absolute magnitude and central velocity dispersion for all local group gcs remains an unsolved puzzle . djorgovski ( 1991 , 1993 ; see also djorgovski & meylan 1994 ) suggested that the relation evolved from a primordial scaling relation , @xmath156 ( assuming a constant m / l ratio among gcs ) , which would be subsequently altered by tidal shocks , leading to mass ( and therefore luminosity ) losses . this would be more efficient for the less massive clusters , thus resulting in a steepening of the relationship to its currently observed form . mclaughlin ( 2003 ) suggests that the relation is linked to the mass - dependent star - formation efficiencies in giant molecular clouds , the progenitors of star clusters . we note that the fact that _ all _ local group gcs are found scattering closely around the relationship implies that its origin must be related to gc - internal processes . the tightness of the relationship rules out significant environmental effects as principal cause for its origin . this is simply because the local group gcs are found in a wide variety of environments , ranging from the high - density environments in the galaxy and m31 , via the intermediate density operating in m33 , to the ( very ) low density environments in the dwarf satellite galaxies ( lmc , smc , fornax dsph ) . a similar conclusion was reached by mclaughlin ( 2000a ) when he noted that galactic gcs at larger galactocentric distances exhibit a smaller scatter about the relationship than those closer to the milky way . the fact that we find that our sample ymcs , when evolved to a common age of 12 gyr using the salpeter imf , may also evolve to loci close to the best - fitting gc relationship implies that the initial conditions governing these ymc must have been very similar to those responsible for the formation of the old local group gcs . this , therefore , provides an argument in favour of the suggestion that most of these ymcs may in fact be proto - gcs . it also suggests that a large number of the present - day young compact lmc ( and smc ) clusters , as well as the large majority of the galactic open clusters , all of which are currently found to occupy regions close to the old gc relationship ( in some cases further toward fainter magnitudes than any of the known gcs , for a given central velocity dispersion ) , are unlikely to survive until they reach gc - type ages of @xmath157 gyr . thus far , we have been dealing predominantly with internal cluster processes that might prevent ( a number of ) the ymcs from surviving for a hubble time . the most likely _ internal _ processes leading to cluster disruption were found to be related to variations in the imf . however , we note that our predictions for the future fate of our sample clusters should only be adopted as first - order approximations . until now , we have only mentioned external disruptive effects in passing , and have assumed our clusters to reside in quiescent galactic disc environments . this assumption is clearly not justified in a number of cases considered in this paper . one should realise that star cluster survivability also and crucially so depends on external factors affecting its stellar content , such as tidal shocking by galactic discs , bulges , spiral arms and giant molecular clouds ( gmcs ) , and the associated ram - pressure stripping . these external effects will accelerate the cluster disruption time - scale relative to that caused by cluster - internal effects . in a recent study , boutloukos & lamers ( 2003 ) derived an empirical expression for the `` characteristic '' cluster disruption time - scale ( i.e. , the time - scale on which a @xmath158 m@xmath36 cluster will dissolve , assuming instantaneous disruption ) , and found that for a given cluster system and environment this time - scale is entirely dependent on the initial mass of the cluster , as @xmath159 ( see also lamers , gieles & portegies zwart 2005 , who confirmed this prediction using _ n_-body simulations ) . boutloukos & lamers ( 2003 ) derived characteristic cluster disruption time - scales for the cluster systems in the solar neighbourhood , the smc , and in selected regions of m33 and the interacting galaxy m51 . in de grijs et al . ( 2003a , c ) , we extended this sample to include the fossil starburst region m82 b , and the interacting systems ngc 3310 and ngc 6745 . in de grijs et al . ( 2003c ) , we concluded that the very short characteristic cluster disruption time - scale for the clusters in m82 b is most likely caused by the very high ambient density of its interstellar medium ( ism ) , leading to cluster disruption on similarly short time - scales as in the high - density centre of m51 . if we place our own results in this context , we see that four of the six clusters that are expected to evolve to beyond the @xmath160 boundary by an age of 12 gyr are in fact located in the high - density overlap region in the antennae galaxies . we would expect these objects to dissolve on shorter - than - average time - scales , simply because of the higher density ism in which they are embedded , and because of the high pressure and tidal shocks expected in the ongoing merger . similarly , the remaining two objects ( ngc 1487 - 3 and ic 342-nc ) are located in high - density galactic centre environments . by the same token , ngc 1487 - 1 and 2 , and ngc 5236 - 805 are located in similarly high - density environments ; their luminosity evolution arrows do , in fact , overshoot the @xmath91 envelope . this is supported by a recent study by lamers et al . ( 2005 ) , based on numerical simulations . we caution that the results for the ngc 1487 clusters should be treated with caution in view of the large photometric uncertainties caused by the passband conversion applied . however , if we take the evolution of the central velocity dispersion into account , all of these objects may well evolve to loci within the @xmath161 boundary by the time they age to 12 gyr . if we assume that the _ initial _ mf of all of these objects was roughly constant for the entire ymc sample , this implies that tidal effects and their location in regions of higher - than - average density must have affected the stellar content of these clusters already on time - scales as short of @xmath162 yr , i.e. , a significant fraction of the low - mass stars in these objects has likely been tidally stripped already during their very short lifetimes . ongoing tidal effects would lead to luminosity evolution to still fainter magnitudes than implied by assuming a salpeter - type imf . now that we have established that a number of our sample clusters are already likely to have been affected significantly by tidal effects and externally induced disruption , despite their young ages , we return to the origin of the tight gc relation . with the remainder of our sample ymcs , except the most massive objects that may be governed by the fj relation rather than the old gc correlation , we can now test the suggestion by djorgovski ( 1991 , 1993 ) and djorgovski & meylan ( 1994 ) that at the time of _ proto - globular _ cluster formation the clusters ( central ) velocity dispersion correlated linearly with their luminosity . for the following arguments , one needs to keep in mind that we have shown ( i ) that the remainder of our ymc sample shows behaviour consistent with their stellar content being described by a salpeter - type present - day ( and presumably initial ) mf ( assuming that they are to obey the @xmath8 relationship at old age ) , ( ii ) that all of these clusters are likely governed by a very similar imf , and ( iii ) that they are possible gc progenitors , in the absence of significant external disruptive processes . with this picture in mind , we can now evolve the present - day luminosities of these ymcs back to a common age corresponding to the youngest age found in this cluster sample , i.e. , 8 myr , again using the galev ssps with a standard salpeter imf . we show the results of this exercise in fig . [ initconds.fig ] . for the first time , we can now assess the almost - initial conditions of proto - gcs in our diagnostic @xmath8 diagram . the best - fitting ( dashed ) relationship corresponds to @xmath163 or @xmath164 with correlation coefficient @xmath165 , when expressed in logarithmic units . this result excludes a linear @xmath166 relation at the @xmath167 level . the exact relationship is somewhat dependent on the exact functional form of the imf adopted . for instance , if we had adopted a kroupa01 imf , the exponents in eq . ( 4 ) and ( 5 ) would have been @xmath168 and @xmath169 , respectively . this relation can be understood in terms of the state of equilibrium of the observed clusters . for a cluster in virial equilibrium we have ( see , e.g. , binney & tremaine 1987 ) @xmath170 assuming that the cluster has a constant m / l ratio , @xmath171 . thus , the observed relation for the youngest ymcs has exactly the form expected for clusters in virial equilibrium , provided that ( i ) the cluster radii are independent of their luminosities , ( ii ) the cluster radii have not changed significantly since the clusters were @xmath172 myr old , and ( iii ) the ratio of central velocity dispersion @xmath2 to the total cluster dispersion is independent of luminosity . with regard to the first point , mclaughlin ( 2000a ) found that the half - light radii of the milky way clusters are indeed independent of their total masses . similarly , harris et al . ( 2002 ) found no significant correlation between cluster sizes and their absolute magnitudes in a sample of clusters surrounding the giant elliptical galaxy ngc 5128 , and neither did we find any such correlation between the half - light radii and absolute magnitudes of our local group gc sample . note that in the @xmath5-body simulations presented in section [ sigma.sec ] the low - mass clusters were systematically smaller in radius than the more massive clusters , which is why the simulated low - mass clusters do not lie on the young ymc relation ( see fig . [ fig : nbody_evol ] ) . the half - light radius of a cluster is most significantly affected by the expulsion of gas immediately following the end of star formation , which results in the expansion of the cluster by up to a factor of @xmath173 ( boily & kroupa 2003 , goodwin 1997b ) . bound clusters rapidly re - establish equilibrium . it is therefore reasonable to expect that the half - light radii have not evolved significantly since an age of 8 myr even if some of the clusters have expanded since that time , eq . ( [ sigma.eq ] ) shows that the magnitude of this effect will be less than 0.35 dex in @xmath174 . finally , the absence of significant luminosity dependence of the ratio of central to total cluster velocity dispersions is expected for clusters in equilibrium . [ initconds.fig ] is thus consistent with the youngest ymcs having rapidly achieved virial equilibrium . in order to strengthen this result , it would be interesting to use accurate determinations of cluster radii to confirm the independence of the cluster sizes and luminosities in the extragalactic ymc sample . the simple , virial @xmath8 relation for the youngest clusters in our ymc sample may be the pre - cursor for the fundamental plane of globular clusters . clearly , quiescent evolution would be expected to transform a primordial linear relation into another linear relation since two clusters which are initially close together in the @xmath175 plane will evolve similarly provided their external environments do not differ too greatly . the change in the slope of the relation is then probably due to the dependence of the @xmath2 evolution on the mass of the cluster . the increased relaxation time of more massive clusters would be expected to lead to less evolution in these clusters than is seen in the lower mass clusters ; we also note that the amount of luminosity evolution is driven by relative age differences and , to first order , independent of a cluster s initial luminosity . this would naturally account for the steeper slope of the late - time relation seen for the local group gcs . further numerical simulations are required to confirm that this picture is consistent in all respects with the observations . in this paper , we have presented a new analysis of the properties and possible evolutionary paths of the ymcs forming profusely in intense starburst environments , such as those associated with galaxy interactions and mergers . the method hinges on the empirical relationship for old galactic and m31 gcs , which occupy a tightly constrained locus in the plane defined by their @xmath0-band luminosities , @xmath1 ( or , equivalently , absolute magnitudes , @xmath9 ) and central velocity dispersions , @xmath2 ( djorgovski et al . 1997 , mclaughlin 2000a , and references therein ) . we added to the galactic and m31 gc sample the old compact magellanic cloud clusters , and the m33 and fornax dsph gcs for which the relevant observational parameters were available in the literature . the relationship between @xmath1 and @xmath2 for this increased gc sample , @xmath176 ( km s@xmath4 ) , is within the uncertainties consistent with djorgovski et al.s ( 1997 ) determination for the smaller galactic and m31 gc sample . the tightness of the relationship for a sample drawn from environments as diverse as those found in the local group , ranging from high to very low ambient densities , implies that its origin must be sought in intrinsic properties of the gc formation process itself , rather than in external factors . this is further supported by mclaughlin s ( 2000a ) result that gcs at greater galactocentric distances exhibit a smaller scatter about the relation than closer objects . encouraged by the tightness of the gc relationship , we also added the available data points for the ymcs in the local universe , including nuclear star clusters , for which velocity dispersion information was readily available . in order to be able to compare them to the ubiquitous old local group gcs , we evolved their luminosities to a common age of 12 gyr , adopting the `` standard '' ( solar neighbourhood ) salpeter imf covering masses from 0.1 to 100 m@xmath36 , and assuming stellar evolution as described by the galev ssps . based on a careful assessment of the uncertainties associated with this luminosity evolution , we concluded that the most important factor affecting the robustness of our conclusions is the adopted form of the stellar imf . we found that if we adopt the salpeter imf as the basis for the ymcs luminosity evolution , the large majority will evolve to loci within twice the observational scatter around the best - fitting gc relationship ( although systematically to somewhat fainter luminosities ) . using more realistic imf descriptions , our ymc sample do , in fact , end up scattering more closely about the improved local group gc relationship . in the absence of significant external disturbances , this implies that these objects may potentially survive to become old gc - type objects by the time they reach a similar age . thus , these results provide additional support to the suggestion that the formation of proto - gcs appears to be continuing until the present , a conclusion we reached independently based on the statistical treatment of the @xmath113 gyr - old intermediate - age star cluster system in m82 s fossil starburst region b ( de grijs et al . 2003b ) . detailed case by case comparisons between our results based on this new method with those obtained previously and independently based on dynamical mass estimates and m / l ratio considerations lend significant support to the feasibility and robustness of our new method , and provide a key insight into the inherent uncertainties associated with any of the methods used in this field . the key characteristic and main advantage of this method compared to the more complex analysis involved in using dynamical mass estimates for this purpose is its simplicity and empirical basis . where dynamical mass estimates require one to obtain accurate size estimates and to make assumptions regarding a system s virialised state and m / l ratio , these complications can now be avoided by using the empirically determined gc relationship as reference . the only observables required are the system s ( central or line - of - sight ) velocity dispersion and photometric properties . mclaughlin ( 2000a ) has shown that this is , in fact , a physically relevant correlation , since ( i ) the @xmath177 diagram ( where @xmath10 is the cluster binding energy ) is composed of physically meaningful quantities , and ( ii ) the scatter about the correlation is of the same order as the observational uncertainties . careful analysis of those ymcs that would overshoot the gc relationship significantly if they were to survive for a hubble time ( and are characterised by a salpeter - type initial or present - day mf ) showed that their unusually high ambient density has probably already had a significant effect on their stellar content , despite their young ages , thus altering their present - day mf in a such a way that they have become unable to survive for any significant length of time . this is , again , supported by independent analyses , thus further strengthening the robustness of our new approach . the expected loci in the @xmath8 plane that these objects would evolve to over a hubble time are well beyond any gc luminosities for a given velocity dispersion , leading us to conclude that they will either dissolve long before reaching gc - type ages , or that they must be characterised by a present - day mf that is significantly depleted in low - mass stars ( or highly mass segregated ) , thus also resulting in fast dispersion . this , therefore , allows us to place moderate limits on the functionality of their present - day mfs . in order to investigate whether dynamical evolution would have a dramatic impact on the evolution of clusters in the @xmath58 plane , we analysed the results of a number of @xmath5-body simulations . the velocity dispersions of the model clusters were calculated in a manner analogous to that used for the observed clusters . we concluded that the evolution of the observed @xmath2 is relatively small for clusters that survive to old age , and thus our conclusions remain unchanged . based on our analysis of the objects with the largest velocity dispersions , including the nuclear star clusters , we conclude that the recently discovered ucds in the fornax cluster may be most closely related to stripped dsph or de nuclei . we also show that the unusual galactic gc ngc 2419 is unlikely to be a similar type of object , despite recent suggestions to the contrary . finally , we evolved those ymcs that appear to be least affected by external disruptive effects and are likely to be well - represented by salpeter - type imfs back to a common young age of 8 myr , in order to assess the @xmath8 relationship in almost - initial conditions . the resulting best - fitting relationship , @xmath178 ( km s@xmath4 ) , implies that these clusters follow a simple virial relation . the evolution of relatively undisturbed star clusters in the @xmath58 plane , as seen in our @xmath5-body simulations , will subsequently transform this relation into the steeper relation displayed by the local group gcs . the existence of a simple , virial @xmath8 relationship for the youngest ymcs may therefore constitute the origin of the gc fundamental plane . rdg acknowledges the stimulating atmosphere at the 2004 guillermo haro workshop at inaoe , tonantzintla , mexico , during which much of this work was done ; he also thanks the royal society for providing travel funding to attend this workshop . we are very grateful to jarrod hurley for providing us with data from his @xmath5-body simulations in advance of publication . we thank peter anders for re - calculating his galev ssp models for the range of imf representations discussed in this paper , and acknowledge stimulating discussions with roberto terlevich and jay gallagher , and a number of insightful comments by the anonymous referee . miw thanks gijs nelemans for valuable discussions and pparc for financial support . this research has made use of the simbad database , operated at cds , strasbourg , france , and of the webda database maintained by jean - claude mermilliod at http://obswww.unige.ch / webda/. bker t. , walcher c .- j . , rix h .- w . , hring n. , schinnerer e. , sarzi m. , van der marel r.p . , ho l.c . , shields j.c . , lisenfeld u. , laine s. , 2005 , in massive young clusters , asp conf . ser . , lamers h.j.g.l.m . , smith l.j . , nota a. , eds . , asp : san francisco , p. 39 walcher c.j . , hring n. , bker t. , rix h .- w . , van der marel r.p . , gerssen j. , ho l. , shields j. , 2004 , in carnegie observatories astrophysics series , vol . 1 : coevolution of black holes and galaxies , ho l.c . , ed . , ( carnegie observatories : pasadena ) , in press

we present a new analysis of the properties of the young massive star clusters forming profusely in intense starburst environments , which demonstrates that these objects are plausible progenitors of the old globular clusters ( gcs ) seen abundantly in the local group . the method is based on the tight relationship for old gcs between their @xmath0-band luminosities , @xmath1 , and ( central ) velocity dispersions , @xmath2 . we improve the significance of the relationship by increasing the gc sample size and find that its functional form , @xmath3 ( km s@xmath4 ) , is fully consistent with previous determinations for smaller galactic and m31 gc samples . the tightness of the relationship for a gc sample drawn from environments as diverse as those found in the local group implies that its origin must be sought in intrinsic properties of the gc formation process itself . + we evolve the luminosities of those young massive star clusters ( ymcs ) in the local universe which have velocity dispersion measurements to an age of 12 gyr , adopting a variety of imf descriptions , and find that most ymcs will evolve to loci close to , or to slightly fainter luminosities than the improved gc relationship . in the absence of significant external disturbances , this implies that these objects may potentially survive to become old gc - type objects over a hubble time . the main advantage of our new method is its simplicity . where alternative methods , based on dynamical mass estimates , require one to obtain accurate size estimates and to make further assumptions , the only observables required here are the system s velocity dispersion and luminosity . the most important factor affecting the robustness of our conclusions is the adopted form of the initial mass function . we use the results of @xmath5-body simulations to confirm that dynamical evolution of the clusters does not significantly alter our conclusions about the likelihood of individual clusters surviving to late times . finally , we find that our youngest observed clusters are consistent with having evolved from a relation of the form @xmath6 ( km s@xmath4 ) . this relation may actually correspond to the origin of the gc fundamental plane . stellar dynamics methods : miscellaneous galaxies : nuclei galaxies : starburst galaxies : star clusters

electron correlations in quantum - dot structures result in many fascinating effects that can be probed in detail with remarkable experimental control of system parameters.@xcite perhaps one of the most interesting regimes occurs when electrons confined in the dot acquire antiferromagnetic correlations with electrons in the leads , giving rise to the well - known kondo effect.@xcite the simplest realization of this phenomenon in a single quantum dot is characterized by just one low - energy scale , set by the kondo temperature , which controls ( among other features ) the width of a many - body resonance at the fermi energy.@xcite recent experimental@xcite studies of the kondo effect in multiple quantum dots have revealed a complex competition between geometry and correlations , making evident that these structures provide a flexible setting in which to explore much novel physics . in this context , double - quantum - dot arrangements exhibit striking manifestations of kondo physics , with conductance signatures of these effects predicted to show up in realistic experimental setups . a telling example is the interplay of kondo physics and quantum interference in `` side - coupled '' or `` hanging - dot '' configurations , @xcite leading to a variety of interesting fano - kondo " effects . @xcite a rather unexpected situation arises when a small , strongly interacting `` dot 1 '' is connected to external leads via a large `` dot 2 '' that is tuned to have a single - particle level in resonance with the common fermi energy of the leads.@xcite in this configuration the kondo resonance , which normally has a single peak at the fermi energy , splits into two peaks a behavior that can be understood as a consequence of interference between the many - body kondo state in dot 1 and a single - particle - like resonance that controls ( or `` filters '' ) its connection to the leads.@xcite the magnitude of the kondo peak splitting is determined by the balance of several important energy scales in the problem : the width and position of the active single - particle level in dot 2 ; the height of the effective single - particle resonance set by the interdot coupling ; and the many - body kondo temperature ( determined by the preceding energy scales in combination with the dot-1 level - position and interaction strength ) . this filtering of the leads preserves a fully screened kondo ground state with a kondo temperature that rises with increasing interdot coupling . in this work , we investigate the effects of an external in - plane magnetic field on such a double - quantum dot - system in the side - dot arrangement . the field which introduces another energy scale , the zeeman energy is known to be detrimental to the kondo state in single - dot systems.@xcite using numerical renormalization - group methods,@xcite we study the interplay between the different energy scales and discuss the behavior of the kondo resonance in the presence of competing interactions . this interplay reveals itself in the fundamental fermi - liquid properties of the system , such as the variation with magnetic field @xmath0 at zero temperature of the fermi - energy ( @xmath1 ) value of the side - dot spectral function @xmath2 . instead of the usual monotonic decay@xcite of @xmath3 with increasing @xmath0 we find a markedly nonuniversal behavior , where @xmath3 passes through a maximum at a nonzero value of the field . this effective _ field - enhancement _ of the kondo spectral function is a consequence of the side - dot geometry . the same behavior can also be understood using an appropriate form of the friedel sum rule , which predicts parameter- and field - dependent phase shifts that impart the unusual nonmonotonicity to the variation of @xmath3 with @xmath0 . in addition , we show that the competition between zeeman splitting of the dot levels and kondo screening results in a dominant exchange - mediated antiferromagnetic coupling of the dots over a range of moderate magnetic fields , before both dots become fully polarized at higher fields . finally , we identify signatures of the aforementioned phenomena in the transport properties . a key result is the generation of _ spin - polarized currents _ through the device , which can be tuned by adjusting gate voltages to achieve _ total polarization . _ the remainder of the paper is organized as follows : in sec . [ sec : system ] we describe the effective anderson impurity model for the double - quantum - dot system . section [ sec : specfeatures ] presents the low - energy spectral properties , while sec . [ sec : genfsr ] interprets the nonuniversal behavior of @xmath4 vs @xmath0 in terms of the friedel sum rule . the transport properties , including spin polarization , are explored in sec.[sec : conductance ] . concluding remarks appear in sec . [ sec : conclusion ] . the system under study , which is depicted schematically in fig.[fig : model ] , contains two quantum dots . dot 1 has a large coulomb repulsion @xmath5 when its single active energy level is doubly occupied . dot 2 has negligible electron - electron interactions ( @xmath6 ) and one active level that can be tuned by gate voltages to be at or near resonance with the common fermi energy @xmath7 of left ( @xmath8 ) and right ( @xmath9 ) leads . electrons can tunnel between dots 1 and 2 with tunneling matrix element @xmath10 , and between dot 2 and lead @xmath11 with tunneling matrix element @xmath12 . ( color online ) schematic representation of the side - coupled double - quantum - dot system . the dot qd1 has a strong coulomb interaction @xmath5 and is coupled only to the second dot , labeled qd2 . the latter dot has negligible local interactions ( i.e. , @xmath13 ) and the energy of its active level is tuned to allow tunneling at or near resonance with the fermi level of the left ( @xmath8 ) and right ( @xmath9 ) leads.,width=249 ] the system can be described by a variant of the two - impurity anderson hamiltonian : @xmath14 with @xmath15 and @xmath16 here , @xmath17 annihilates an electron in dot @xmath18 with spin @xmath19 component @xmath20 ( @xmath21 or equivalently @xmath22 ) and energy @xmath23 ; @xmath24 is the corresponding number operator ; and @xmath25 annihilates an electron in lead @xmath11 with spin @xmath19 component @xmath20 and energy @xmath26 . the magnetic field @xmath27 with @xmath28 is assumed to lie in the plane of the two - dimensional electron gas in which the dots and leads are defined , so that it produces no kinematic effects and enters only through zeeman level splittings . this hamiltonian differs from a generic two - impurity anderson model through the absence of dot-1 hybridizations @xmath29 , a consequence of the side - dot geometry . throughout the greater part of the paper , we also take @xmath30 , a case that is particularly convenient for algebraic analysis . the effect of nonvanishing dot-2 interactions is addressed at the end of sec.[sec : conductance ] . without loss of generality , we take all tunneling matrix elements to be real . we consider local ( @xmath31-independent ) dot - lead tunneling and assume that the dots have equal effective @xmath32 factors @xmath33 , simplifications that do not qualitatively affect the physics . the leads are taken to have featureless band structures near the fermi energy , modeled by the flat - top densities of states @xmath34 where @xmath35 is the half - bandwidth and @xmath36 is the heaviside function . the equilibrium and linear - response properties of the system may be calculated@xcite by considering the coupling of dot 2 via hybridization matrix element @xmath37 to a single effective conduction band described by annihilation operators @xmath38 and a density of states @xmath39 . the zeeman splitting of this conduction band produces only very small effects near the band edges , so for convenience we set the bulk @xmath32 factor to @xmath40 throughout what follows . the primary quantities of interest in this work are the retarded dot green s functions @xmath41 for @xmath42 , where @xmath43 and @xmath44 denotes an appropriate thermal average.@xcite in particular , we are interested in the spectral functions @xmath45 and the system s linear ( zero - bias ) conductance , given by the meir - wingreen formula@xcite as @xmath46 with @xmath47\ , ( -\partial f/\partial\omega ) \ , d\omega,\ ] ] where @xmath48 is the fermi distribution function at temperature @xmath49 and @xmath50 ^ 2 ( 2e^2 / h)$ ] represents the unitary conductance of a single channel of electrons multiplied by a factor@xcite that varies between 1 ( for @xmath51 ) and 0 ( in the limit of extreme left - right asymmetry of the dot - tunneling ) . in the side - connected geometry , the transmission is@xcite @xmath52 where @xmath53 . thus , @xmath54 which reduces at zero temperature to @xmath55 in order calculate the dot spectral functions @xmath56 taking full account of the electronic correlations arising from the @xmath5 term in eq . , we employ the numerical renormalization - group ( nrg ) method , performing a logarithmic discretization of the conduction band and iteratively solving the discretized hamiltonian . in evaluating the spectral functions , we perform a gaussian - logarithmic broadening of discrete poles obtained by the procedure described in ref . . at temperatures @xmath57 we use the density - matrix variant of the nrg,@xcite which has better spectral resolution at high frequencies and nonzero fields.@xcite although these schemes are not totally free from systematic errors , @xcite the main results of the paper do not depend crucially on the broadening procedure . all numerical results were obtained for a symmetric dot 1 described by @xmath58 , for dot 2 width @xmath59 , and for nrg discretization parameter @xmath60 . except where it is stated otherwise , we consider a strongly correlated dot 1 with @xmath61 and situations in which a noninteracting dot 2 is tuned to be in resonance with the leads , i.e. , @xmath62 . we adopt units in which @xmath63 . ( color online ) ( a ) spin - averaged dot-1 spectral function @xmath64 vs frequency @xmath65 at zero temperature for @xmath66 , @xmath67 , @xmath68 , and ( from bottom to top curve , offset for clarity ) @xmath69 , @xmath70 , @xmath71 , and @xmath72 . the spectral function is multiplied by @xmath73 where @xmath74 . insets : expanded views of @xmath64 vs @xmath65 around the fermi level @xmath1 for the same system , with @xmath0 ranging from @xmath75 to @xmath76 ( bottom to top , curves offset for clarity ) in steps of @xmath77 in the left inset , and from @xmath78 to @xmath79 ( bottom to top , curves offset for clarity ) in steps of @xmath70 in the right inset . ( b ) spin - up ( black squares ) and spin - down ( red circles ) dot-1 spectral functions @xmath80 for @xmath81 with all other parameters as in ( a ) . ( c ) same as ( b ) , except for @xmath82.,width=316 ] in single quantum dots , the presence of an in - plane magnetic field @xcite or connection to ferromagnetic leads@xcite modifies coherent spin fluctuations and weakens the kondo effect . the spin - averaged spectral function exhibits a kondo - peak splitting that grows with increasing applied field , while the value of the spectral function at the fermi energy decreases monotonically . in this section we investigate the effects of a zeeman field on the side - dot spectral function in the double - dot system defined in sec . [ sec : system ] . figure [ fig : a_vs_w](a ) shows the spin - averaged spectral function @xmath83\ ] ] for a side - dot setup at zero temperature with @xmath61 and @xmath68 . the different curves , vertically offset for clarity , correspond to four different values of @xmath0 . for zero field ( the bottom curve ) , @xmath84 , so each spin - resolved spectral function shows a symmetric kondo - peak splitting due to the interdot coupling @xmath10 . with increasing @xmath0 , the split peaks merge into a single peak at @xmath1 , clearly seen for @xmath85 ( top curve ) . the left inset to fig . [ fig : a_vs_w](a ) shows in greater detail the convergence of the peaks near the fermi energy , with the maximum in @xmath64 vs @xmath65 at @xmath1 being best defined at @xmath86 , a field where , incidentally , the absolute value of @xmath3 exceeds that at @xmath69 by nearly a factor of two . for slightly larger fields , the central peak again splits into two before all low - energy features become flattened out at fields @xmath87 [ right inset to fig . [ fig : a_vs_w](a ) ] . the field - induced merging of the peaks in @xmath88 arises from opposite displacements of @xmath89 and @xmath90 along the @xmath65 axis . in a nonzero magnetic field , @xmath91 but @xmath92 . this is illustrated for @xmath81 in fig . [ fig : a_vs_w](b ) , which also shows that the heights of the two peaks in each spin - resolved spectral function @xmath93 are no longer equal . upon further increase in the field to @xmath82 [ fig . [ fig : a_vs_w](c ) ] , the double - peak structure is replaced by a single peak near @xmath1 in each spin - resolved spectral function . for larger values of @xmath0 , these peaks move away from the fermi energy and the usual zeeman - splitting of the kondo peak with decreasing amplitude becomes evident in the spin - averaged spectral function [ right inset in fig . [ fig : a_vs_w](a ) ] . this behavior can be qualitatively understood by considering the evolution with @xmath0 of the level energies found@xcite in the `` atomic limit '' @xmath94 where the dots are isolated from the leads . we now focus on the field dependence of @xmath4 , a quantity that acts as a sensitive measure of the interplay of the different energy scales in the problem : the single - particle resonance width @xmath95 , the zero - field kondo temperature @xmath96 , and the zeeman energy @xmath97 . figure [ fig : a_vs_b](a ) plots @xmath98 vs @xmath99 ( taking @xmath100 ) for six values of @xmath10 . the energy scale @xmath101 , introduced for normalization purposes , is defined in eq . below . for now , it suffices to note that @xmath101 is proportional to @xmath102^{-1}$ ] , i.e. , it is a decreasing function of the field . the figure reveals two distinct regimes of behavior : ( 1 ) for @xmath103 , @xmath98 decreases monotonically from its zero - field value @xmath104 over a characteristic field scale that increases with @xmath10 ( and is not simply @xmath96 , as it is in the single - dot case ) . ( 2 ) for @xmath105 , @xmath98 has a nonmonotonic variation with increasing @xmath0 , reaching a second maximum @xmath106 at @xmath107 , beyond which field it decreases . in view of the field dependence of @xmath101 , the value of @xmath3 at @xmath108 is @xmath109 $ ] times its zero - field counterpart . the two regimes seen in fig . [ fig : a_vs_b](a ) are in sharp contrast with the monotonically decreasing and universal dependence of the fermi - energy spectral function on @xmath99 in the conventional single - impurity kondo@xcite and anderson@xcite models . the next section discusses these behaviors in terms of the friedel sum rule . ( color online ) ( a ) spin - averaged dot-1 spectral function at the fermi level @xmath110 vs scaled magnetic field @xmath99 for @xmath66 , @xmath67 , and six values of @xmath10 . @xmath3 has been multiplied by the field - dependent quantity @xmath111 [ eq . ] to yield @xmath112 defined in eq . . the larger @xmath10 values produce a nonmonotonic field variation of @xmath3 , with a peak around @xmath113 . inset : corresponding plot for the noninteracting case @xmath114 , with the field scaled by the interdot coupling @xmath10 . ( b ) phase factor @xmath115 corresponding to the data in ( a ) , determined from the friedel sum rule [ eq . ] using the magnetization data plotted in fig . [ fig : m_vs_b].,width=316 ] in sec . [ subsec:1imp ] we review the fermi - liquid relation known as the friedel sum rule@xcite that sets the fermi - energy value of the zero - temperature spectral function in the one - impurity anderson model , and write down a form of the sum rule valid for systems featuring both a zeeman field and nontrivial structure in the density of states . section [ subsec:2imp ] shows how the variation of @xmath4 in our double - quantum - dot system can also be understood in terms of the friedel sum rule . we consider a single - impurity anderson model @xmath116 where @xmath117 and @xmath118 . the conduction - band dispersion @xmath119 and the hybridization @xmath120 enter the impurity properties only in a single combination : the zero - field hybridization function @xmath121 . we denote the fully interacting retarded impurity green s for this problem by @xmath122 where @xmath123 is the retarded impurity self - energy . in the conventional anderson model , where the hybridization function is assumed to take a flat - top form @xmath124 , the friedel sum rule relates the fermi - energy value of the zero - temperature , zero - field impurity spectral function @xmath125 to the average impurity occupancy @xmath126 as @xmath127 in the wide - band limit where @xmath35 greatly exceeds all other energy scales in the problem , eq . has been extended @xcite to show that in a zeeman field @xmath0 , the spin - averaged impurity spectral function @xmath128 $ ] satisfies @xmath129,\ ] ] where @xmath130 is the impurity magnetization in units of @xmath131 . our goal is to extend eqs . and to allow for finite values of @xmath35 and any form of @xmath132 . one can show@xcite that provided the system is in a fermi - liquid regime [ where the imaginary part of @xmath133 varies as @xmath134 for @xmath135 , the spin - resolved spectral functions at zero temperature satisfy @xmath136 where @xmath137 and @xmath138 is a spin - dependent phase shift . in eq . , @xmath139 is the fully interacting retarded impurity green s function specified in eq . , but @xmath140 is the retarded impurity self - energy for the noninteracting system [ eq . with @xmath141 , which satisfies @xmath142 . in situations where @xmath143 , it is convenient to focus on a dimensionless , hybridization - weighted average of the spin - resolved spectral functions : @xmath144 in terms of this quantity , the linear conductance is @xmath145 with a zero - temperature limit @xmath146 here , @xmath147 ^ 2 ( 2e^2 / h)$ ] is the maximum possible conductance through the dot for hybridizations @xmath148 and @xmath149 with the left and right leads , respectively . we note that the hybridization - weighted , spin - averaged spectral function reduces to @xmath150 for ( i ) all values of @xmath65 in zero magnetic field , and ( ii ) at @xmath1 for any field @xmath0 such that @xmath151 . inserting eq . ( [ eq : fsrnonconsthyb ] ) into eq . , rewriting @xmath152 , and defining @xmath153 , one obtains @xmath154 .\ ] ] this form of the friedel sum rule relates the value of the hybridization - weighted spin - averaged spectral function at @xmath1 and @xmath155 to the impurity occupancy , the impurity magnetization , and spin - dependent phase factors that account for the energy dependence of the hybridization function . the right - hand side of eq . has a maximum possible value of 1 , implying through eq . that @xmath156 , as one would expect for a problem with a single transmission mode in the left and right leads . in general , each of the phase factors @xmath157 and @xmath158 has a complicated dependence on @xmath159 , the impurity parameters @xmath160 and @xmath161 , and the magnetic field @xmath0 . this makes it highly improbable that for a generic choice of model parameters there exists a value of @xmath0 for which the system satisfies the requirements @xmath162 for achieving @xmath163 and , hence , a maximum conductance @xmath164 . however , under conditions where both the impurity and the conduction band exhibit particle - hole symmetry , the hamiltonian is invariant under the transformation @xmath165 , @xmath166 , @xmath167 . this invariance leads to the relations @xmath168 , @xmath169^*$ ] and @xmath170^*$ ] , which in turn imply that @xmath171 and @xmath172 ( or @xmath173 ) . since particle - hole symmetry also ensures @xmath174 and @xmath175 , it follows that @xmath176 and the friedel sum rule reduces to @xmath177 in situations described by eq . , the conductance will reach its maximum possible value @xmath178 whenever @xmath179 equals an integer . it is much more likely that this single condition can be met at some value of @xmath0 than that a system away from particle - hole symmetry can be tuned to satisfy both parts of eq . . the conventional flat - top hybridization function @xmath180 is not only particle - hole symmetric , but yields vanishingly small values of @xmath181 , thereby simplifying eq . to the previously derived@xcite eq . . one expects @xmath182 to be an increasing function of @xmath0 with a limiting value @xmath183 , and therefore [ via eq . ] both @xmath184 and @xmath185 should decrease monotonically with increasing @xmath0 . we now return to the double - quantum - dot setup defined in eq . . it has been shown@xcite that for the special case @xmath30 , the properties of dot 1 are identical to those of the impurity in a single - impurity anderson model [ eq . ] with @xmath186 , @xmath187 , and a zero - field hybridization function @xmath188 where @xmath189 describes a unit - normalized lorentzian resonance of width @xmath95 [ defined after eq . ] centered on energy @xmath190 . in a zeeman field @xmath0 , where the spin - dependent hybridization function of the effective one - impurity problem is @xmath191 a quantity of interest is @xmath192 the value of the hybridization - weighted spin - averaged dot-1 spectral function at @xmath193 . for the resonant case @xmath67 considered in figs.[fig : a_vs_w ] and [ fig : a_vs_b ] , @xmath194 in units where @xmath195 . taking into account also the particle - hole symmetry present for @xmath196 and @xmath67 , the friedel sum rule [ eq . ] gives ( after translation back into the variables of the double - dot problem ) @xmath197 where @xmath198 is the magnetic moment on dot @xmath18 , and @xmath199 with @xmath200 . ( color online ) magnetization of dot 1 ( empty symbols ) and dot 2 ( filled symbols ) vs scaled magnetic field @xmath99 at zero temperature for the same parameters as in the main panels of fig . [ fig : a_vs_b ] . the dot-1 magnetization @xmath201 decreases monotonically from zero over a characteristic field scale that grows with @xmath10 and approaches @xmath202 for sufficiently large interdot couplings . the dot-2 magnetization @xmath203 is of opposite sign to @xmath201 for @xmath204 , pointing to the dominance of the antiferromagnetic interdot exchange interaction over this field range . both dots become fully polarized antiparallel to the field for @xmath205 . inset : @xmath203 vs @xmath206 for the noninteracting system with the same parameters as in the inset of fig.[fig : a_vs_b](a ) . in contrast to the interacting case , @xmath203 decreases monotonically from zero with increasing field.,width=316 ] figure [ fig : m_vs_b ] shows the variation of @xmath201 with @xmath0 for the same model parameters used in fig . [ fig : a_vs_w ] . as expected , @xmath201 decreases monotonically from zero over a field scale that grows with @xmath10 . for small @xmath10 , this scale is identical to that characterizing the initial decrease of @xmath207 from 1 [ see fig . [ fig : a_vs_b](a ) ] , while for larger @xmath10 , @xmath208 grows on the scale @xmath209 of the second peak in @xmath112 . in all cases , dot 1 is essentially fully polarized for @xmath210 . that the monotonic evolution of @xmath201 does not accompany a monotonic decrease in @xmath112 is an indication of the importance of the phase factor @xmath211 on the right - hand side of eq . . it is difficult to evaluate @xmath212 directly from eq . using the nrg because this task requires accurate determination of both the real and imaginary parts of @xmath213 for all @xmath214 , whereas the nrg is well - suited only to compute @xmath215 for @xmath216 . at particle - hole symmetry , however , one can use eq . to work backward from the nrg values of @xmath207 and @xmath201 to find @xmath115 . [ fig : a_vs_b](b ) plots the phase obtained in this manner from the data in figs . [ fig : a_vs_b](a ) and [ fig : m_vs_b ] . for all values of @xmath10 , @xmath211 is zero at @xmath69 ( as expected ) and approaches @xmath217 at large field values . for larger values of @xmath10 , @xmath211 shows a pronounced kink at @xmath108 . this kink is related , via eq . , to the peak in @xmath112 at @xmath209 , since @xmath218 is a smooth function of @xmath0 ( as shown in fig . [ fig : m_vs_b ] ) . figure [ fig : m_vs_b ] also plots the field dependence of the dot-2 magnetization . the fact that @xmath203 is of _ opposite _ sign to @xmath201 for @xmath204 indicates that the interactions in dot 1 combine with the interdot hopping to yield a dominant antiferromagnetic interdot exchange interaction . over this range of @xmath0 , it appears that the system minimizes its energy by first aligning the partially kondo - screened magnetic moment of the strongly interacting dot 1 along the direction favored by the field , and then orienting the less - developed moment on dot 2 to minimize the interdot exchange energy even at a cost in zeeman energy . the data show that this tendency becomes weaker for stronger interdot couplings , presumably because the interdot exchange @xmath219 grows more slowly than the energy scale @xmath96 for breaking the kondo singlet . for all values of @xmath10 , once @xmath210 , the zeeman field has largely destroyed the kondo effect , and both dots are fully polarized for @xmath220 . one can gain further insight into the results presented in figs.[fig : a_vs_b ] and [ fig : m_vs_b ] by considering the limit where both dots are noninteracting . equations and hold equally well for interacting and noninteracting problems . however , the case @xmath221 offers the advantage that @xmath3 can also be calculated directly from the imaginary part of @xmath222 where at zero temperature the noninteracting self - energy is @xmath223\delta_{\sigma}(\omega),\ ] ] giving @xmath224 ^ 2 + [ \delta_{\sigma}(0)]^2}.\ ] ] the hybridization - weighted spin - average of @xmath225 satisfies @xmath226 where @xmath227 and @xmath228 . it should be noted that @xmath229 depends on @xmath0 both through the zeeman shift of @xmath230 and the value of @xmath231 . from eq . it is apparent that @xmath112 attains its maximum value of @xmath104 only if @xmath232 for both spin orientations , a condition that can be satisfied only for @xmath233 and either @xmath69 or ( if @xmath234 ) @xmath235 . for @xmath236 and/or @xmath237 , @xmath112 may have zero , one or two maxima at nonzero fields , but @xmath238 for all @xmath0 . these observations are consistent with the conclusion drawn from the friedel sum rule that @xmath239 is likely to be achieved only under conditions of strict particle - hole symmetry . the inset of fig . [ fig : a_vs_b](a ) illustrates the field variation of @xmath207 for the particle - hole - symmetric case @xmath240 , with all other parameters as in the main panel . for each of the @xmath10 values illustrated ( all of which lie in the range @xmath234 ) , @xmath207 reaches 1 at a magnetic field consistent with the value @xmath209 derived in the previous paragraph . note that @xmath209 approaches @xmath241 from below in the limit of strong interdot coupling . the inset of fig . [ fig : m_vs_b ] plots @xmath203 vs @xmath0 for the same noninteracting cases . for each @xmath10 value , @xmath242 shows a purely monotonic field variation , with a rather sudden increase around @xmath243 , a behavior that is mimicked in the interacting system for @xmath244 , especially at large interdot coupling @xmath10 . the variation of the interacting @xmath207 and @xmath203 for @xmath210 seen in the main panels of figs . [ fig : a_vs_b](a ) and [ fig : m_vs_b ] , particularly for the larger values of @xmath10 , may perhaps be interpreted as a many - body analog of the noninteracting behavior in the insets , with @xmath96 serving as a renormalized value of the single - particle scale @xmath10 . while the spectral functions discussed in the preceding sections are difficult to access directly in experiments , they may be probed indirectly through transport measurements . in this section , we show that the zero - bias electrical conductance through the double - dot device contains clear signatures of the nonuniversal variation of @xmath98 with applied field . in particular , we demonstrate the feasibility of generating currents through the system that are strongly or even completely spin - polarized . although the linear conductance is given most compactly by eq . , it is also useful to express @xmath245 in terms of the green s function for the interacting dot 1 by combining eq . with a generalization of eq . ( 6 ) in ref . to include the zeeman field : @xmath246 \ , \pi \delta_{\sigma}(\omega ) \ , a_{1\sigma}(\omega , t ) + \pi\delta_2\rho_{2\sigma}(\omega ) \notag \\ & \qquad + 2\pi(\omega-{\varepsilon}_{2\sigma})\ , \rho_{2\sigma}(\omega ) \ , \delta_{\sigma}(\omega ) \ : \mathrm{re}\ , \mathcal{g}_{1\sigma}(\omega , t ) , \end{aligned}\ ] ] where @xmath247 , with @xmath248 and @xmath249 as defined in eqs . and , the term @xmath250 describes the bare transmission through dot 2 in the absence of dot 1 , while the remaining terms represent additional contributions arising from conductance paths that include dot 1 . in the special case @xmath251 where the latter contributions necessarily vanish , the zero - temperature conductance reduces to @xmath252 where @xmath253 is defined after eq . . ( color online ) linear conductance @xmath245 vs scaled magnetic field @xmath99 at zero temperature for the same parameters as in the main panels of fig . [ fig : a_vs_b ] . @xmath245 rises from zero over the same characteristic field scale as governs the rise of @xmath208 in the main panel of fig . [ fig : m_vs_b].,width=316 ] figure [ fig : g_vs_b ] plots the zero - temperature linear conductance @xmath245 as a function of scaled field @xmath99 for the same parameters used in fig.[fig : a_vs_b ] . for the case @xmath67 considered here , the conductance of dot 2 alone , @xmath254^{-1}$ ] , decreases monotonically from @xmath178 as the zeeman field detunes the dot level from the fermi energy of the leads . for any @xmath255 and @xmath69 , kondo correlations in dot 1 produce zero conductance through the double - dot system.@xcite figure [ fig : g_vs_b ] shows that with increasing field , the double - dot conductance initially increases , then peaks at its maximum possible value @xmath256 for a field value @xmath257 that for large @xmath10 approaches @xmath258 from above , and finally drops back toward zero for @xmath259 . the field @xmath257 is distinct from that characterizing the peak in @xmath98 . in general @xmath260 , but these three scales converge for @xmath261 . the initial rise in @xmath245 with increasing field can be attributed to the progressive suppression of the kondo effect allowing dot 1 to become partially polarized and reducing the destructive interference between the kondo resonance and the dot-2 resonant state . this change takes place in agreement with the evolution seen in @xmath98 and @xmath201over a field scale that increases with @xmath10 but is not just a constant multiple of @xmath96 . by the point that the conductance reaches its peak at @xmath262 , the interchannel interference is clearly constructive since eq . would predict a much lower conductance for dot 2 alone . at still larger fields , the destruction of the kondo resonance becomes complete and the dot-2 resonance is shifted far from the fermi level , leading to a decrease of the conductance . ( color online ) linear conductance @xmath245 vs dot-2 level energy @xmath263 at zero temperature for @xmath66 , @xmath68 , and five different magnetic field values . the conductance is symmetric about the point @xmath67 of particle - hole symmetry . in nonzero fields , @xmath245 peaks at some @xmath264 , apart from the special case @xmath265 , for which the conductance is maximal at @xmath67 ( as already seen in fig.[fig : g_vs_b ] ) . inset : conductance vs magnetic field @xmath0 for @xmath67 and @xmath71.,width=316 ] figure [ fig : condteq0 ] illustrates aspects of the transport away from particle - hole symmetry . the main panel shows the variation of the @xmath155 linear conductance at several different fixed magnetic fields as the value of @xmath263 is swept by varying the voltage on a plunger gate near dot 2 . for @xmath69 , the conductance increases from zero at @xmath67 and approaches @xmath178 for @xmath266 as the dot-2 resonance is tuned away from the fermi energy , thereby permitting perfect conduction through the kondo many - body resonance . for fixed @xmath267 , competition between zeeman splitting of the dot-2 resonance and partial destruction of the kondo effect leads in most cases to an initial rise in @xmath245 for small @xmath268 followed by a fall - off at larger @xmath268 . as the magnetic field increases from zero , the conductance peaks initially move to smaller @xmath268 , then merge into a single peak at @xmath256 for @xmath269 for the case @xmath68 here , before separating and moving to larger @xmath268 as @xmath0 moves to still higher values . thus @xmath257 can in principle be located as the only field at which @xmath245 has a single peak vs @xmath263 . the inset to figure [ fig : condteq0 ] compares the field variation of @xmath185 for @xmath67 and for @xmath270 . it is only in the former case ( i.e. , under conditions of strict particle - hole symmetry ) , that the conductance has a single peak vs @xmath0 and attains @xmath256 , whereas for @xmath237 one finds a pair of peaks at @xmath271 . the presence of a single peak under field sweeps can therefore be used to identify the particle - hole - symmetric point in experiments . to better understand these results , we again turn to the noninteracting case @xmath272 , where the linear conductance can be calculated by substituting the noninteracting green s function given by eqs . and for the full green s function @xmath273 in eq . . at @xmath155 , this results in a conductance contribution @xmath274 [ 1+(e_{2\sigma}-e_{1\sigma})^2 ] } \,\ ] ] where @xmath229 and @xmath253 are defined after eq .. equation correctly reduces to eq . in the limit @xmath275 where dot 1 can play no role in the conductance . at particle - hole symmetry ( @xmath233 ) , eq . gives @xmath276 ^ 2 } { [ 1+(b/2\delta_2)^2]\{1+[b/2\delta_2 - b/2\delta(0)]^2\ } } \,\ ] ] which peaks at @xmath256 for @xmath277 , a characteristic field greater than the one @xmath278 at which @xmath98 reaches 1 . since we have seen above that @xmath185 for the interacting case at particle - hole symmetry reaches @xmath178 for some @xmath279 , with @xmath280 for large @xmath10 , the field dependence of the conductance reinforces the parallels between the large-@xmath10 interacting problem and the noninteracting limit , with the many - body scale @xmath96 playing the role of a renormalized @xmath10 . an interesting feature of eq . is that it predicts conduction contributions @xmath281 when particle - hole symmetry and time - reversal symmetry are both broken . in particular , for @xmath282 ( or @xmath283 ) , the conductance polarization measured by @xmath284 grows from @xmath285 for @xmath69 to reach @xmath286 ( or @xmath287 ) for @xmath288 , at which field @xmath289 ( or @xmath290 ) , before decreasing toward zero for still larger fields . by contrast , keeping @xmath291 but allowing @xmath237 results in variation of @xmath292 with field , but does not allow one to achieve perfect polarization of the conductance . ( color online ) conductance spin - polarization @xmath292 ( a ) vs dot-2 level energy @xmath263 at six fixed magnetic fields @xmath0 , and ( b ) vs @xmath0 for two values of @xmath263 . all data are for @xmath66 , @xmath68 , and zero temperature . in ( a ) , @xmath292 is odd about the point @xmath67 of particle - hole symmetry . complete spin polarization of the conductance is achieved in the case @xmath293 . panel ( b ) shows a strong , nonuniversal variation of @xmath292 with @xmath0 for different values of @xmath263.,width=316 ] spin - dependent conductance is also exhibited when dot 1 has strong interactions . figure [ fig : polconducteq0](a ) shows the variation of @xmath292 with the dot 2 level energy @xmath263 in different fields @xmath294 for a symmetric dot 1 ( @xmath58 ) and fixed @xmath10 . the conductance spin - polarization is odd about the point @xmath67 of particle - hole symmetry where the condition @xmath92 ensures [ via eq . ] that @xmath285 . for fields @xmath295 , @xmath292 has the same sign as @xmath263 , whereas for @xmath210 , @xmath292 and @xmath263 have opposite signs . for each field value , @xmath296 peaks at a nonzero value of @xmath268 . one sees that a field @xmath293 combines with a level energy @xmath297 to achieve complete destructive interference of the conduction for one spin species , allowing passage only of a fully spin - polarized current through the device . the fact that reaching @xmath298 in this manner by varying @xmath263 while dot 1 is held at particle - hole symmetry ( @xmath291) is impossible to achieve in the noninteracting case @xmath272 indicates that the interference effects are more complex in the presence of strong interactions . it is important to emphasize that , in contrast to the maximal conductance value @xmath178 , the polarization @xmath292 is unaffected by asymmetry between the left and right dot - lead couplings . complete spin polarization ( @xmath298 ) can be achieved even in setups where @xmath299 . figure [ fig : polconducteq0](b ) shows the variation of @xmath292 under field sweeps at two different values @xmath300 . for each position of the dot-2 level , @xmath292 changes sign at a nonzero @xmath0 . for @xmath301 , @xmath292 reaches @xmath302 at a small field and then dips to nearly @xmath303 at a larger field before increasing back toward zero . for @xmath304 , by contrast , a small positive peak in @xmath292 is followed at larger fields by a dip at ( or very close to ) @xmath303 . this nonuniversal behavior reflects the subtlety of the interplay between the field and particle - hole asymmetry in controlling the constructive or destructive interference between transmission of electrons directly through dot 2 and paths involving one or more detours to dot 1 . similar `` spin - filtering '' effects in a magnetic field have been investigated previously@xcite in the context of a single - mode wire , coupled near its midpoint via a tunnel junction to a quantum dot ( the `` side dot '' ) . a number of experiments and models using different geometries for spin - dependent transport have also been reported in the literature.@xcite reference showed that conductance polarizations @xmath286 and @xmath287 ( in the language of the present paper ) occur at values of the dot energy @xmath305 and @xmath306 differing by a large scale exceeding the dot coulomb interaction strength @xmath160 . thus , the change in gate voltage needed to switch the polarizations is so large that all traces of the kondo effect are suppressed . these behaviors should be contrasted with those found here , where the @xmath263 values that lead to @xmath307 differ only by an energy of order @xmath95 ( much smaller than @xmath5 ) . what is more , the complete spin filtering achieved in our setup depends crucially on the presence of kondo many - body correlations . this point will become particularly clear in the next section , where we consider the effect of nonzero temperatures . reference considered a side - coupled quantum dot in regime of much smaller kondo temperatures . in contrast to our results for double quantum dots , complete polarization of the conductance was reported to occur quite generically due to a mechanism very similar to that we find in the noninteracting limit @xmath272 described by eq . . to this point , only zero - temperature results have been presented . this subsection addresses the effect of finite temperatures on the zero - bias conductance @xmath245 and its spin polarization @xmath292 . throughout the discussion , temperatures are expressed as multiples of a characteristic many - body scale @xmath308 , the system s kondo temperature for @xmath67 , @xmath69 , and the representative value @xmath68 that we have used in all our @xmath57 calculations . ( color online ) linear conductance @xmath245 vs dot-2 level position @xmath263 for @xmath66 and @xmath68 at four temperatures @xmath49 for ( a ) @xmath69 , and ( b ) @xmath309 . temperatures are expressed as multiples of @xmath308.,width=316 ] figure [ fig : finitet_cond ] plots @xmath245 vs @xmath263 for @xmath68 in fields @xmath69 ( panel a ) and @xmath310 ( panel b ) . for @xmath69 , the effect of increasing temperature is a progressive suppression of the kondo effect and hence of the conductance channel involving the many - body kondo resonance . as a result , @xmath245 rises near @xmath67 due to a lessening of the destructive interference between the kondo channel and the single - particle resonance on dot 2 ( discussed above in connection with fig.[fig : condteq0 ] ) , but there is a decrease in the conductance at @xmath311 , which is dominated by transmission through the kondo channel . this trend results in a conductance peak at some @xmath264 for temperatures @xmath312 , which evolves into a peak centered at @xmath67 for @xmath313 , in which regime transmission is dominated by the single - particle , lorentzian - like contribution from dot 2 . figure [ fig : finitet_cond](b ) reveals a very different behavior for @xmath314 . as described above , the @xmath155 conductance attains its maximum possible value @xmath178 at @xmath67 due to constructive interference between the kondo and single - particle conductance channels , and @xmath245 decreases monotonically with increasing @xmath268 . raising the temperature over the range @xmath315 leads to suppression of the kondo conductance channel but has little effect on the single - particle channel , leading to a decrease in @xmath245 that is strongest for @xmath67 . once the temperature passes @xmath316 , the variation of @xmath245 with @xmath263 increasingly reflects the field splitting of the dot-2 energy level , with peaks centered at @xmath317 . ( color online ) ( a ) conductance spin - polarization @xmath292 , and ( b ) conductance @xmath245 vs dot-2 level energy @xmath263 at different temperatures for @xmath66 , @xmath68 , and @xmath293 . temperatures are expressed as multiples of @xmath308.,width=316 ] the influence of temperature on the spin polarization of the conductance is shown in fig . [ fig : finitet_pol](a ) , which focuses on the case @xmath293 that we know from fig . [ fig : polconducteq0 ] yields full spin polarization ( @xmath307 ) at zero temperature for @xmath318 . as @xmath49 increases from zero , the peak spin polarization is lowered , presumably due to a combination of two effects : ( i ) a reduction in the destructive interference between the kondo and single - particle conduction channels for one spin species @xmath319 leading to an increase in @xmath320 entering eq . ; and ( ii ) thermal broadening of @xmath321 in eq . leading to sampling of @xmath65 values having nonzero @xmath322 . at higher temperatures , @xmath323 , the suppression of the kondo conductance channel unmasks oscillations in @xmath292 vs @xmath263 that result from shifts in the spin - resolved energy levels in dot 2 . these oscillations are much less pronounced than the polarization variations at lower temperatures and the maximum values of @xmath296 are about an order of magnitude smaller than those obtained in the kondo regime . figure [ fig : finitet_pol](b ) plots the total conductance @xmath245 vs @xmath263 corresponding to each of the @xmath292 vs @xmath263 traces in fig.[fig : finitet_pol](a ) . there is a close correlation ( although not a perfect match ) between the @xmath263 values of the peaks in @xmath245 and of those in @xmath296 . this suggests that measurements of the total conductance can provide a useful starting point for experiments seeking to optimize the system s spin - filtering performance . ( color online ) effect of a nonzero dot-2 interaction ( @xmath324 ) on ( a ) the conductance spin - polarization @xmath292 , and ( b ) the conductance @xmath245 , both plotted vs dot-2 level energy @xmath263 for the same parameters as in fig.[fig : finitet_pol ] . open symbols correspond to @xmath30 and filled symbols to @xmath325 . temperatures are expressed as multiples of @xmath308.,width=316 ] although we have focused on the special case @xmath30 , the conductance features described above by no means depend on this condition . in fact , qualitatively similar results are obtained for an interacting dot 2 provided that @xmath326 is small compared to the level broadening @xmath95 . this is illustrated in fig . [ fig : u2neq0_finitet_pol ] , which compares the @xmath263 dependence of the conductance and of its spin - polarization for @xmath30 ( data from fig . [ fig : finitet_pol ] ) and @xmath327 , both for the lowest ( @xmath155 ) and highest ( @xmath328 ) temperatures shown in fig.[fig : finitet_pol ] . apart from a small shift in the point of particle - hole symmetry , which moves from @xmath67 to @xmath329 , the other essential features ( such as the complete spin polarization at zero temperature ) are unaffected by the presence of coulomb repulsion within dot 2 . in this work , we have investigated the effect of an applied magnetic field on a strongly interacting quantum dot side - coupled to external leads via a weakly interacting dot . our numerical renormalization - group results show that the interplay of electronic interference , the kondo effect , and zeeman splitting brings about qualitative changes in the spectral and transport properties of this system . we have found , for instance , that the value of the interacting dot s zero - temperature spectral function at the fermi energy does not decay monotonically with increasing field , as it does in single - dot setups . instead , the presence of the extra energy scale determined by the interdot coupling introduces nonuniversal behavior , and in some cases leads to the appearance of one or two maxima in the fermi - energy spectral function at nonzero values of @xmath0 . these features can be understood by the presence of a parameter - dependent phase appearing in the friedel sum rule for energy- and spin - dependent hybridization functions . one of the signatures of the interplay of site and spin degrees of freedom in this double - dot device is the appearance of spin - polarized currents between the two leads . we have shown that the degree of spin polarization can be tuned up to 100% by changing gate voltages and/or small magnetic fields in the system . these results underscore the flexibility of quantum - dot systems for exploration of novel effects in correlated electron physics . we thank a. seridonio and d. logan for helpful discussions . this work was supported in part under nsf materials world network grants dmr-0710540 and dmr-1107814 ( florida ) , and dmr-0710581 and dmr-1108285 ( ohio ) . , s.e.u . , and e.v . acknowledge the hospitality of the kitp , and support under nsf grant phy-0551164 . acknowledges support from brazilian agencies cnpq ( grant no . 482723/2010 - 6 ) and fapesp ( grant no . 2010/20804 - 9 ) . e.v . acknowledges support from cnpq ( grant no . 493299/2010 - 3 ) and fapemig ( grant no . cex - apq-02371 - 10 ) . n.s . and s.e.u . acknowledge the hospitality of the dahlem center and the support of the a. von humboldt foundation . k. g. wilson , rev . phys . * 47 * , 773 ( 1975 ) ; h. r.krishna-murthy , j. w. wilkins , and k. g. wilson , phys . b * 21 * 1003 ( 1980 ) ; r. bulla , t. a. costi , t. pruschke , rev . phys.*80 * , 395 ( 2008 ) .

we study the effect of a magnetic field in the kondo regime of a double - quantum - dot system consisting of a strongly correlated dot ( the `` side dot '' ) coupled to a second , noninteracting dot that also connects two external leads . we show , using the numerical renormalization group , that application of an in - plane magnetic field sets up a subtle interplay between electronic interference , kondo physics , and zeeman splitting with nontrivial consequences for spectral and transport properties . the value of the side - dot spectral function at the fermi level exhibits a nonuniversal field dependence that can be understood using a form of the friedel sum rule that appropriately accounts for the presence of an energy- and spin - dependent hybridization function . the applied field also accentuates the exchange - mediated interdot coupling , which dominates the ground state at intermediate fields leading to the formation of antiparallel magnetic moments on the dots . by tuning gate voltages and the magnetic field , one can achieve complete spin polarization of the linear conductance between the leads , raising the prospect of applications of the device as a highly tunable spin filter . the system s low - energy properties are qualitatively unchanged by the presence of weak on - site coulomb repulsion within the second dot .

after the discovery of ferromagnetism in ( ga , mn)as , diluted magnetic semiconductors ( dmss ) have received considerable attention owing to potential applications based on the use of both their charge and spin degrees of freedom in electronic devices @xcite . thus far , the highest curie temperature of ( ga , mn)as has been t@xmath1 = 190 k @xcite . the substitution of divalent mn atoms into trivalent ga sites introduces hole carriers ; thus , ( ga , mn)as is a p - type dms . the valence mismatch between mn and ga leads to severely limited chemical solubility for mn in gaas . moreover , owing to simultaneous doping of charge and spin induced by mn substitution , it is difficult to individually optimize charge and spin densities . to overcome these difficulties , a new type of dms , i.e. , li(zn , mn)as was proposed @xcite and later fabricated with t@xmath1 = 50 k @xcite . it is based on liznas , a i@xmath2ii@xmath2v semiconductor . spin is introduced by isovalent ( zn@xmath3 , mn@xmath3 ) substitution , which is decoupled from carrier doping with excess / deficient li concentration . although li(zn , mn)as was proposed as a promising n - type dms with excess li@xmath4 , p - type carriers were obtained in the experiment with excess li . the introduction of holes was presumably because of the excess li@xmath4 in substitutional zn@xmath3 sites @xcite . later , another i@xmath2ii@xmath2v dms , i.e. , li(zn , mn)p was reported in an experiment with t@xmath1 = 34 k @xcite . li(zn , mn)p with excess li was determined to be of the p - type as well in the experiment . according to first - principles calculations , the reason for this is the same as that for li(zn , mn)as @xcite . although such p - type i@xmath2ii@xmath2v dmss have a few distinct advantages over ( ga , mn)as , the achievable t@xmath1 is much lower than that of ( ga , mn)as . another type of dms ( ba , k)(zn , mn)@xmath0as@xmath0 was observed in experiments with t@xmath1 up to 230 k @xcite , which is higher than that for ( ga , mn)as . based on the semiconductor bazn@xmath0as@xmath0 , holes were doped by ( ba@xmath3 , k@xmath4 ) substitutions , and spins by isovalent ( zn@xmath3 , mn@xmath3 ) substitutions . it was a p - type dms . motivated by the high t@xmath1 , density functional theory ( dft ) calculations @xcite and photoemission spectroscopy experiments @xcite were conducted to understand the microscopic mechanism of ferromagnetism of p - type dms ( ba , k)(zn , mn)@xmath0as@xmath0 . by contrast , an n - type dms , i.e. , ba(zn , mn , co)@xmath0as@xmath0 was recently reported in an experiment with t@xmath1 @xmath5 80 k @xcite . in this material , electrons are doped because of the substitution of zn with co , and spins are generated mainly because of ( zn@xmath3 , mn@xmath3 ) substitutions . in mn - doped bazn@xmath0as@xmath0 , why is the ferromagnetic ( fm ) coupling observed in both p- and n - type cases ? why is t@xmath1 much lower in the n - type case than that in the p - type case ? in general , can p- and n - type dmss be realized ? the answers will be helpful for fabricating spin p - n junctions in the future . in this study , we attempt to address such issues . in previous studies on dms materials with wide band gap @xmath6 , we found that the position of the impurity bound state ( ibs ) @xmath7 was close to the top of the valence band ( vb ) owing to the strong mixing between the impurity and the vb , and usually no ibs appeared below the bottom of the conduction band ( cb ) because of weak mixing between the impurity and the cb @xcite . thus , we have 0 @xmath8 @xmath7 @xmath9 @xmath6 , as shown in fig . [ f - schematic](a ) . the magnetic correlation @xmath10 between two impurities with fm coupling ( positive @xmath10 ) can be determined when the chemical potential @xmath11 is tuned to be close to the ibs : @xmath11 @xmath5 @xmath7 @xcite . therefore , for p - type carriers ( @xmath11 @xmath5 0 ) , fm coupling can be obtained as @xmath11 @xmath5 @xmath7 , and for n - type carriers ( @xmath11 @xmath5 @xmath6 ) , no magnetic coupling is obtained between impurities because @xmath11 @xmath12 @xmath7 . a schematic diagram describing p - type dms materials with a wide band gap , including ( zn , mn)o @xcite , ( ga , mn)as @xcite , and mg(o , n ) @xcite , is shown in fig . [ f - schematic ] ( b ) . here , we propose a method for realizing p- and n - type dms . the key is choosing host semiconductors with a narrow band gap @xmath13 . by selecting suitable host semiconductors and impurities , the condition 0 @xmath8 @xmath7 @xmath8 @xmath6 is satisfied , as shown in fig . [ f - schematic](c ) . we show that for both the p - type ( @xmath14 ) and the n - type ( @xmath15 ) cases , the condition for developing fm coupling , that is @xmath11 @xmath5 @xmath7 , can be fulfilled , as shown in fig . [ f - schematic](d ) . in the following , we realistically calculate the electronic and magnetic properties of the mn - doped bazn@xmath0as@xmath0 dms , which has a narrow band gap @xmath6 (= 0.2 ev ) @xcite . we use a combination of the dft @xcite and the hirsch@xmath2fye quantum monte carlo ( qmc ) simulation @xcite . our combined dft+qmc method can be used for an in - depth treatment of the band structures of materials and strong electron correlations of magnetic impurities on an equal footing ; thus , it can be applied for designing functional semiconductor- @xcite and metal - based @xcite materials . the method involves two calculations steps . first , the haldane@xmath2anderson impurity model @xcite is formulated within the local density approximation for determining the host band structure and impurity - host mixing . second , magnetic correlations of the haldane - anderson impurity model at finite temperatures are calculated using the hirsch@xmath2fye qmc technique @xcite . the haldane@xmath2anderson impurity model is defined as follows : @xmath16 c^{\dag}_{\textbf{k}\alpha\sigma}c_{\textbf{k}\alpha\sigma } + \sum_{\textbf{k},\alpha,\textbf{i},\xi,\sigma}(v_{\textbf{i}\xi\textbf{k}\alpha } d^{\dag}_{\textbf{i}\xi\sigma } c_{\textbf{k}\alpha\sigma } \notag\\ & + & h.c . ) + ( \epsilon_d-\mu)\sum_{\textbf{i},\xi,\sigma } d^{\dag}_{\textbf{i}\xi\sigma}d_{\textbf{i}\xi\sigma } + u\sum_{\textbf{i},\xi}n_{\textbf{i}\xi\uparrow}n_{\textbf{i}\xi\downarrow } , \label{e - ham}\end{aligned}\ ] ] where @xmath17 ( @xmath18 ) is the creation ( annihilation ) operator for a host electron with wave vector @xmath19 and spin @xmath20 in the vb ( @xmath21 ) or the cb ( @xmath22 ) , and @xmath23 ( @xmath24 ) is the creation ( annihilation ) operator for a localized electron at impurity site @xmath25 in orbital @xmath26 and spin @xmath20 with @xmath27 . here , @xmath28 is the host band dispersion , @xmath11 is the chemical potential , @xmath29 denotes mixing between the impurity and the host , @xmath30 is the impurity @xmath31 orbital energy , and @xmath32 is the on - site coulomb repulsion of the impurity . considering the condition of hund coupling @xmath33 , @xmath34 is neglected and the single - orbital approximation is used to describe the magnetic sates of impurities . the parameters @xmath28 and @xmath29 are obtained by dft calculations using the wien2k package @xcite . to reproduce the experimental narrow band gap of 0.2 ev in bazn@xmath0as@xmath0 @xcite , we use the modified becke@xmath2johnsom exchange potential ( mbj ) @xcite , which has been implemented in the wien2k package . the obtained energy band @xmath36 ( * k * ) is shown in fig . [ f - band - mix ] ( a ) , where bazn@xmath0as@xmath0 has space group i4/mmm . we obtained an indirect gap band @xmath6 = 0.2 ev , which is in good agreement with the experimental @xcite and previous calculated @xcite values . the mixing parameter between the @xmath26 orbitals of an mn impurity and the bazn@xmath0as@xmath0 host is defined as @xmath29@xmath37@xmath38@xmath37@xmath39 , which can be expressed as @xmath40 where @xmath41 is the impurity @xmath31 state at site @xmath25 , and @xmath42 is the host state with wave vector @xmath19 and band index @xmath43 , which is expanded by atomic orbitals @xmath44 having orbital index @xmath45 and site index @xmath46 . here , @xmath47 is the total number of host lattice sites , and @xmath48 is an expansion coefficient . to obtain the mixing integrals of @xmath49 , we consider a supercell ba@xmath50zn@xmath51mnas@xmath52 , which is comprised of 2x2x2 primitive cells , where each primitive cell consists of a bazn@xmath0as@xmath0 , and a zn atom is replaced by an mn atom . the results of the mixing function @xmath53 are shown in fig . [ f - band - mix ] ( b ) for valence bands , and in fig . [ f - band - mix ] ( c ) for conduction bands . the parameters @xmath32 and @xmath30 are determined as follows . for ( ga , mn)as , the reasonable parameters are estimated as @xmath32 = 4 ev and @xmath30 = -2 ev @xcite . a recent resonance photoemission spectroscopy experiment showed that the mn @xmath31 partial density of states in ( ba , k)(zn , mn)@xmath0as@xmath0 and ( ga , mn)as are quite similar , excepted that the peak of ( ga , mn)as is approximately 0.4 ev deeper than that of ( ba , k)(zn , mn)@xmath0as@xmath0 @xcite . thus , the reasonable parameters of mn - doped bazn@xmath0as@xmath0 are @xmath32 = 4 ev and @xmath30 = -1.5 ev . on the basis of the parameters obtained above , magnetic correlations of the impurities are calculated using the hirsch@xmath2fye qmc technique with more than 10@xmath54 monte carlo sweeps and a matsubara time step @xmath55 = 0.25 . figure [ f - ibs ] ( a ) shows a plot of the occupation number @xmath56 of a @xmath26 orbital of an mn impurity in bazn@xmath0as@xmath0 against the chemical potential @xmath11 at 360 k. the top of the vb was taken to be 0 , and the bottom of the cb to be 0.2 ev . operator @xmath57 is defined as follows : @xmath58 the orbitals @xmath59 and @xmath60 of mn subsitutional impurities at the zn site degenerate owing to the crystal field of bazn@xmath0as@xmath0 , which has a group space of i4/mmm @xcite . sharp increases in @xmath57 are observed around -0.5 , -0.4 , -0.2 , and 0.0 ev for the orbitals @xmath26 = @xmath61 , @xmath62 , @xmath63 , and @xmath64 , respectively . this implies the existence of an ibs at this energy @xmath7 @xcite . in order to make the ibs clearer , we show the partial density of state of an mn impurity , @xmath65 , in fig . [ f - ibs ] ( b ) . the peaks in @xmath65 correspond to the positions of ibs . figure [ f - ibs ] ( c ) shows the magnetic correlation @xmath66 between the @xmath26 orbitals of two mn impurities with fixed distance @xmath67 of the first - nearest neighbor . the operator @xmath68 of the @xmath26 orbital at impurity site @xmath25 is defined as follows : @xmath69 for each @xmath26 orbital , fm coupling is obtained when the chemical potential @xmath11 is close to the ibs position , and fm correlations become weaker and eventually disappear when @xmath11 moves away from the ibs . this role of the ibs in determining the strength of fm correlations between impurities is consistent with the hartree@xmath2fock and qmc results of various dms systems @xcite . for mn - doped bazn@xmath0as@xmath0 with p - type carriers , a recent angle - resolved photoemission spectroscopy ( arpes ) experiment showed that the fermi level ( @xmath11 ) is below the top of the vb by several tenths of an ev and a non - dispersive mn 3@xmath70 impurity band is present slightly below the fermi level @xcite . on the basis of the results in fig . [ f - ibs ] ( a ) , we take @xmath11 = -0.3 ev as an estimate for the p - type case . we argue that the ibs of orbitals @xmath62 and @xmath61 , whose positions are below the @xmath11 = -0.3 ev , can account for the non - dispersive mn 3@xmath70 impurity band below the fermi level observed in the arpes experiment . figure [ f - c2 ] ( a ) shows the distance @xmath67 dependence of the magnetic correlation @xmath66 between the @xmath26 orbitals of two mn impurities for the p - type case with @xmath11 = -0.3 ev . long - range fm coupling up to approximately 6 (the third nearest neighbor ) is obtained for the orbitals @xmath26 = @xmath63 and @xmath61 , while short - range fm coupling is obtained for the other three orbitals . thus , our theoretical results are consistent with the fm observed in the experiment involving mn - doped bazn@xmath0as@xmath0 with p - type carriers . for mn - doped bazn@xmath0as@xmath0 with n - type carriers , a recent experiment showed fm coupling below t@xmath1 = 80 k @xcite . because no information about the fermi level has been reported , we take @xmath11 = 0.15 ev as an estimate for the n - type case , which is below the bottom of the cb by 0.05 ev . as shown in fig . [ f - c2 ] ( b ) , long - range fm coupling up to approximately 6 (the 3rd nearest neighbor ) is obtained for the orbitals @xmath26 = @xmath59 and @xmath60 . no fm is obtained for the other three orbitals , shown in fig . [ f - ibs ] ( b ) as well . a comparison of figs . [ f - c2](a ) and [ f - c2](b ) shows that the magnitude of fm coupling @xmath66 in the n - type case is smaller than that in the p - type case , which can qualitatively explain why the t@xmath1 in the n - type case @xcite is lower than that in the p - type case @xcite in the experiments . to understand the long - range fm correlation function @xmath71 between two mn impurities in the bazn@xmath0as@xmath0 host , we have calculated the impurity - host magnetic correlation function @xmath72 . here , @xmath73 is the site of the host electron and the impurity mn is located at site @xmath74 = 0 . the magnetization @xmath75 of the host electron at site @xmath76 is defined as @xmath77 where @xmath78 is the number operator for host electrons with band index @xmath43 and site @xmath73 and spin @xmath20 . in fig . [ f - host](a ) , for p - type carriers with @xmath11 = -0.3 ev , the long - range antiferromagnetic ( afm ) correlation is obtained between the orbitals @xmath26 = @xmath63 and @xmath61 of mn impurity and host electrons . in fig . [ f - host](b ) , for n - type carriers with @xmath11 = 0.15 ev , the long - range afm correlation is obtained between the orbitals @xmath26 = @xmath59 and @xmath60 of mn impurity and host electrons . thus , the long - range fm coupling between impurities is mediated by the polarization of host electron spin . such carrier - mediated fm is already discussed in previous dms materials with a wide band gap , such as ( zn , mn)o @xcite , ( ga , mn)as @xcite , and mg(o , n ) @xcite . we made similar calculations for mn - doped bazn@xmath0sb@xmath0 , where a distinct advantage was the replacement of as with nontoxic sb . bazn@xmath0sb@xmath0 , too , has a narrow band gap @xmath6 = 0.2 ev , but a different space group pnma @xcite . a direct band gap of 0.2 ev was obtained by the dft calculation as shown in fig . [ f - bazn2sb2-mix ] ( a ) , which agrees well with the experimental value . figure [ f - ibs - bazn2sb2 ] ( a ) shows the occupation number @xmath56 of the @xmath26 orbital of the mn impurity in bazn@xmath0sb@xmath0 versus chemical potential @xmath11 at temperature 360 k. the @xmath31 orbitals of mn did not degenerate owing to the low symmetry of the crystal field of bazn@xmath0sb@xmath0 . sharp increases in @xmath56 , which imply the position of ibs @xmath7 , were observed around -0.6 ev for the @xmath62 orbital , -0.4 ev for the @xmath60 , @xmath63 , and @xmath61 orbitals , and -0.2 ev for the @xmath59 orbital . the ibs can be seen more clearly in the partial density of state of an mn impurity , @xmath65 , in fig . [ f - ibs - bazn2sb2 ] ( b ) . the peaks in @xmath65 correspond to the positions of the ibs . figure [ f - ibs - bazn2sb2 ] ( c ) shows the magnetic correlation @xmath66 between the @xmath26 orbitals of two mn impurities with fixed distance @xmath67 as the first nearest neighbor . the role of the ibs in determining the strength of the fm correlations between impurities is the same as that discussed for mn - doped bazn@xmath0as@xmath0 in fig . [ f - ibs ] . for mn - doped bazn@xmath0sb@xmath0 with p - type carriers , we take @xmath11 = -0.3 ev , the same value as that used for mn - doped bazn@xmath0as@xmath0 with p - type carriers . figure [ f - c2-bazn2sb2 ] shows the distance @xmath67 dependence of the magnetic correlation @xmath66 between the @xmath26 orbitals of two mn impurities for the p - type case . long - range fm coupling up to approximately 10 (the 14th nearest neighbor ) was obtained for the @xmath26 = @xmath59 , @xmath60 , @xmath63 , and @xmath61 orbitals , while relatively short - range fm coupling is obtained for the @xmath62 orbital . this is considerably longer than 6 (the third nearest neighbor ) obtained for mn doped bazn@xmath0as@xmath0 with p - type carriers , as shown in fig . [ f - c2 ] ( a ) . such long - range fm coupling arises from the short distance between the neighboring zn sites in bazn@xmath0sb@xmath0 , as is clear from comparison of the first- , second- , and third - nearest neighbors in fig . [ f - c2](a ) and those neighbors in fig . [ f - c2-bazn2sb2 ] , respectively . we predict that the t@xmath1 of mn - doped bazn@xmath0sb@xmath0 with p - type carriers should be higher than that of mn - doped bazn@xmath0as@xmath0 with p - type carriers , in which t@xmath1 = 230 k was reported in a recent experiment @xcite . for mn - doped bazn@xmath0sb@xmath0 with n - type carriers , we take @xmath11 = 0.15 ev , the same value as that used for mn - doped bazn@xmath0as@xmath0 with n - type carriers . no fm coupling is obtained with @xmath11 = 0.15 ev . this is because @xmath11 = 0.15 ev is far from the ibs position @xmath80 -0.2 ev of the @xmath59 orbital , as shown in figs . [ f - ibs - bazn2sb2](a ) and [ f - ibs - bazn2sb2](b ) . it is consistent with previous studies that no magnetic coupling is obtained between impurities when @xmath81@xcite . in the above qmc calculations , we fix the model parameters of impurity level @xmath30 = -1.5 ev and coulomb repulsion @xmath32 = 4 ev , which are reasonable values for mn - doped bazn@xmath0as@xmath0 and bazn@xmath0sb@xmath0 as discussed in sec . iii . in this section , we will discuss how the uncertainty of these values affects the outcome of the calculations . for mn - doped bazn@xmath0as@xmath0 with the same impurity level parameter @xmath30 = -1.5 ev and a larger coulomb repulsion parameter @xmath32 = 5 ev , the occupation number @xmath56 , the partial density of state @xmath65 of the @xmath26 orbital of an mn impurity , and the magnetic correlation @xmath66 between the @xmath26 orbitals of two mn impurities with fixed distance of the first - nearest neighbor are shown in figs . [ f - ibs - u5](a)-[f - ibs - u5](c ) , respectively . compared with the results obtained with parameters @xmath30 = -1.5 ev and @xmath32 = 4 ev in fig . [ f - ibs ] , no essential difference is observed . for mn - doped bazn@xmath0as@xmath0 with a deeper impurity level parameter @xmath30 = -2 ev and the same coulomb repulsion parameter @xmath32 = 4 ev , @xmath56 , @xmath65 , and @xmath66 are shown in figs . [ f - ibs - em2](a)-[f - ibs - em2](c ) , respectively . compared with the results in fig . [ f - ibs ] , the ibs positions @xmath7 of @xmath26 orbitals of mn impurity shift down by about 0.1 ev . as a result , the fm correlation @xmath66 is obtained for p - type carriers with @xmath11 = -0.3 ev , while no fm correlation @xmath66 is obtained for n - type carriers with @xmath11 = 0.15 ev . this result does not agree with the recent experiment of mn - doped bazn@xmath0as@xmath0 with n - type carriers , where fm coupling is observed below t@xmath1 = 80 k @xcite . thus , the impurity level parameter @xmath30 = -2 ev may be too deep for mn - doped bazn@xmath0as@xmath0 , as we have also discussed in sec . for mn - doped bazn@xmath0sb@xmath0 with the same impurity level parameter @xmath30 = -1.5 ev and a larger coulomb repulsion parameter @xmath32 = 5 ev , @xmath56 , @xmath65 , and @xmath66 are shown in figs . [ f - ibs - bazn2sb2-u5](a)-[f - ibs - bazn2sb2-u5](c ) , respectively . compared with the results obtained with parameters @xmath30 = -1.5 ev and @xmath32 = 4 ev in fig . [ f - ibs - bazn2sb2 ] , no essential difference is observed . for mn - doped bazn@xmath0sb@xmath0 with a deeper impurity level parameter @xmath30 = -2.0 ev and the same coulomb repulsion parameter @xmath32 = 4 ev , @xmath56 , @xmath65 , and @xmath66 are shown in figs . [ f - ibs - bazn2sb2-em2](a)-[f - ibs - bazn2sb2-em2](c ) , respectively . compared with the results in fig . [ f - ibs - bazn2sb2 ] , the ibs positions @xmath7 of @xmath26 orbitals of mn impurity shift down by about 0.1 ev . the fm correlation @xmath66 is obtained for p - type carriers with @xmath11 = -0.3 ev , and no fm correlation @xmath66 is obtained for n - type carriers with @xmath11 = 0.15 ev . the conclusion is unchanged . in summary , we have proposed a method to realize dms with p- and n - type carriers by choosing host semiconductors with a narrow band gap . using the combined method of dft and qmc , we describe dms mn - doped bazn@xmath0as@xmath0 , which has a narrow band gap of 0.2 ev . in addition , we find a nontoxic dms mn - doped bazn@xmath0sb@xmath0 , whose t@xmath1 is expected to be higher than that of mn - doped bazn@xmath0as@xmath0 , for which t@xmath1 = 230 k , as reported in a recent experiment . the authors acknowledge h. y. man , f. l. ning , c. q. jin , h. suzuki , and a. fujimori for many valuable discussions about the experiments of mn - doped bazn@xmath0as@xmath0 . z. deng , c. q. jin , q. q. liu , x. c. wang , j. l. zhu , s. m. feng , l. c. chen , r. c. yu , c. arguello , t. goko , f. ning , j. zhang , y. wang , a. a. aczel , t. munsie , t. j. williams , g. m. luke , t. kakeshita , s. uchida , w. higemoto , t. u. ito , b. gu , s. maekawa , g. d. morris , and y. j. uemura , nat . commun . * 2 * , 422 ( 2011 ) . z. deng , k. zhao , b. gu , w. han , j. l. zhu , x. c. wang , x. li , q. q. liu , r. c. yu , t. goko , b. frandsen , l. liu , j. zhang , y. wang , f. l. ning , s. maekawa , y. j. uemura , and c. q. jin , phys . rev . b * 88 * , 081203(r ) ( 2013 ) . k. zhao , z. deng , x. c. wang , w. han , j. l. zhu , x. li , q. q. liu , r. c. yu , t. goko , b. frandsen , l. liu , f. l. ning , y. j. uemura , h. dabkowska , g. m. luke , h. luetkens , e. morenzoni , s. r. dunsiger , a. senyshyn , p. boni , and c. q. jin , nat . commun . * 4 * , 1442 ( 2013 ) . h. suzuki , k. zhao , g. shibata , y. takahashi , s. sakamoto , k. yoshimatsu , b. j. chen , h. kumigashira , f. h. chang , h. j. lin , d. j. huang , c. t. chen , b. gu , s. maekawa , y. j. uemura , c. q. jin , and a. fujimori , phys . b * 91 * , 140401(r ) ( 2015 ) . h. suzuki , g. q. zhao , k. zhao , b. j. chen , m. horio , k. koshiishi , j. xu , m. kobayashi , m. minohara , e. sakai , k. horiba , h. kumigashira , b. gu , s. maekawa , y. j. uemura , c. q. jin , and a. fujimori , phys . b * 92 * , 235120 ( 2015 ) . m. ichimura , k. tanikawa , s. takahashi , g. baskaran , and s. maekawa , _ foundations of quantum mechanics in the light of new technology _ , edited by s. ishioka and k. fujikawa . ( world scientific , singapore , 2006 ) , pp . 183 - 186 . p. blaha , k. schwart , g. k. h. hadsen , d. kvasnicka , and j. luitz , wien2k , an augmented plane wave plus local orbitals program for calculating crystal properties , vienna university of technology , vienna , 2001 .

we propose a method to realize diluted magnetic semiconductors ( dmss ) with p- and n - type carriers by choosing host semiconductors with a narrow band gap . by employing a combination of the density function theory and quantum monte carlo simulation , we demonstrate such semiconductors using mn - doped bazn@xmath0as@xmath0 , which has a band gap of 0.2 ev . in addition , we found a nontoxic dms mn - doped bazn@xmath0sb@xmath0 , of which the curie temperature t@xmath1 is predicted to be higher than that of mn - doped bazn@xmath0as@xmath0 , the t@xmath1 of which was up to 230 k in a recent experiment .

early interferometric observations of radio source structure were typically analyzed by examining how the amplitude of the fringe visibility varied with projected interferometer spacing ( e.g. , * ? ? ? * ) . although these techniques for conventional connected interferometers were later replaced by full synthesis imaging incorporating fourier inversion , clean ( e.g. , * ? ? ? * ) , and self calibration ( see review by * ? ? ? * ) , the interpretation of the early vlbi observations , again , was based on the examination of fringe amplitudes alone ( see * ? ? ? * ) . indeed , the discovery of superluminal motion in the source @xcite was based on single baseline observations of the change in spacing of the first minimum of the fringe visibility . however , after the development of phase - closure techniques , reliable full synthesis images have been produced from vlbi observations for more than 25 years . however , these tend to hide the information on the smallest scale structures , because of the convolution with the synthesized beam ( e.g. , figure [ radplot - map ] ) . a more thorough discussion of non - imaging vlbi data analysis is given by @xcite . the best possible angular resolution is needed to study the environment close to supermassive black holes where relativistic particles are accelerated and collimated to produce radio jets . the greatest angular resolution to date was obtained in observations of interstellar scintillations ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the resolution achievable with vlbi can be improved by observing at shorter wavelengths ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) or by increasing the physical baseline lengths using earth - to - space interferometry . the first space vlbi missions @xcite increased the available baseline lengths by a factor of about 3 . planned space vlbi observations such as radioastron @xcite , vsop2 @xcite , and arise @xcite , will extend the baselines further . for simple source structures , a direct study of the fringe visibilities can give a better angular resolution than an analysis of the images reconstructed from these data ( e.g. , * ? ? ? * ) . in principle , a careful deconvolution of the images should give equivalent results . however , experience has shown that when confronted with even moderately complex images , it is dangerous to attempt to increase the resolution significantly beyond that of the clean restoring beam size ; a procedure referred to in early radio astronomy literature as `` super resolution '' . in two previous papers ( @xcite , hereafter , and @xcite , hereafter ) we have described the sub - milliarcsecond scale structure of 171 active galactic nuclei , based on naturally weighted images made from observations with the vlba @xcite at 15 ghz . in addition , in we have placed more restrictive limits on the sizes of unresolved sources by direct analysis of the fringe visibilities . in a third paper ( @xcite , hereafter ) we have reported on the observed motions in the jets of 110 of these sources during the period 1994 to 2001 . in this paper , we analyze 15 ghz vlba observations of the central regions of 250 extragalactic radio sources . we use the visibility function data to study the most compact structures and the way they change with time . the smallest features we are able to discern from these data have an extent of about 0.020.06 mas . for the nearest object in our study , ( , ) , this corresponds to a linear size of @xmath3 cm , or several tens of schwarzschild radii , if the mass of the central object is @xmath4 solar masses and the distance is 17.5 mpc . we define our sample in [ sampledef ] , describe the visibility data in [ visdata ] , and the model fitting and analysis in [ modelfitting ] . in [ results ] we discuss the results , and the conclusions are summarized in [ summary ] . throughout this paper we use the following cosmological parameters : @xmath5 kms@xmath6mpc@xmath6 , @xmath7 , and @xmath8 . we adopt the convention of using the term `` quasar '' to describe optical counterparts brighter than absolute magnitude @xmath9 , and `` active galaxy '' for the fainter objects . our analysis is based on data obtained during the period 19942003 as part of the vlba 15 ghz monitoring survey of extragalactic sources ( papers i , ii , and iii , @xcite , e. ros et al . , in preparation ) . we have also used additional observations made in 1998 and 1999 by l. i. gurvits et al . ( in preparation ) as part of a separate program to compare 15 ghz source structure measured with the vlba to 5 ghz structure measured in the framework of the vsop survey program @xcite . the program by gurvits et al . used the same observing and data reduction procedures as the vlba 15 ghz monitoring survey , and provides both additional sources and additional epochs . our dataset consists of 1204 vlba observations of 250 different compact extragalactic radio sources . the initial calibration of the data was carried out with the nrao @xmath10 package @xcite , and was followed by imaging with the difmap program @xcite , mostly with the use of an automatic script . the clean images as well as the visibility function data are available on our web sites . most of the radio sources contained in our `` full sample '' of 250 sources have flat radio spectra ( @xmath11 , @xmath12 ) , and a total flux density at 15 ghz ( often originally estimated by extrapolation from lower frequency data ) greater than 1.5 jy for sources with declination @xmath13 , or greater than 2 jy for sources with @xmath14 . however , additional sources which did not meet these criteria but are of special interest were also included in the full sample . our full sample is useful for investigating fine scale structure in a cross - section of known extragalactic radio source classes , and for planning future ( space ) vlbi observations . however , in order to compare observations with the theoretical predictions of relativistic beaming models , it is also useful to have a well - defined sub - sample selected on the basis of beamed , rather than total , flux density . we have therefore formed a flux density limited complete sample which has been used as the basis of our jet monitoring program since mid 2002 , called `` the mojave program : * m*onitoring * o*f * j*ets in * a*gn with * v*lba * e*xperiments '' ( , @xcite ) . there are 133 sources in the mojave sub - sample . the redshift distribution for these sources ranges up to 3.4 ( quasar ) , although most sources have redshifts less than 2.5 , with a peak in the distribution near 0.8 . table [ sample ] summarizes the properties of each source . columns 1 and 2 give the iau source designation , and where appropriate , a commonly used alias ; j2000.0 coordinates are in columns 3 and 4 . the optical classification and redshift are shown in columns 5 and 6 , respectively ; these were obtained mainly from @xcite , as discussed below . in column 7 we give a radio spectral classification for each source based on the radio telescope observations of broad - band instantaneous spectra from 1 to 22 ghz @xcite . these spectra are available on our web site . for the few sources which were not observed at , we used published ( non - simultaneous ) radio flux densities taken from the literature . we consider a radio spectrum to be `` flat '' if any portion of its spectrum in the range 0.6 ghz to 22 ghz has a spectral index flatter than @xmath15 and `` steep '' if the radio spectral index is steeper than @xmath15 over this entire region . in column 8 we indicate whether or not the radio source is associated with a gamma - ray detection by egret @xcite . columns 9 and 10 indicate whether or not the source is a member of the complete correlated flux density limited mojave sample and the vsop 5 ghz agn survey source sample @xcite . column 11 gives references to papers reporting intra - day variability ( idv ) of the source total flux density . of the 250 sources in the full sample , there are 179 quasars , 37 bl lacertae objects , 23 active galaxies , and 11 sources which are optically unidentified . the mojave complete sample of 133 sources includes 94 quasars , 22 bl lacertae objects , 8 active galaxies , and 9 unidentified objects . these classifications come from @xcite , who defined a quasar as a star - like object , or an object with a star - like nucleus , with broad emission lines , brighter than absolute magnitude @xmath16 . @xcite provide a list of bl lacertae objects , which historically were defined as bright galactic nuclei which are highly polarized in the optical regime , and for which no emission or absorption lines have been detected . the precise delineation between bl lacs and ovv quasars remains controversial @xcite , since the original proposed 5 limit @xcite is arbitrary @xcite , and individual emission line equivalent widths are now known to be highly variable over time . indeed the prototype , bl lac itself , shows broad and narrow emission lines , as well as stellar absorption lines , in modern spectra : it no longer meets the classical definition of a bl lac @xcite . the detectability of narrow and broad emission lines and absorption lines is set to a very significant degree by the variable continuum level , signal - to - noise ratio ( snr ) , starlight contribution , and other extrinsic and time - dependent factors ( see , e.g. , * ? ? ? the situation is complicated further by proposed unification schemes @xcite , which apparently led @xcite to re - classify many bl lacs as quasars solely on the basis of their extended 5 ghz luminosity being above the fr i / ii division . many of the objects in our sample are blazars , which are defined as the union of the original categories of bl lacertae objects and optically violently variable ( ovv ) quasars . both groups are highly polarized and variable in the optical spectral region ( e.g. , * ? ? ? * ) . since we are interested in comparing the radio properties of strong and weak - lined blazars , we retain the original bl lac classifications for these objects and indicate these and other controversial classifications in the notes to table [ sample ] . for our analysis , it would have been preferable to directly use optical line equivalent width data ; however , high - quality , multi - epoch spectra are currently available for only a small fraction of our sample . nevertheless , the objects originally classified as bl lacs do , on average , have lower equivalent width spectral lines than classical quasars . this seems to be ( i ) partly the effect of dilution by a beamed non - thermal continuum ( e.g. , * ? ? ? * ) , and ( ii ) partly because many of these objects are intrinsically different from classical quasars as shown by their diffuse radio emission which is similar to that of fr i radio galaxy ( e.g. , * ? ? ? * ; * ? ? ? these two effects can not be clearly separated using only vlbi data . we show here that on average objects historically called `` bl lacs '' differ statistically from classical quasars in their parsec scale radio properties . physical interpretation depends on separating the above two effects . our 15 ghz vlba images , made with natural weighting of the visibility data , have a nominal resolution of 0.5 mas in the east west direction and 0.61.3 mas in the north - south direction . the fringe spacing of the vsop survey at 5 ghz @xcite is similar to that of the vlba at 15 ghz , but the effective resolution of the vlba is better , thanks to the relatively high snr on the longest baselines , and to the good relative calibration of the fringe visibilities , which can be determined with self calibration using higher quality images based on many more interferometer baselines , and full hour angle coverage . typically the dynamic range of the vlba images ( the ratio of the peak flux density to the rms noise level ) is better than 1000:1 ( , @xcite ) . figure [ auv ] shows the visibility function amplitudes ( correlated flux density versus projected baseline length ) for each source in the full sample at the epoch when the amplitude is the highest at the longest projected spacings . with plots for all of the epochs observed in the 15 ghz vlba monitoring program until 2003/08/28 is published in the electronic version only . ] these plots are independent of any assumptions about the source structure , imaging artifacts , or beam smoothing ; and , more directly , they can show the presence of structure on scales smaller than the synthesized beam . also , as illustrated in figure [ radplot_var ] , many of these sources are variable . changes with time in the observed visibility data , especially those on the longest baselines , corresponding to flux density variations in the unresolved components , are not easily seen in the synthesized images constructed from these data ( see , e.g. , figure [ radplot - map ] ) , but they are apparent when comparing visibility function plots . variability characteristics will be discussed in more detail in [ var_discussion ] . examination of the observed amplitudes of the visibility functions in figure [ auv ] suggests that they can be divided into the following categories : \(i ) barely resolved sources where the fringe visibility decreases only slowly with increasing spacing ( e.g. , 0235 + 164 , 0716 + 714 , 1726 + 455 ) . for sources with good snr , we can confidently determine that these sources are resolved even if the fractional fringe visibility on the longest baselines is as large as 0.950.98 , which corresponds to an angular size of only 0.0560.036 mas in the direction corresponding to the largest spacings ( see the detailed discussion of the resolution criterion in [ modelfitting ] ) . there are no sources which are completely unresolved . however , the maximum resolution of the vlba is obtained within a narrow range of position angles close to the east - west direction . in other directions , the resolution is poorer by a factor of two to three . \(ii ) sources with a well resolved component plus an unresolved or barely resolved component . in these , the fringe visibility initially decreases with increasing spacing , and then remains constant or decreases slowly ( e.g. , 0106 + 013 , 0923 + 392 , 1213@xmath17172 ) . for these sources we can place comparable limits on the size of an unresolved feature as in case ( i ) above . \(iii ) more complex or multi - component sources have visibility functions which vary significantly with baseline . if there is an upper envelope to the visibility function , which decreases only slowly to larger spacings , then the structure is primarily one - dimensional , and the upper envelope indicates the smallest dimension ( e.g. , 1045@xmath17188 , 1538 + 149 , 2007 + 777 ) . if there is a well - defined lower envelope , which monotonically decreases to larger spacings ( e.g. , 0014 + 813 , 0917 + 624 , 1656 + 053 ) , this may be used as a measure of the overall dimensions of the source . if minima are observed in the lower envelope ( e.g. , 0224 + 671 , 2131@xmath17021 , 2234 + 282 ) , they correspond to the spacing of the major components . the total flux density of each image , @xmath18 , is the sum of the flux densities of all components of the clean model ; this should be equivalent to the visibility function amplitude , @xmath19 ( the correlated flux density ) , on the shortest projected baselines . in most cases in our sample , @xmath19 at the shortest spacings and @xmath18 are equal to within a few percent , which is a consequence of the hybrid imaging procedure . we define the ( @xmath20,@xmath21)-radius as @xmath22 . the unresolved ( `` compact '' ) flux density @xmath23 is defined as the upper envelope ( with 90% of the visibilities below it ) of the visibility function amplitude @xmath19 at projected baselines @xmath24360m@xmath25 , which is approximately 0.8@xmath26 . the overall uncertainty in @xmath18 and @xmath23 is determined mainly by the accuracy of the flux density ( amplitude ) calibration , which we estimate to be about 5% ( consistent with estimates of * ? ? ? we have used the program difmap @xcite to fit the complex visibility functions with simple models , consisting of two elliptical gaussian components , one representing the vlba core , and the other the inner part of a one - sided jet . by the `` core '' we mean the bright unresolved feature typically found at the end of so - called `` core - jet '' sources ; this is usually thought to be the base of a continuous jet , and does not necessarily correspond to the nucleus of the object . the main objective of our procedure was to obtain a robust characterization of the core . we have verified the suitability of our method for that purpose in several ways . varying the initial values for the iterative fitting procedure did not significantly change the final core parameter values . using more complex models consisting of three or four components also did not significantly change the parameter values for the core component in most sources , even fairly complex ones ; instead , the additional components tend to cover additional parts of the jets . we have also compared the modeling results obtained in this study with the more elaborate models obtained for all mojave sources in the work of @xcite ; those models were built up by adding new components until the thermal noise level was reached in the residual image . for about 90 % of the sources in which the core was modeled by @xcite as an elliptical gaussian component , the parameters derived by the two methods agree to within 10 % . however , for 23 sources with complex structure we found that a two - component model overestimates the flux density and the angular size of the core , and we have added more components to model these sources . we do not present or use any modeling results for an additional 11 sources , which have very complex structure , such as the two - sided radio galaxy ( @xcite , see also the sources in figure 2 , ) . we conclude that , for most of the sources in our sample , the core can be characterized accurately and robustly with the two - component modeling method used , because the beamed emission of the compact core dominates the 15 ghz structure ( median value of @xmath27 for the full sample ) . in about one quarter of the datasets the core is only slightly resolved on the longest baselines . following @xcite we derive a resolution criterion for vlbi core components , by considering a visibility distribution @xmath28 corresponding to the core . @xmath29 is normalized by the flux density @xmath30 , so that @xmath31 . the core is resolved if @xmath32 here , @xmath33 is the rms noise level in the area of the image occupied by the core component ( to exclude possible contamination from the jet ) . in order to measure @xmath33 for each dataset , we have first subtracted the derived model from the image and then used the residual pixel values in the area of the core component convolved with the synthesized beam ( truncated at the half - power level ) . for naturally weighted vlbi data , the beam size , with major and minor axes @xmath34 and @xmath35 measured at the half - power point , yields the largest observed ( @xmath20,@xmath21)-spacing as follows : @xmath36 . the visibility distribution corresponding to a gaussian feature of angular size @xmath37 is given by @xmath38 $ ] . with these relations , the resolution criterion given by equation ( [ eq : res - crit ] ) yields for the minimum resolvable size of a gaussian component fitted to naturally weighted vlbi data : @xmath39 here , @xmath40 is the half power beam size measured along an arbitrary position angle @xmath41 . for all datasets we have derived @xmath42 corresponding to the position angles of the major and minor axis ( @xmath43 , @xmath44 ) of the fitted gaussian core component . whenever either one or both of the two axes were smaller than the respective @xmath42 , the gaussian component was considered to be unresolved . @xmath42 was then used as an upper limit to the size of the component , which yields a lower limit to its brightness temperature . it should be noted that , at high snr , @xmath42 can be significantly smaller than the size of the resolving beam and the rayleigh limit . this is the result of applying a specific _ a priori _ hypothesis about the shape of the emitting region ( a two - dimensional gaussian , in our case ) to fit the observed brightness distribution . a similar approach is employed to provide the theoretical basis for the technique of super resolution @xcite . in our analysis we have also used the total flux density @xmath45 at 15 ghz , determined from observations with single antennas . we have incorporated the data from the university of michigan radio astronomy observatory monitoring program ( umrao , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) as well as instantaneous 122 ghz broad - band radio spectra obtained during the long - term monitoring of compact extragalactic sources with the radio telescope of the special astrophysical observatory @xcite . we have interpolated the umrao observations in time , and the data in both frequency ( between 11 and 22 ghz ) and time to obtain the effective filled aperture total flux density , @xmath45 , at the epoch and frequency of the vlba observations . the main contribution to the total uncertainty on the @xmath45 values comes from the non - simultaneity of the vlba and the single dish observations . this can give errors up to 2030% , but the typical uncertainties are below 5% . figure [ hist_s ] shows the distributions in our sample of the total flux densities , @xmath45 , from single dish measurements , and the correlated flux densities , @xmath23 , from long vlba spacings . the peak in the distribution of @xmath45 corresponds to our nominal flux density limit of 1.5 or 2 jy ( depending on declination ) . the tail to lower flux densities in both panels is due to variability , and in the full sample ( left hand panel ) the tail also includes some sources of particular interest , which we included in the observations , but which did not meet our flux density criteria . figure [ hist_ss ] gives the distributions of the `` indices of compactness '' on arcsecond scales , @xmath46 , and sub - mas scales , @xmath47 , as well as of the vlba core dominance , @xmath48 . many sources in our sample have considerable flux on spatial scales sampled by the longest vlba baselines . in the lower left hand panel of figure [ hist_s ] , we see that more than 90% of the sources have an unresolved flux density greater than 0.1 jy at projected baselines longer than 360 million wavelengths , while the middle left panel of figure [ hist_ss ] shows that 68% of the sources have a median @xmath49 . table [ data_table ] lists , for each source , flux densities and model fitting results ( as well as some other data ) at the epoch for which its unresolved flux density @xmath23 was greatest . these data will be of value for various purposes , including planning future vlbi observations using earth - space baselines . for 163 of these sources , the median flux density of the most compact component is greater than 0.5 jy . we have compared the measured values of @xmath18 and @xmath45 . figure [ hist_ss ] indicates that there are no significant systematic errors in the independently - constructed vlba / ratan / umrao flux density scales . the median compactness index on arcsecond scales , @xmath46 , is 0.91 for the full sample and 0.93 for the mojave sample , which indicates that for most sources the vlba image contains nearly all of the flux density . some sources have an apparent compactness on arcsecond scales @xmath50 . most likely , this is due to source variability and the non - simultaneity of the vlba and single antenna observations . sources with compactness index close to unity ( see table [ data_table ] ) are well - suited as calibrators for other vlba observations . the curves in figure [ relative_radplot ] show the mean visibility amplitude versus projected ( @xmath20,@xmath21 ) spacing , averaged over all sources in the mojave sample , and averaged over the mojave quasars , bl lacs , and active galaxies , separately . the best fitting parameter values for a model consisting of two gaussian components are listed in table [ radplot - model ] for each of these mean visibility curves . the active galaxies are , on average , the least vlba core dominated and the least compact on arcsecond ( figure [ hist_ss ] ) and sub - mas ( figures [ hist_ss ] and [ relative_radplot ] , and table [ radplot - model ] ) scales . the fact that the relative contribution of an extended component ( i.e. a jet ) is significantly greater for active galaxies is consistent with unification models in which radio galaxies are viewed at larger angles to the line of sight than bl lacs or quasars ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , so that the latter have higher doppler factors , their cores are more boosted , and thus they appear more core dominated . the kolmogorov - smirnov ( k - s ) test confirms that for both the full and the mojave sample the probability is less than 1% that the active galaxies have the same parent distribution as the quasars or the bl lacs with regard to their compactness on arcsecond scales , @xmath46 , or their compactness on sub - mas scales , @xmath47 ; with regard to their core dominance , @xmath48 , the probability is less than 2% . for the bl lacs versus quasars , k - s tests were inconclusive . however , figure [ relative_radplot ] and table [ radplot - model ] show that the bl lacs are , on average , even more compact on sub - mas scales than the quasars . we have also found this distinction between sub - samples of quasars and bl lacs chosen to have statistically indistinguishable redshift distributions . the differences between the quasars and the bl lacs in angular size at sub - mas scales in the sample as a whole are therefore not related to the different overall redshift distributions of these groups . as discussed in [ sampledef ] , classifying objects is a complex issue , particularly with regard to bl lacs . with our tabulated data , others could repeat the analysis using their own classification procedure if desired . however , the optical classification scheme we have used is evidently `` clean '' enough that , after the fact , it turns out to correspond to differences in radio compactness . we can not image any hypothetical optical classification bias which could be fully responsible for the correspondence with radio compactness , and we conclude that it is an actual physical phenomenon . our sample does not show a significant dependence on redshift of the index of compactness on sub - mas scales , @xmath47 , although the few heavily resolved sources are mostly active galaxies at low redshift . @xcite have presented a plot , similar to figure [ relative_radplot ] , based on 5 ghz vlba and vsop observations of 189 radio sources which cover a comparable range of spatial frequencies as our 15 ghz vlba data . they find that the average fringe visibility in the range 400 to 440 million wavelengths is 0.210.24 . for the 116 sources in common to the two samples ( see table [ sample ] ) , we find an average fringe visibility at 15 ghz of about 0.6 . the compact component emission dominates at 15 ghz ( @xmath5175% , see table [ radplot - model ] ) , but not at 5 ghz ( 40% , * ? ? ? this reflects the fact that the 5 ghz observations detect a larger contribution from steep spectrum optically thin large scale components . the 5 ghz vsop survey sample and our 15 ghz vlba sample are not identical , but this result is confirmed if only the subset of overlapping pearson readhead vsop survey sources @xcite is used for comparison . figure [ ft_difmap ] shows distributions of the core parameters . the ratio of the major axis of the core to the beam width in the same direction varies by more than an order of magnitude , so , in most cases , we believe that our measured core dimensions are not an artifact of the finite beam size . the cores are always resolved along their major axis . however , for 158 sources in our full sample , there is at least one epoch at which the core component appears unresolved along the minor axis , where it is then typically less than 0.05 mas in size . in 19 of these sources , including 5 bl lacs , the core is unresolved along the minor axis at all observed epochs . figure [ hist_pabjcdiff ] shows the distribution of the difference between the position angle of the major axis of the core , @xmath52 , and the jet direction , @xmath53 ; the latter was taken to be the median over all epochs of the position angle of the jet component with respect to the core . all possible values of @xmath54 between @xmath55 and @xmath56 are observed . not unexpectedly , the peak of the distribution is close to zero ; that is , the gaussian component representing the core is typically extended along the jet direction . for the majority of sources which have multi - epoch modeling data , the orientation of the core is stable in time , with a scatter around the average @xmath52 of less than @xmath57 . figure [ hist_pabjcdiff ] also demonstrates that the position angle of the core is not correlated with the position angle of the vlba beam , so that the measured core orientation is , in most cases , not distorted by the orientation of the vlba beam . the brightness temperature of a slightly resolved component in the rest frame of the source is given by @xmath58 where @xmath30 is the flux density of a vlba core , @xmath43 and @xmath44 are the full width at half maximum ( fwhm ) of an elliptical gaussian component along the major and the minor axis , @xmath25 is the wavelength of observation , @xmath59 is the redshift , and @xmath60 is the boltzmann constant . observing at @xmath61 cm with @xmath30 measured in jy , and @xmath43 and @xmath44 in mas , we can write @xmath62 the brightness temperature can also be represented in terms of an effective baseline @xmath63 . if @xmath64 is measured in km and @xmath30 in jy , we have @xmath65 which is independent of wavelength and depends only on the physical length of the effective projected baseline and on the core flux density . for sources without measured redshift ( see table [ sample ] ) we use @xmath66 to define a limit to @xmath67 . and gave conservative estimates of the observed peak brightness temperature based on the observed angular size , which is the intrinsic size convolved with the vlba beam width . here , we derive the core brightness temperature using the dimensions or upper limits obtained from direct modeling of the complex visibility functions . for many sources , the effective resolution is an order of magnitude better than given by and , so the corresponding derived brightness temperatures are as much as a factor of 100 greater . the median value of these vlba core brightness temperatures , shown in figure [ hist_tb ] , is near @xmath68 k ; they extend up to @xmath2 k. this is comparable with brightness temperatures derived from vsop space vlbi observations @xcite . in many cases our measurement refers only to the upper limit of the angular size , corresponding to our minimum resolvable size derived using equation ( [ eq : resolution ] ) . the effective resolution depends on the maximum baseline and on the signal - to - noise ratio near the maximum resolution . the true brightness temperatures of many sources may extend to a much higher value , beyond the equipartition value of @xmath0 k @xcite or the inverse compton limit of @xmath68 k @xcite . these high brightness temperatures are probably due to doppler boosting , but transient non - equilibrium events , coherent emission , emission by relativistic protons , or a combination of these effects ( e.g. , * ? ? ? * ; * ? ? ? * ) may also play a role . if the high observed brightness temperatures are due to doppler boosting , and if the range of intrinsic brightness temperatures , @xmath69 , is small , there should be a correlation between the apparent jet velocity , @xmath70 , and the observed brightness temperature . for those sources listed in , figure [ beta_maxtb ] shows the fastest observed jet velocity against the maximum observed brightness temperature of their core . while this plot contains mostly lower limits to the brightness temperature , there are no sources with a low brightness temperature and a high observed speed ; conversely , the highest speeds are observed only in sources with a high brightness temperature . this is the trend which we would expect if the observed brightness temperatures are doppler boosted with @xmath71 , where @xmath72 is the doppler factor . at the optimum angle @xmath73 to the observer s line of sight for superluminal motion , @xmath74 , @xmath75 and therefore @xmath76 . as shown in figure [ beta_maxtb ] , with @xmath77k , this curve tracks the trend of the data . of course , the actual jet orientations deviate from the optimum viewing angle given by @xmath78 , and many of our brightness temperature estimates are lower limits . both of these factors lead to a spread in the data , so we should not expect a tight correlation along the plotted line ; however , the general agreement between the trend of the data and this simple model supports the idea that the intrinsic brightness temperatures have been doppler boosted by the same relativistic motion that gives us the observed component speeds . we note that there are some sources with high brightness temperatures but low speeds . this is expected , as some sources will have an angle to the line of sight much smaller than @xmath78 , and those sources will have a small apparent motion but will still be highly beamed ( m. h. cohen et al . , in preparation ) . for those sources where there are multiple epochs of observation , the core parameters , in particular the observed brightness temperatures , vary significantly with time . population modeling of the distribution of brightness temperature , as well as comparisons with the results of other vlbi surveys ( e.g. , * ? ? ? * ) may give insight into the distributions of intrinsic brightness temperatures and doppler factors . we have not found any significant correlation between redshift and brightness temperature . this is in agreement with the 5 ghz vsop results of @xcite . the long term cm - wave monitoring data on our sources from umrao and ratan show complex light curves with frequent flux density outbursts ( e.g. , * ? ? ? * ; * ? ? ? these outbursts are thought to be associated with the birth of new compact features , which are often not apparent in vlbi images until they have moved sufficiently far down the jet . changes usually appear sooner in the visibility function ( figure [ radplot_var ] ) , which in the case of our data , probes angular scales roughly 10 times smaller than the typical image restoring beam . most new jet features typically increase in size and/or decrease in flux density after a few months to years as a result of adiabatic expansion and/or synchrotron losses . however , an interesting exception is m87 ( 1228 + 126 ) , where the most compact feature appears to remain constant ( figure [ radplot_var ] ) , although the larger scale jet structure shows changes by up to a factor of two in correlated flux density . this unusual behavior , which was first noted by @xcite , is remarkable in that the dimensions of this compact stable feature are only of the order of ten lightdays or less . this weak ( @xmath79 jy ) stable feature in the center of may be closely associated with the accretion region . more sensitive observations with comparable linear resolution might show similar phenomena in more distant sources , but such observations will only be possible with large antennas in space . we define a variability index as @xmath80 . figure [ vindex ] shows distributions of the variability indices @xmath81 , @xmath82 , and @xmath83 , as well as , to represent the cores , @xmath84 , @xmath85 , and @xmath86 . as expected , the variability indices for the flux density become larger with improved resolution , going from median values of @xmath87 and @xmath88 to @xmath89 and @xmath90 . for about 68% of the sources , the flux density of the core has varied by a factor of 2 or more , that is @xmath91 . similarly , the size of the core major axis , @xmath43 , changed by as much as a factor of 5 , or @xmath92 , in some cases . probably , this is due to the creation or ejection of a new component , which initially is not resolved from the core , but then separates from it , causing first an apparent increase , and then an apparent decrease in the strength and size of the core . the observed strong variability of the brightness temperatures ( median @xmath93 ) may reflect strong variations of particle density ( due to ejections ) and/or magnetic field strength . the most variable sources tend to have the most compact structure . a variability index @xmath94 is observed for nine sources , all but one of which have sub - mas compactness @xmath95 . we have used the results of several idv search and monitoring programs at the effelsberg 100 meter telescope , the vla , and the atca at 1.4 to 15 ghz @xcite , to identify idv sources in our sample ( table [ sample ] ) . the biggest and most complete idv survey so far , the 5 ghz masiv survey @xcite , started at the vla in 2002 ; the first results reported by @xcite suggest that 85 of 710 compact flat - spectrum sources are idvs . the masiv data , however , are not yet fully published . we have labeled a source in our sample as an idv if there is a published statistically significant detection of flux density variations on a time scale of less that 3 days ( 72 hours ) . however , we are not able to identify all of the potential idv sources in our sample consistently , because some sources are not ( yet ) listed in any of the published idv survey results , and also because intra - day variability is a transient phenomenon , and not all sources were monitored equally well . figure [ idv ] shows the distribution of the sub - mas compactness index , @xmath47 ; the median value over all observing epochs was taken for each source . the median vlba core dominance , @xmath48 , and the maximum brightness temperature are also shown . the full and the mojave sample are shown separately , and we have separated idvs with high modulation index ( @xmath96 observed at least once ) from ones with low modulation index ( measured values of modulation index always less than 0.02 , or not reported ) . we find that idv sources have more compact and more core dominant structure on sub - milliarcsecond scales ( table [ idv - table ] ) than non - idv sources . idvs with a higher amplitude of intra - day variation tend to have a higher unresolved flux density , @xmath23 . the results for core dominance are in agreement with previous findings ( e.g. , * ? ? ? * ; * ? ? ? a k - s test yields a probability of less than 1% for both the full and the mojave sample that the sub - mas compactness ( figure [ idv ] ) has the same parent distribution for idv and non - idv sources . one might expect idv behavior in almost all the sources with high visibility amplitude at long vlbi spacings . however , this was not observed by idv surveys , perhaps because of the intermittent nature of the idv phenomenon . some idv observations have suggested apparent brightness temperatures up to @xmath97k if they are due to interstellar scintillations , and up to @xmath98k if they are intrinsic ( e.g. , * ? ? ? more recently , idv observations of @xcite have shown typical brightness temperature values of the order of @xmath68k , consistent with our results ( figure [ idv ] ) . however , as seen from equation ( [ tb_eqn_notheta ] ) , the highest brightness temperature which we can reliably discern is of the order of @xmath1k , so we are not able to comment on the evidence for the extremely high brightness temperatures and the corresponding high lorentz factors inferred for some idv . the third catalog of high energy gamma - ray sources detected by the egret telescope of the compton gamma ray observatory @xcite includes 66 high - confidence identifications of blazars @xcite . while the gamma - ray sources were identified with flat - spectrum extragalactic radio sources ( @xmath11 , not all flat - spectrum sources have been detected as gamma - ray sources . this is not necessarily indicative of a bi - modality in the gamma - ray loudness distribution of extragalactic radio sources ( such as that found at radio wavelengths ) , since the sensitivity level of egret was such that many sources were only detected in their flaring state . with the next generation of gamma - ray telescopes , such as glast @xcite , the sensitivity may be sufficient to actually separate the classes of gamma - ray loud and gamma - ray quiet objects and to define the relationship between radio and gamma - ray emission of the sources properly . for the purpose of our test we have grouped the `` highly probable '' and `` probable '' egret identifications ( table [ sample ] ) together , which yields 52 `` egret detections '' out of 250 objects for the full and 35 out of 133 for the mojave sample . in we did not find any clear differences in the sub - milliarcsecond scale structure between the egret ( 20% of our radio sample ) and non - egret sources . however , we find here that the sub - mas compactness , @xmath47 , for egret detections is , on average , greater than for the egret non - detections . this can be seen in figure [ relative_radplot_egret ] which shows the mean visibility function amplitudes versus projected spacing for egret detected and non - detected sources , for the full and mojave samples . for both samples , the egret detected sources have , on average , a higher contribution of compact vlba structure ( see table [ radplot - model ] for the parameters of the two - component fit ) . this comparison is valid because the egret detected and non - detected blazars in our sample have indistinguishable redshift distributions . this result still holds if we exclude the gps and steep spectrum sources , which are generally gamma - ray weak ( see , e.g. , * ? ? ? the difference is more pronounced for the full sample , which is not selected on the basis of the vlbi flux density . the difference for the mojave sample is small , but remains significant . these results suggest that a connection may exist between gamma - ray and beamed radio emission from extragalactic sources on sub - milliarcsecond scales , as has already being argued by others ( e.g. , * ? ? ? we have analyzed visibility function data of 250 extragalactic radio sources , obtained with the vlba at 15 ghz . almost all of the radio sources in our sample have unresolved radio emission brighter than 0.1 jy on the longest vlba baselines . for 171 objects , more than half of the flux density comes from unresolved features . we have compiled a list of 163 sources with unresolved structure stronger than 0.5 jy , which will form a target list of special interest for planned space vlbi observations such as radioastron , vsop2 , and arise . a few of the sources have an overall radio structure which is only slightly resolved at the longest spacings . their total angular size is less than about 0.05 mas , at least in one dimension , at some epochs . however , even though most sources in our full sample are extended overall , there are 158 sources in which the vlba core component appears unresolved , usually smaller than 0.05 mas , again in one direction , at least at one epoch . for 19 of these , the core was unresolved at all epochs . the distribution of the brightness temperature of the cores peaks at @xmath68k and extends up to @xmath2k ; this is close to the limit set by the dimensions of the vlba . however , for many sources we only measure a lower limit to the brightness temperature . there is evidence that the observed brightness temperatures can be explained as the result of doppler boosting , but transient phenomena , coherent emission , or synchrotron emission by relativistic protons may also be important . on sub - milliarcsecond scales , active galaxies are on average larger and less core dominated than quasars , which is consistent with unification models in which the latter are viewed at smaller angles to the line of sight . additionally , the weak - lined objects classified as bl lacs tend to be smaller than the broad - lined quasars in our sample . idv sources show a higher compactness and core dominance on sub - mas scales than non - idv ones . idvs with a higher amplitude of intra - day variation tend to have a higher flux density in an unresolved component . the most variable sources tend to have the most compact structure . egret - detected radio sources show a higher degree of sub - mas compactness than non - egret sources , supporting emission models which relate the radio and gamma - ray emission , such as inverse compton scattering ( see , e.g. , * ? ? ? the national radio astronomy observatory is a facility of the national science foundation operated under cooperative agreement by associated universities , inc . the university of michigan radio astronomy observatory has been supported by the university of michigan department of astronomy and the national science foundation . part of this work was done by mll and dch during their jansky postdoctoral fellowships at the national radio astronomy observatory . this work was supported partially by the nasa jurriss program ( w19611 ) , nsf ( 0406923ast ) , and by the russian foundation for basic research ( 020216305 , 050217377 ) . mk was supported through a stipend from the international max planck research school for radio and infrared astronomy at the university of bonn . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . we thank the referee , ski antonucci , for many helpful comments , which have led us in particular to clarify the relationship between source classifications and our statistical results . hirabayashi , h. , murata , y. , edwards , p. g. , asaki , y. , mochizuki , n. , inoue , m. , umemoto , t. , kameno , s. , & kono , y.2004 , in proceedings of the 7th european vlbi network symposium , ed . r. bachiller , f. colomer , j .- f . desmurs , p. de vicente ( observatorio astronomico nacional ) , 285 ; astro - ph/0501020 llccllccclc 0003@xmath17066 & nrao 5 & 00 06 13.8929 & @xmath1706 23 35.3353 & b & 0.347 & flat & & y & a & + 0007@xmath99106 & iii zw 2 & 00 10 31.0059 & @xmath9910 58 29.5041 & g & 0.089 & flat & & y & & + 0014@xmath99813 & & 00 17 08.4750 & @xmath9981 35 08.1360 & q & 3.387 & flat & & & & + 0016@xmath99731 & & 00 19 45.7864 & @xmath9973 27 30.0175 & q & 1.781 & flat & & y & a , pr & + 0026@xmath99346 & & 00 29 14.2425 & @xmath9934 56 32.2466 & g & 0.517 & flat & & & c & + 0035@xmath99413 & & 00 38 24.8436 & @xmath9941 37 06.0006 & q & 1.353 & flat & & & c & + 0039@xmath99230 & & 00 42 04.5451 & @xmath9923 20 01.0610 & u & & peaked & & & a & + 0048@xmath17097 & & 00 50 41.3174 & @xmath1709 29 05.2102 & b & & flat & & y & & + 0055@xmath99300 & ngc 315 & 00 57 48.8834 & @xmath9930 21 08.8119 & g & 0.016 & flat & & & & + 0059@xmath99581 & & 01 02 45.7624 & @xmath9958 24 11.1366 & u & & flat & & y & & * 5,*6 + 0106@xmath99013 & & 01 08 38.7711 & @xmath9901 35 00.3171 & q & 2.107 & flat & & y & b & + 0108@xmath99388 & & 01 11 37.3192 & @xmath9939 06 27.9986 & g & 0.669 & peaked & & & c & + 0109@xmath99224 & & 01 12 05.8247 & @xmath9922 44 38.7862 & b & & flat & & y & & + 0112@xmath17017 & & 01 15 17.1000 & @xmath1701 27 04.5772 & q & 1.365 & flat & & & a & + 0113@xmath17118 & & 01 16 12.5220 & @xmath1711 36 15.4340 & q & 0.672 & flat & & & a & + 0119@xmath99041 & & 01 21 56.8617 & @xmath9904 22 24.7344 & q & 0.637 & flat & np & & a & + 0119@xmath99115 & oc + 131 & 01 21 41.5950 & @xmath9911 49 50.4131 & q & 0.570 & flat & & y & a & + 0122@xmath17003 & & 01 25 28.8427 & @xmath1700 05 55.9630 & q & 1.070 & flat & & & a & + 0133@xmath17203 & & 01 35 37.5086 & @xmath1720 08 45.8870 & q & 1.141 & flat & & & & + 0133@xmath99476 & da 55 & 01 36 58.5948 & @xmath9947 51 29.1001 & q & 0.859 & flat & & y & a , pr & 1 + 0138@xmath17097 & & 01 41 25.8320 & @xmath1709 28 43.6730 & b & 0.733 & flat & & & b & + 0146@xmath99056 & & 01 49 22.3709 & @xmath9905 55 53.5680 & q & 2.345 & flat & & & a & + 0149@xmath99218 & & 01 52 18.0590 & @xmath9922 07 07.7000 & q & 1.32 & flat & & & b & + 0153@xmath99744 & & 01 57 34.9649 & @xmath9974 42 43.2300 & q & 2.341 & flat & & & c , pr & + 0201@xmath99113 & & 02 03 46.6571 & @xmath9911 34 45.4096 & q & 3.61 & peaked & & & & + 0202@xmath99149 & 4c + 15.05 & 02 04 50.4139 & @xmath9915 14 11.0435 & q & 0.405 & flat & yy & y & b & + 0202@xmath99319 & & 02 05 04.9254 & @xmath9932 12 30.0956 & q & 1.466 & flat & & y & b & + 0212@xmath99735 & & 02 17 30.8134 & @xmath9973 49 32.6218 & q & 2.367 & flat & & y & a , pr & + 0215@xmath99015 & & 02 17 48.9547 & @xmath9901 44 49.6991 & b & 1.715 & flat & & y & a & + 0218@xmath99357 & & 02 21 05.4740 & @xmath9935 56 13.7315 & q & 0.944 & flat & & & c & + 0221@xmath99067 & 4c + 07.11 & 02 24 28.4282 & @xmath9906 59 23.3416 & q & 0.511 & flat & & & b & + 0224@xmath99671 & 4c + 67.05 & 02 28 50.0515 & @xmath9967 21 03.0293 & u & & flat & & y & & + 0234@xmath99285 & ctd 20 & 02 37 52.4057 & @xmath9928 48 08.9901 & q & 1.207 & flat & yp & y & b & + 0235@xmath99164 & & 02 38 38.9301 & @xmath9916 36 59.2747 & b & 0.940 & flat & yy & y & c & * 1,*2,*6 + 0238@xmath17084 & ngc 1052 & 02 41 04.7985 & @xmath1708 15 20.7518 & g & 0.005 & flat & & y & c & + 0248@xmath99430 & & 02 51 34.5368 & @xmath9943 15 15.8290 & q & 1.310 & flat & & & a & + 0300@xmath99470 & 4c + 47.08 & 03 03 35.2422 & @xmath9947 16 16.2755 & b & & flat & & y & & + 0310@xmath99013 & & 03 12 43.6028 & @xmath9901 33 17.5380 & q & 0.664 & flat & & & b & + 0316@xmath99162 & cta 21 & 03 18 57.8016 & @xmath9916 28 32.7048 & g & & peaked & & & & + 0316@xmath99413 & 3c 84 & 03 19 48.1601 & @xmath9941 30 42.1031 & g & 0.017 & flat & & y & a , pr & + 0333@xmath99321 & nrao 140 & 03 36 30.1076 & @xmath9932 18 29.3424 & q & 1.263 & flat & & y & b & 2 + 0336@xmath17019 & cta 26 & 03 39 30.9378 & @xmath1701 46 35.8040 & q & 0.852 & flat & yy & y & a & * 2 + 0355@xmath99508 & nrao 150 & 03 59 29.7473 & @xmath9950 57 50.1615 & q & & flat & & & & + 0402@xmath17362 & & 04 03 53.7499 & @xmath1736 05 01.9120 & q & 1.417 & peaked & & & a & + 0403@xmath17132 & & 04 05 34.0034 & @xmath1713 08 13.6911 & q & 0.571 & flat & & y & a & + 0405@xmath17385 & & 04 06 59.0353 & @xmath1738 26 28.0421 & q & 1.285 & flat & & & b & * 3 + 0415@xmath99379 & 3c 111 & 04 18 21.2770 & @xmath9938 01 35.9000 & g & 0.049 & steep & & y & & + 0420@xmath17014 & & 04 23 15.8007 & @xmath1701 20 33.0653 & q & 0.915 & flat & yy & y & a & + 0420@xmath99022 & & 04 22 52.2146 & @xmath9902 19 26.9319 & q & 2.277 & flat & & & & + 0422@xmath99004 & & 04 24 46.8421 & @xmath9900 36 06.3298 & b & & flat & & y & a & * 3 + 0429@xmath99415 & 3c 119 & 04 32 36.5026 & @xmath9941 38 28.4485 & q & 1.023 & steep & & & & + 0430@xmath99052 & 3c 120 & 04 33 11.0955 & @xmath9905 21 15.6194 & g & 0.033 & flat & & y & a & + 0438@xmath17436 & & 04 40 17.1800 & @xmath1743 33 08.6030 & q & 2.852 & flat & & & & + 0440@xmath17003 & nrao 190 & 04 42 38.6608 & @xmath1700 17 43.4191 & q & 0.844 & flat & yp & & & * 3 + 0446@xmath99112 & & 04 49 07.6711 & @xmath9911 21 28.5966 & q & & flat & py & y & b & + 0454@xmath17234 & & 04 57 03.1792 & @xmath1723 24 52.0180 & b & 1.003 & flat & yy & & a & + 0454@xmath99844 & & 05 08 42.3635 & @xmath9984 32 04.5440 & b & @xmath511.34 & flat & & & b & * 4 + 0458@xmath17020 & & 05 01 12.8099 & @xmath1701 59 14.2562 & q & 2.291 & flat & yy & y & a & + 0521@xmath17365 & & 05 22 57.9846 & @xmath1736 27 30.8516 & g & 0.055 & steep & & & & + 0524@xmath99034 & & 05 27 32.7030 & @xmath9903 31 31.4500 & b & & flat & & & & + 0528@xmath99134 & & 05 30 56.4167 & @xmath9913 31 55.1495 & q & 2.07 & flat & yy & y & b & + 0529@xmath99075 & & 05 32 38.9985 & @xmath9907 32 43.3459 & u & & flat & & y & c & + 0529@xmath99483 & & 05 33 15.8658 & @xmath9948 22 52.8078 & q & 1.162 & flat & py & y & & + 0537@xmath17286 & & 05 39 54.2814 & @xmath1728 39 55.9460 & q & 3.104 & flat & np & & a & + 0552@xmath99398 & da 193 & 05 55 30.8056 & @xmath9939 48 49.1650 & q & 2.363 & peaked & & y & & + 0602@xmath99673 & & 06 07 52.6716 & @xmath9967 20 55.4098 & q & 1.97 & flat & & & b & 4 + 0605@xmath17085 & oh @xmath17010 & 06 07 59.6992 & @xmath1708 34 49.9781 & q & 0.872 & flat & & y & a & + 0607@xmath17157 & & 06 09 40.9495 & @xmath1715 42 40.6726 & q & 0.324 & flat & & y & a & * 3 + 0615@xmath99820 & & 06 26 03.0062 & @xmath9982 02 25.5676 & q & 0.71 & flat & & & b & + 0642@xmath99449 & oh 471 & 06 46 32.0260 & @xmath9944 51 16.5901 & q & 3.408 & peaked & & y & b & + 0648@xmath17165 & & 06 50 24.5819 & @xmath1716 37 39.7250 & u & & flat & & y & & + 0707@xmath99476 & & 07 10 46.1049 & @xmath9947 32 11.1427 & q & 1.292 & flat & & & & + 0710@xmath99439 & & 07 13 38.1641 & @xmath9943 49 17.2051 & g & 0.518 & peaked & & & b & + 0711@xmath99356 & oi 318 & 07 14 24.8175 & @xmath9935 34 39.7950 & q & 1.620 & peaked & & & a , pr & 1 + 0716@xmath99714 & & 07 21 53.4485 & @xmath9971 20 36.3634 & b & & flat & yy & y & & * 1,*2,*4,*6 + 0723@xmath17008 & & 07 25 50.6400 & @xmath1700 54 56.5444 & b & 0.127 & flat & & & & + 0727@xmath17115 & & 07 30 19.1125 & @xmath1711 41 12.6005 & q & 1.591 & flat & & y & & + 0730@xmath99504 & & 07 33 52.5206 & @xmath9950 22 09.0621 & q & 0.720 & flat & & y & & + 0735@xmath99178 & & 07 38 07.3937 & @xmath9917 42 18.9983 & b & @xmath510.424 & flat & yy & y & a & * 1 + 0736@xmath99017 & & 07 39 18.0339 & @xmath9901 37 04.6180 & q & 0.191 & flat & & y & a & + 0738@xmath99313 & oi 363 & 07 41 10.7033 & @xmath9931 12 00.2286 & q & 0.630 & flat & & y & a & + 0742@xmath99103 & & 07 45 33.0595 & @xmath9910 11 12.6925 & q & 2.624 & peaked & & y & b & + 0745@xmath99241 & & 07 48 36.1093 & @xmath9924 00 24.1102 & q & 0.409 & flat & & & a & + 0748@xmath99126 & & 07 50 52.0457 & @xmath9912 31 04.8282 & q & 0.889 & flat & & y & b & + 0754@xmath99100 & & 07 57 06.6429 & @xmath9909 56 34.8521 & b & 0.266 & flat & & y & & * 5,*6 + 0804@xmath99499 & oj 508 & 08 08 39.6663 & @xmath9949 50 36.5305 & q & 1.432 & flat & & y & b & * 1,*2,*4,*6 + 0805@xmath17077 & & 08 08 15.5360 & @xmath1707 51 09.8863 & q & 1.837 & flat & & y & c & + 0808@xmath99019 & & 08 11 26.7073 & @xmath9901 46 52.2200 & b & 0.93 & flat & & y & a & * 3 + 0814@xmath99425 & oj 425 & 08 18 15.9996 & @xmath9942 22 45.4149 & b & & flat & & y & a , pr & 1 + 0821@xmath99394 & 4c + 39.23 & 08 24 55.4839 & @xmath9939 16 41.9043 & q & 1.216 & flat & & & b & + 0823@xmath99033 & & 08 25 50.3384 & @xmath9903 09 24.5201 & b & 0.506 & flat & & y & a & + 0827@xmath99243 & & 08 30 52.0862 & @xmath9924 10 59.8205 & q & 0.941 & flat & yy & y & & + 0829@xmath99046 & & 08 31 48.8770 & @xmath9904 29 39.0853 & b & 0.18 & flat & yy & y & b & + 0831@xmath99557 & 4c + 55.16 & 08 34 54.9041 & @xmath9955 34 21.0710 & g & 0.240 & flat & & & c & + 0834@xmath17201 & & 08 36 39.2152 & @xmath1720 16 59.5035 & q & 2.752 & flat & & & & + 0836@xmath99710 & 4c + 71.08 & 08 41 24.3653 & @xmath9970 53 42.1731 & q & 2.218 & flat & yy & y & a , pr & + 0838@xmath99133 & 3c 207 & 08 40 47.6848 & @xmath9913 12 23.8790 & q & 0.684 & flat & & & b & + 0850@xmath99581 & 4c + 58.17 & 08 54 41.9964 & @xmath9957 57 29.9393 & q & 1.322 & flat & & & b & + 0851@xmath99202 & oj 287 & 08 54 48.8749 & @xmath9920 06 30.6409 & b & 0.306 & flat & yy & y & a & + 0859@xmath17140 & & 09 02 16.8309 & @xmath1714 15 30.8757 & q & 1.339 & steep & & & & + 0859@xmath99470 & 4c + 47.29 & 09 03 03.9901 & @xmath9946 51 04.1375 & q & 1.462 & steep & & & a , pr & + 0906@xmath99015 & 4c + 01.24 & 09 09 10.0916 & @xmath9901 21 35.6177 & q & 1.018 & flat & & y & a & + 0917@xmath99449 & & 09 20 58.4585 & @xmath9944 41 53.9851 & q & 2.180 & flat & np & & a & + 0917@xmath99624 & & 09 21 36.2311 & @xmath9962 15 52.1804 & q & 1.446 & flat & & y & b & * 1,*2,*4 + 0919@xmath17260 & & 09 21 29.3538 & @xmath1726 18 43.3850 & q & 2.300 & peaked & & & b & + 0923@xmath99392 & 4c + 39.25 & 09 27 03.0139 & @xmath9939 02 20.8520 & q & 0.698 & flat & & y & a , pr & + 0945@xmath99408 & 4c + 40.24 & 09 48 55.3381 & @xmath9940 39 44.5872 & q & 1.252 & flat & & y & b & + 0953@xmath99254 & ok 290 & 09 56 49.8754 & @xmath9925 15 16.0498 & q & 0.712 & flat & & & b & + 0954@xmath99658 & & 09 58 47.2451 & @xmath9965 33 54.8181 & b & 0.367 & flat & yy & & b & 1,2,*4 + 0955@xmath99476 & ok 492 & 09 58 19.6716 & @xmath9947 25 07.8425 & q & 1.873 & flat & & y & b & * 6 + 1012@xmath99232 & 4c + 23.24 & 10 14 47.0654 & @xmath9923 01 16.5709 & q & 0.565 & flat & & & b & * 6 + 1015@xmath99359 & & 10 18 10.9877 & @xmath9935 42 39.4380 & q & 1.226 & flat & & & & + 1032@xmath17199 & & 10 35 02.1553 & @xmath1720 11 34.3597 & q & 2.198 & flat & & & a & + 1034@xmath17293 & & 10 37 16.0797 & @xmath1729 34 02.8120 & q & 0.312 & flat & & & a & * 3 + 1036@xmath99054 & & 10 38 46.7799 & @xmath9905 12 29.0854 & u & & flat & & y & & + 1038@xmath99064 & 4c + 06.41 & 10 41 17.1625 & @xmath9906 10 16.9238 & q & 1.265 & flat & & y & b & + 1045@xmath17188 & & 10 48 06.6206 & @xmath1719 09 35.7270 & q & 0.595 & flat & & y & b & + 1049@xmath99215 & 4c + 21.28 & 10 51 48.7891 & @xmath9921 19 52.3142 & q & 1.300 & flat & & & b & + 1055@xmath99018 & 4c + 01.28 & 10 58 29.6052 & @xmath9901 33 58.8237 & q & 0.888 & flat & & y & a & + 1055@xmath99201 & 4c + 20.24 & 10 58 17.9025 & @xmath9919 51 50.9018 & q & 1.11 & flat & & & b & + 1101@xmath99384 & mrk 421 & 11 04 27.3139 & @xmath9938 12 31.7991 & b & 0.031 & flat & yy & & & + 1116@xmath99128 & & 11 18 57.3014 & @xmath9912 34 41.7180 & q & 2.118 & flat & & & a & + 1124@xmath17186 & om @xmath17148 & 11 27 04.3924 & @xmath1718 57 17.4417 & q & 1.048 & flat & & y & b & + 1127@xmath17145 & & 11 30 07.0526 & @xmath1714 49 27.3882 & q & 1.187 & flat & & y & & + 1128@xmath99385 & & 11 30 53.2826 & @xmath9938 15 18.5470 & q & 1.733 & flat & & & & + 1144@xmath99402 & & 11 46 58.2979 & @xmath9939 58 34.3046 & q & 1.089 & flat & & & & + 1145@xmath17071 & & 11 47 51.5540 & @xmath1707 24 41.1411 & q & 1.342 & flat & & & a & + 1148@xmath17001 & 4c @xmath1700.47 & 11 50 43.8708 & @xmath1700 23 54.2049 & q & 1.980 & peak ? & & & b & + 1150@xmath99812 & & 11 53 12.4991 & @xmath9980 58 29.1545 & q & 1.25 & flat & & y & b & 1 + 1155@xmath99251 & & 11 58 25.7875 & @xmath9924 50 17.9640 & q & 0.202 & flat & & & c & + 1156@xmath99295 & 4c + 29.45 & 11 59 31.8339 & @xmath9929 14 43.8269 & q & 0.729 & flat & yp & y & b & * 5,*6 + 1213@xmath17172 & & 12 15 46.7518 & @xmath1717 31 45.4029 & u & & flat & & y & a & + 1219@xmath99044 & on + 231 & 12 22 22.5496 & @xmath9904 13 15.7763 & q & 0.965 & flat & & y & & + 1219@xmath99285 & w comae & 12 21 31.6905 & @xmath9928 13 58.5002 & b & 0.102 & flat & pp & & & + 1222@xmath99216 & & 12 24 54.4584 & @xmath9921 22 46.3886 & q & 0.435 & flat & yy & y & b & + 1226@xmath99023 & 3c 273 & 12 29 06.6997 & @xmath9902 03 08.5982 & q & 0.158 & flat & yy & y & a & + 1228@xmath99126 & m87 & 12 30 49.4234 & @xmath9912 23 28.0439 & g & 0.004 & steep & & y & & + 1244@xmath17255 & & 12 46 46.8020 & @xmath1725 47 49.2880 & q & 0.638 & flat & & & a & + 1253@xmath17055 & 3c 279 & 12 56 11.1666 & @xmath1705 47 21.5246 & q & 0.538 & flat & yy & y & a & + 1255@xmath17316 & & 12 57 59.0608 & @xmath1731 55 16.8520 & q & 1.924 & flat & & & a & + 1302@xmath17102 & & 13 05 33.0150 & @xmath1710 33 19.4282 & q & 0.286 & flat & & & a & + 1308@xmath99326 & & 13 10 28.6638 & @xmath9932 20 43.7830 & q & 0.997 & flat & & y & a & + 1313@xmath17333 & & 13 16 07.9859 & @xmath1733 38 59.1720 & q & 1.21 & flat & np & & a & + 1323@xmath99321 & 4c + 32.44 & 13 26 16.5122 & @xmath9931 54 09.5154 & g & 0.370 & peaked & & & & + 1324@xmath99224 & & 13 27 00.8613 & @xmath9922 10 50.1631 & q & 1.40 & flat & np & y & & + 1328@xmath99307 & 3c 286 & 13 31 08.2880 & @xmath9930 30 32.9588 & q & 0.846 & steep & & & & + 1334@xmath17127 & & 13 37 39.7828 & @xmath1712 57 24.6932 & q & 0.539 & flat & yy & y & a & + 1345@xmath99125 & 4c + 12.50 & 13 47 33.3616 & @xmath9912 17 24.2390 & g & 0.121 & peaked & & & c & + 1354@xmath17152 & & 13 57 11.2450 & @xmath1715 27 28.7864 & q & 1.890 & flat & & & & + 1354@xmath99195 & da 354 & 13 57 04.4368 & @xmath9919 19 07.3640 & q & 0.719 & flat & & & a & + 1402@xmath99044 & & 14 05 01.1198 & @xmath9904 15 35.8190 & q & 3.211 & flat & & & b & + 1404@xmath99286 & oq 208 & 14 07 00.3944 & @xmath9928 27 14.6899 & g & 0.077 & peaked & & & a & + 1413@xmath99135 & & 14 15 58.8175 & @xmath9913 20 23.7126 & b & 0.247 & flat & & y & b & + 1417@xmath99385 & & 14 19 46.6138 & @xmath9938 21 48.4752 & q & 1.832 & flat & np & y & & + 1418@xmath99546 & oq 530 & 14 19 46.5974 & @xmath9954 23 14.7872 & b & 0.152 & flat & & & c & 4 + 1424@xmath99366 & & 14 26 37.0875 & @xmath9936 25 09.5739 & q & 1.091 & flat & np & & & + 1458@xmath99718 & 3c 309.1 & 14 59 07.5839 & @xmath9971 40 19.8677 & q & 0.904 & steep & & y & & + 1502@xmath99106 & 4c + 10.39 & 15 04 24.9798 & @xmath9910 29 39.1986 & q & 1.833 & flat & & y & a & + 1504@xmath17166 & or @xmath17102 & 15 07 04.7870 & @xmath1716 52 30.2673 & q & 0.876 & flat & & y & a & + 1504@xmath99377 & & 15 06 09.5300 & @xmath9937 30 51.1324 & g & 0.674 & flat & & & b & * 5 + 1508@xmath17055 & & 15 10 53.5914 & @xmath1705 43 07.4172 & q & 1.191 & steep & & & & + 1510@xmath17089 & & 15 12 50.5329 & @xmath1709 05 59.8295 & q & 0.360 & flat & yy & y & a & + 1511@xmath17100 & or @xmath17118 & 15 13 44.8934 & @xmath1710 12 00.2644 & q & 1.513 & flat & & & b & + 1514@xmath99004 & & 15 16 40.2190 & @xmath9900 15 01.9100 & g & 0.052 & flat & & & a & + 1514@xmath17241 & ap librae & 15 17 41.8131 & @xmath1724 22 19.4759 & b & 0.048 & flat & pp & & a & + 1519@xmath17273 & & 15 22 37.6760 & @xmath1727 30 10.7854 & b & 1.294 & peaked & & & a & * 3 + 1532@xmath99016 & & 15 34 52.4537 & @xmath9901 31 04.2066 & q & 1.420 & flat & & & c & + 1538@xmath99149 & 4c + 14.60 & 15 40 49.4915 & @xmath9914 47 45.8849 & b & 0.605 & flat & & y & b & + 1546@xmath99027 & & 15 49 29.4368 & @xmath9902 37 01.1635 & q & 0.412 & flat & & y & a & + 1548@xmath99056 & 4c + 05.64 & 15 50 35.2692 & @xmath9905 27 10.4482 & q & 1.422 & flat & & y & b & + 1555@xmath99001 & da 393 & 15 57 51.4340 & @xmath1700 01 50.4137 & q & 1.772 & flat & & & b & + 1606@xmath99106 & 4c + 10.45 & 16 08 46.2032 & @xmath9910 29 07.7759 & q & 1.226 & flat & yy & y & b & + 1607@xmath99268 & ctd 93 & 16 09 13.3208 & @xmath9926 41 29.0364 & g & 0.473 & peaked & & & & + 1611@xmath99343 & da 406 & 16 13 41.0642 & @xmath9934 12 47.9091 & q & 1.401 & flat & yy & y & a & 1 + 1622@xmath17253 & os @xmath17237 & 16 25 46.8916 & @xmath1725 27 38.3269 & q & 0.786 & flat & yp & & b & 7 + 1622@xmath17297 & & 16 26 06.0208 & @xmath1729 51 26.9710 & q & 0.815 & flat & yy & & a & * 3 + 1624@xmath99416 & 4c + 41.32 & 16 25 57.6697 & @xmath9941 34 40.6300 & q & 2.55 & flat & & & c , pr & + 1633@xmath99382 & 4c + 38.41 & 16 35 15.4930 & @xmath9938 08 04.5006 & q & 1.807 & flat & yy & y & a , pr & + 1637@xmath99574 & os 562 & 16 38 13.4563 & @xmath9957 20 23.9792 & q & 0.751 & flat & & y & a , pr & 4 + 1638@xmath99398 & nrao 512 & 16 40 29.6328 & @xmath9939 46 46.0285 & q & 1.666 & flat & & y & a & + 1641@xmath99399 & 3c 345 & 16 42 58.8100 & @xmath9939 48 36.9939 & q & 0.594 & flat & & y & a , pr & + 1642@xmath99690 & 4c + 69.21 & 16 42 07.8485 & @xmath9968 56 39.7564 & q & 0.751 & flat & & & a , pr & 1,2,*6 + 1652@xmath99398 & mrk 501 & 16 53 52.2167 & @xmath9939 45 36.6089 & b & 0.033 & flat & y & & a , pr & + 1655@xmath99077 & & 16 58 09.0115 & @xmath9907 41 27.5407 & q & 0.621 & flat & & y & a & + 1656@xmath99053 & & 16 58 33.4473 & @xmath9905 15 16.4442 & q & 0.879 & flat & & & a & + 1656@xmath99477 & & 16 58 02.7796 & @xmath9947 37 49.2310 & q & 1.622 & flat & & & a & + 1726@xmath99455 & & 17 27 27.6508 & @xmath9945 30 39.7314 & q & 0.714 & flat & & y & b & + 1730@xmath17130 & nrao 530 & 17 33 02.7058 & @xmath1713 04 49.5482 & q & 0.902 & flat & yy & y & a & + 1739@xmath99522 & 4c + 51.37 & 17 40 36.9779 & @xmath9952 11 43.4075 & q & 1.379 & flat & yy & y & a , pr & * 4 + 1741@xmath17038 & ot @xmath17068 & 17 43 58.8561 & @xmath1703 50 04.6167 & q & 1.057 & flat & yy & y & b & 1 + 1749@xmath99096 & 4c + 09.57 & 17 51 32.8186 & @xmath9909 39 00.7285 & b & 0.320 & flat & & y & a & + 1749@xmath99701 & & 17 48 32.8402 & @xmath9970 05 50.7688 & b & 0.770 & flat & & & pr & 1,2,4 + 1751@xmath99288 & & 17 53 42.4736 & @xmath9928 48 04.9391 & u & & flat & & y & & + 1758@xmath99388 & & 18 00 24.7654 & @xmath9938 48 30.6976 & q & 2.092 & peaked & & y & & + 1800@xmath99440 & & 18 01 32.3149 & @xmath9944 04 21.9003 & q & 0.663 & flat & & y & b & * 6 + 1803@xmath99784 & & 18 00 45.6839 & @xmath9978 28 04.0185 & b & 0.680 & flat & & y & a , pr & 1,*2,*4 + 1807@xmath99698 & 3c 371 & 18 06 50.6806 & @xmath9969 49 28.1085 & b & 0.050 & flat & & & a , pr & 4 + 1821@xmath99107 & & 18 24 02.8552 & @xmath9910 44 23.7730 & q & 1.364 & peaked & & & b & + 1823@xmath99568 & 4c + 56.27 & 18 24 07.0684 & @xmath9956 51 01.4909 & b & 0.663 & flat & & y & a , pr & + 1828@xmath99487 & 3c 380 & 18 29 31.7388 & @xmath9948 44 46.9710 & q & 0.692 & steep & & y & pr & + 1845@xmath99797 & 3c 390.3 & 18 42 08.9900 & @xmath9979 46 17.1280 & g & 0.057 & steep & & & & + 1849@xmath99670 & & 18 49 16.0723 & @xmath9967 05 41.6799 & q & 0.657 & flat & & y & & + 1901@xmath99319 & 3c 395 & 19 02 55.9389 & @xmath9931 59 41.7021 & q & 0.635 & steep & & & a & + 1908@xmath17201 & & 19 11 09.6528 & @xmath1720 06 55.1080 & q & 1.119 & flat & yy & & b & + 1921@xmath17293 & ov @xmath17236 & 19 24 51.0560 & @xmath1729 14 30.1212 & q & 0.352 & flat & & & a & + 1928@xmath99738 & 4c + 73.18 & 19 27 48.4952 & @xmath9973 58 01.5700 & q & 0.303 & flat & & y & a , pr & + 1936@xmath17155 & & 19 39 26.6577 & @xmath1715 25 43.0583 & q & 1.657 & flat & py & y & & + 1937@xmath17101 & & 19 39 57.2566 & @xmath1710 02 41.5210 & q & 3.787 & flat & & & & + 1954@xmath17388 & & 19 57 59.8192 & @xmath1738 45 06.3560 & q & 0.630 & flat & & & b & + 1954@xmath99513 & ov 591 & 19 55 42.7383 & @xmath9951 31 48.5462 & q & 1.223 & flat & & & b , pr & 4 + 1957@xmath99405 & cygnus a & 19 59 28.3567 & @xmath9940 44 02.0966 & g & 0.056 & steep & & y & & + 1958@xmath17179 & & 20 00 57.0904 & @xmath1717 48 57.6725 & q & 0.652 & flat & & y & a & + 2000@xmath17330 & & 20 03 24.1163 & @xmath1732 51 45.1320 & q & 3.783 & peaked & & & b & + 2005@xmath99403 & & 20 07 44.9449 & @xmath9940 29 48.6041 & q & 1.736 & flat & & y & & + 2007@xmath99777 & & 20 05 31.0035 & @xmath9977 52 43.2248 & b & 0.342 & flat & & & a & 1,2,4 + 2008@xmath17159 & & 20 11 15.7109 & @xmath1715 46 40.2538 & q & 1.180 & peaked & & y & a & + 2010@xmath99463 & & 20 12 05.6374 & @xmath9946 28 55.7770 & u & & flat & & & & + 2021@xmath99317 & 4c + 31.56 & 20 23 19.0174 & @xmath9931 53 02.3059 & u & & flat & & y & & + 2021@xmath99614 & ow 637 & 20 22 06.6817 & @xmath9961 36 58.8047 & g & 0.227 & flat & & y & a , pr & + 2029@xmath99121 & & 20 31 54.9942 & @xmath9912 19 41.3400 & q & 1.215 & flat & np & & b & + 2037@xmath99511 & 3c 418 & 20 38 37.0348 & @xmath9951 19 12.6627 & q & 1.687 & flat & & y & & + 2059@xmath99034 & & 21 01 38.8341 & @xmath9903 41 31.3200 & q & 1.015 & flat & & & a & + 2113@xmath99293 & & 21 15 29.4135 & @xmath9929 33 38.3669 & q & 1.514 & flat & & & b & + 2121@xmath99053 & ox 036 & 21 23 44.5174 & @xmath9905 35 22.0932 & q & 1.941 & flat & & y & b & * 6 + 2126@xmath17158 & & 21 29 12.1758 & @xmath1715 38 41.0400 & q & 3.28 & flat & & & a & + 2128@xmath99048 & da 550 & 21 30 32.8775 & @xmath9905 02 17.4747 & g & 0.99 & peaked & & & & + 2128@xmath17123 & & 21 31 35.2618 & @xmath1712 07 04.7959 & q & 0.501 & flat & & y & b & + 2131@xmath17021 & 4c @xmath1702.81 & 21 34 10.3096 & @xmath1701 53 17.2389 & b & 1.285 & flat & & y & & + 2134@xmath99004 & & 21 36 38.5863 & @xmath9900 41 54.2133 & q & 1.932 & peaked & & y & a & + 2136@xmath99141 & ox 161 & 21 39 01.3093 & @xmath9914 23 35.9920 & q & 2.427 & flat & & y & a & + 2144@xmath99092 & & 21 47 10.1630 & @xmath9909 29 46.6723 & q & 1.113 & flat & & & b & + 2145@xmath99067 & 4c + 06.69 & 21 48 05.4587 & @xmath9906 57 38.6042 & q & 0.999 & flat & & y & b & + 2155@xmath17152 & ox @xmath17192 & 21 58 06.2819 & @xmath1715 01 09.3281 & q & 0.672 & flat & & y & a & + 2200@xmath99420 & bl lac & 22 02 43.2914 & @xmath9942 16 39.9799 & b & 0.069 & flat & yy & y & a , pr & 4 + 2201@xmath99171 & & 22 03 26.8937 & @xmath9917 25 48.2478 & q & 1.076 & flat & & y & & + 2201@xmath99315 & 4c + 31.63 & 22 03 14.9758 & @xmath9931 45 38.2699 & q & 0.298 & flat & & y & b & + 2209@xmath99236 & & 22 12 05.9663 & @xmath9923 55 40.5439 & q & 1.125 & flat & yy & y & a & + 2216@xmath17038 & & 22 18 52.0377 & @xmath1703 35 36.8794 & q & 0.901 & flat & & y & a & + 2223@xmath17052 & 3c 446 & 22 25 47.2593 & @xmath1704 57 01.3907 & q & 1.404 & flat & & y & a & + 2227@xmath17088 & phl 5225 & 22 29 40.0843 & @xmath1708 32 54.4354 & q & 1.562 & flat & & y & a & + 2230@xmath99114 & cta 102 & 22 32 36.4089 & @xmath9911 43 50.9041 & q & 1.037 & flat & yp & y & & + 2234@xmath99282 & ctd 135 & 22 36 22.4709 & @xmath9928 28 57.4133 & q & 0.795 & flat & & & a & + 2243@xmath17123 & & 22 46 18.2320 & @xmath1712 06 51.2773 & q & 0.630 & flat & & y & a & + 2251@xmath99158 & 3c 454.3 & 22 53 57.7479 & @xmath9916 08 53.5609 & q & 0.859 & flat & yy & y & a & + 2255@xmath17282 & & 22 58 05.9629 & @xmath1727 58 21.2567 & q & 0.927 & peaked & y & & b & + 2318@xmath99049 & & 23 20 44.8566 & @xmath9905 13 49.9527 & q & 0.623 & flat & & & a & + 2329@xmath17162 & & 23 31 38.6524 & @xmath1715 56 57.0080 & q & 1.153 & flat & & & a & + 2331@xmath99073 & & 23 34 12.8282 & @xmath9907 36 27.5520 & u & & flat & & y & & + 2345@xmath17167 & & 23 48 02.6085 & @xmath1716 31 12.0220 & q & 0.576 & flat & & y & & + 2351@xmath99456 & 4c + 45.51 & 23 54 21.6803 & @xmath9945 53 04.2365 & q & 1.986 & flat & pp & y & b & + * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * llrrrcrrrrrc 0003@xmath17066 & 2000/01/11 & 2.51 & 2.28 & 1.97 & 0.88 & 1.85 & 0.50 & @xmath1000.06 & @xmath514.60e+11 & @xmath1779 & 7 / 1 + 0007@xmath99106 & 1998/10/30 & 1.21 & 1.28 & 1.11 & 1.01 & 1.28 & 0.15 & 0.06 & 9.05e+11 & & 6 / 4 + 0014@xmath99813 & 1999/01/02 & 0.50 & 0.47 & 0.24 & 0.94 & 0.37 & 0.54 & 0.14 & 1.20e+11 & 167 & 1 / 0 + 0016@xmath99731 & 2003/08/28 & 1.11 & 1.06 & 0.93 & 0.85 & 1.02 & 0.13 & 0.09 & 1.39e+12 & 130 & 6 / 2 + 0026@xmath99346 & 1995/04/07 & 0.70 & 0.65 & 0.21 & 0.66 & & & & & 55 & 7 / + 0035@xmath99413 & 1998/03/19 & 0.50 & 0.47 & 0.29 & 0.93 & 0.38 & 0.43 & 0.18 & 6.16e+10 & 101 & 6 / 3 + 0039@xmath99230 & 1998/12/05 & 0.42 & 0.45 & 0.13 & 1.07 & 0.16 & 0.31 & 0.22 & @xmath511.34e+10 & @xmath17164 & 1 / 0 + 0048@xmath17097 & 1996/10/27 & 2.09 & 1.60 & 1.60 & 0.85 & 1.61 & 0.08 & @xmath1000.05 & @xmath512.37e+12 & @xmath1712 & 5 / 2 + 0055@xmath99300 & 1995/04/07 & 0.70 & 0.82 & 0.33 & 0.98 & 0.47 & 0.48 & @xmath1000.05 & @xmath511.11e+11 & @xmath1753 & 6 / 3 + 0059@xmath99581 & 2002/11/23 & 3.12 & 3.32 & 2.72 & 1.06 & 3.25 & 0.13 & 0.11 & @xmath511.22e+12 & @xmath17126 & 1 / 0 + 0106@xmath99013 & 1999/05/21 & 2.99 & 2.98 & 2.09 & 0.90 & 2.32 & 0.16 & 0.10 & 2.47e+12 & @xmath17121 & 9 / 4 + 0108@xmath99388 & 2002/06/12 & 0.53 & 0.46 & 0.17 & 0.83 & 0.13 & 0.29 & 0.25 & 1.57e+10 & @xmath17106 & 2 / 0 + 0109@xmath99224 & 2002/06/15 & 1.02 & 1.01 & 0.78 & 1.03 & 0.96 & 0.15 & 0.04 & @xmath517.71e+11 & 88 & 2 / 1 + 0112@xmath17017 & 1998/12/05 & 0.93 & 0.82 & 0.46 & 0.88 & 0.48 & 0.32 & @xmath1000.02 & @xmath518.28e+11 & 120 & 6 / 4 + 0113@xmath17118 & 2002/10/20 & 1.39 & 1.32 & 0.86 & 0.96 & 1.00 & 0.35 & 0.05 & 5.27e+11 & @xmath1731 & 2 / 1 + 0119@xmath99041 & 1995/07/28 & 1.25 & 1.28 & 0.72 & 0.95 & 1.11 & 0.29 & 0.20 & 1.65e+11 & 79 & 6 / 3 + 0119@xmath99115 & 1998/10/30 & 1.35 & 1.35 & 0.95 & 0.94 & 1.20 & 0.36 & 0.12 & 2.33e+11 & 0 & 3 / 0 + 0122@xmath17003 & 1998/06/05 & 1.59 & 1.56 & 0.43 & 0.98 & 1.16 & 0.33 & 0.11 & 3.58e+11 & @xmath1796 & 3 / 0 + 0133@xmath17203 & 1998/06/05 & 0.49 & 0.41 & 0.20 & 0.84 & 0.38 & 0.67 & 0.17 & 3.77e+10 & 16 & 1 / 0 + 0133@xmath99476 & 2003/02/05 & 4.74 & 4.97 & 4.33 & 0.90 & 4.71 & 0.12 & 0.08 & 4.99e+12 & @xmath1729 & 8 / 2 + 0138@xmath17097 & 1998/12/05 & 0.48 & 0.48 & 0.26 & 1.06 & 0.37 & 0.33 & 0.21 & 5.08e+10 & @xmath17113 & 2 / 1 + 0146@xmath99056 & 1998/12/05 & 1.06 & 1.10 & 0.53 & 1.02 & 0.78 & 0.30 & 0.18 & 2.67e+11 & 112 & 2 / 0 + 0149@xmath99218 & 1995/04/07 & 1.20 & 1.26 & 1.22 & 1.02 & 1.19 & 0.28 & @xmath1000.04 & @xmath511.40e+12 & @xmath1715 & 7 / 1 + 0153@xmath99744 & 1996/07/10 & 0.49 & 0.37 & 0.20 & 0.77 & 0.19 & 0.20 & 0.07 & 2.55e+11 & 72 & 7 / 4 + 0201@xmath99113 & 1998/11/01 & 0.66 & 0.65 & 0.51 & 1.00 & 0.49 & 0.40 & @xmath1000.03 & @xmath519.44e+11 & @xmath1729 & 2 / 1 + 0202@xmath99149 & 2001/10/31 & 2.73 & 2.01 & 1.59 & 0.76 & 1.76 & 0.30 & 0.09 & 4.82e+11 & @xmath1753 & 6 / 1 + 0202@xmath99319 & 2003/03/29 & 2.26 & 2.25 & 1.81 & 1.00 & 2.21 & 0.19 & 0.12 & 1.29e+12 & 9 & 7 / 1 + 0212@xmath99735 & 1994/08/31 & 3.53 & 2.70 & 1.75 & 0.90 & 2.40 & 0.32 & 0.13 & 1.08e+12 & 114 & 6 / 0 + 0215@xmath99015 & 1998/06/05 & 1.03 & 0.93 & 0.79 & 0.90 & 0.83 & 0.12 & 0.06 & 1.74e+12 & 111 & 3 / 1 + 0218@xmath99357 & 1995/04/07 & 1.14 & 1.13 & 0.31 & 0.76 & 0.54 & 0.46 & 0.24 & 5.13e+10 & 50 & 3 / 0 + 0221@xmath99067 & 1999/11/06 & 0.84 & 0.83 & 0.39 & 0.99 & 0.74 & 0.23 & @xmath1000.05 & @xmath515.87e+11 & @xmath1767 & 3 / 3 + 0224@xmath99671 & 2002/11/23 & 1.20 & 1.30 & 0.82 & 1.08 & 0.59 & 0.26 & @xmath1000.03 & @xmath513.94e+11 & 4 & 1 / 1 + 0234@xmath99285 & 2002/11/23 & 4.04 & 4.06 & 2.22 & 0.96 & 0.55 & 0.33 & @xmath1000.04 & @xmath514.51e+11 & @xmath1713 & 7 / 2 + 0235@xmath99164 & 2001/03/15 & 1.51 & 1.60 & 1.36 & 0.93 & 1.56 & 0.16 & 0.11 & 9.08e+11 & @xmath1792 & 5 / 0 + 0238@xmath17084 & 1995/12/15 & 2.60 & 2.09 & 0.37 & 0.80 & & & & & 68 & 13 / + 0248@xmath99430 & 1999/01/02 & 0.80 & 0.65 & 0.25 & 0.81 & 0.28 & 0.37 & 0.09 & 1.08e+11 & 149 & 1 / 0 + 0300@xmath99470 & 2002/11/23 & 1.36 & 1.28 & 1.01 & 0.94 & 1.03 & 0.19 & 0.04 & @xmath518.46e+11 & 147 & 1 / 0 + 0310@xmath99013 & 1998/11/01 & 0.20 & 0.16 & 0.10 & 0.80 & 0.12 & 0.58 & 0.04 & 4.20e+10 & 143 & 1 / 0 + 0316@xmath99162 & 1997/08/28 & 0.85 & 0.26 & 0.06 & 0.31 & & & & & & 3 / + 0316@xmath99413 & 2003/03/01 & 16.35 & 10.39 & 1.73 & 0.64 & 3.63 & 0.66 & 0.21 & 1.45e+11 & @xmath17136 & 1 / 0 + 0333@xmath99321 & 2003/03/29 & 2.54 & 2.23 & 1.66 & 0.90 & 1.84 & 0.27 & 0.06 & 1.29e+12 & 123 & 11 / 4 + 0336@xmath17019 & 1997/03/13 & 2.50 & 2.24 & 1.50 & 0.93 & 1.78 & 0.21 & @xmath1000.03 & @xmath513.24e+12 & 65 & 6 / 3 + 0355@xmath99508 & 2001/03/04 & 7.21 & 7.09 & 3.77 & 0.98 & 6.96 & 0.23 & 0.22 & @xmath517.43e+11 & 48 & 5 / 0 + 0402@xmath17362 & 1998/06/05 & 2.20 & 1.60 & 1.25 & 0.73 & 1.54 & 0.44 & 0.10 & 4.66e+11 & 23 & 1 / 0 + 0403@xmath17132 & 2002/05/31 & 2.37 & 1.40 & 1.28 & 0.52 & 1.34 & 0.24 & 0.06 & 8.18e+11 & @xmath17177 & 2 / 1 + 0405@xmath17385 & 2002/03/09 & 1.40 & 1.21 & 0.80 & 0.85 & 1.15 & 0.26 & 0.16 & 3.40e+11 & @xmath1790 & 2 / 0 + 0415@xmath99379 & 1997/03/10 & 6.96 & 5.98 & 1.84 & 0.61 & 2.60 & 0.42 & @xmath1000.05 & @xmath516.74e+11 & 64 & 13 / 6 + 0420@xmath17014 & 2003/03/01 & 11.22 & 10.62 & 7.62 & 0.94 & 8.75 & 0.09 & @xmath1000.02 & @xmath515.18e+13 & @xmath17165 & 6 / 4 + 0420@xmath99022 & 1999/11/06 & 1.18 & 1.16 & 0.78 & 0.96 & 1.05 & 0.54 & 0.06 & 5.62e+11 & @xmath17100 & 2 / 0 + 0422@xmath99004 & 2002/06/15 & 1.65 & 1.76 & 1.58 & 1.01 & 1.57 & 0.20 & 0.03 & @xmath511.38e+12 & 4 & 2 / 1 + 0429@xmath99415 & 2002/10/09 & 1.79 & 1.06 & 0.14 & 0.59 & & & & & & 1 / + 0430@xmath99052 & 1998/03/07 & 4.93 & 3.09 & 1.27 & 0.77 & 1.71 & 0.26 & @xmath1000.04 & @xmath519.19e+11 & @xmath17117 & 15 / 11 + 0438@xmath17436 & 1998/06/05 & 2.50 & 1.77 & 1.01 & 0.71 & 1.79 & 0.37 & 0.18 & 5.58e+11 & & 1 / 0 + 0440@xmath17003 & 1998/03/07 & 1.40 & 0.94 & 0.52 & 0.82 & 0.62 & 0.36 & 0.07 & 2.31e+11 & @xmath17119 & 6 / 0 + 0446@xmath99112 & 2002/05/31 & 2.51 & 2.29 & 1.74 & 0.95 & 2.04 & 0.17 & @xmath1000.02 & @xmath512.74e+12 & 116 & 4 / 2 + 0454@xmath17234 & 1998/06/05 & 2.67 & 2.42 & 0.51 & 0.91 & 2.33 & 0.31 & @xmath1000.04 & @xmath512.13e+12 & & 1 / 1 + 0454@xmath99844 & 2001/12/30 & 0.35 & 0.34 & 0.23 & 0.79 & 0.29 & 0.33 & 0.13 & @xmath513.57e+10 & 153 & 6 / 0 + 0458@xmath17020 & 1995/07/28 & 2.85 & 2.41 & 1.62 & 0.85 & 2.04 & 0.30 & 0.12 & 1.06e+12 & @xmath1747 & 4 / 0 + 0521@xmath17365 & 1999/11/06 & 4.45 & 1.79 & 1.20 & 0.40 & 1.40 & 0.17 & @xmath1000.15 & @xmath513.09e+11 & @xmath1741 & 3 / 2 + 0524@xmath99034 & 1998/12/05 & 0.70 & 0.78 & 0.70 & 1.01 & 0.72 & 0.22 & 0.04 & @xmath514.37e+11 & @xmath1737 & 2 / 0 + 0528@xmath99134 & 1995/07/28 & 7.51 & 7.95 & 4.21 & 1.00 & 7.07 & 0.22 & @xmath1000.03 & @xmath512.06e+13 & 48 & 6 / 3 + 0529@xmath99075 & 2002/05/31 & 1.37 & 1.34 & 0.10 & 0.95 & 1.07 & 0.70 & 0.56 & @xmath511.48e+10 & @xmath1738 & 2 / 0 + 0529@xmath99483 & 2002/10/09 & 1.04 & 1.03 & 0.80 & 0.99 & 0.72 & 0.09 & @xmath1000.01 & @xmath517.89e+12 & 33 & 1 / 1 + 0537@xmath17286 & 1998/09/29 & 1.45 & 0.86 & 0.60 & 0.54 & 0.83 & 0.20 & 0.05 & 1.95e+12 & 89 & 2 / 0 + 0552@xmath99398 & 1999/01/02 & 4.50 & 4.51 & 1.60 & 0.97 & 3.21 & 0.28 & 0.19 & 1.12e+12 & @xmath1772 & 5 / 0 + 0602@xmath99673 & 1999/07/19 & 0.90 & 0.99 & 0.66 & 0.99 & 0.93 & 0.43 & 0.13 & 2.78e+11 & 158 & 4 / 1 + 0605@xmath17085 & 1996/10/27 & 2.78 & 1.89 & 0.83 & 0.80 & 1.79 & 0.69 & @xmath1000.11 & @xmath512.44e+11 & 122 & 5 / 5 + 0607@xmath17157 & 1998/10/30 & 7.89 & 7.28 & 4.95 & 0.77 & 6.92 & 0.23 & 0.16 & 1.34e+12 & 66 & 7 / 4 + 0615@xmath99820 & 1994/08/31 & 0.40 & 0.43 & 0.14 & 0.97 & 0.24 & 0.43 & 0.10 & 5.34e+10 & & 5 / 1 + 0642@xmath99449 & 2003/03/29 & 4.45 & 4.27 & 1.67 & 0.97 & 2.92 & 0.21 & 0.08 & 4.32e+12 & 90 & 6 / 4 + 0648@xmath17165 & 2002/11/23 & 2.84 & 2.68 & 1.38 & 0.76 & 2.33 & 0.22 & @xmath1000.02 & @xmath512.55e+12 & @xmath1790 & 2 / 1 + 0707@xmath99476 & 1994/08/31 & 0.70 & 0.61 & 0.47 & 0.95 & 0.49 & 0.39 & 0.04 & 3.83e+11 & @xmath1724 & 5 / 0 + 0710@xmath99439 & 1996/10/27 & 0.80 & 0.50 & 0.14 & 0.74 & & & & & 179 & 4 / + 0711@xmath99356 & 1999/01/02 & 0.40 & 0.40 & 0.09 & 1.00 & 0.26 & 0.72 & 0.58 & 8.79e+09 & 158 & 1 / 0 + 0716@xmath99714 & 2003/08/28 & 2.37 & 2.55 & 2.51 & 0.69 & 2.46 & 0.08 & @xmath1000.01 & @xmath511.85e+13 & 16 & 6 / 5 + 0723@xmath17008 & 1997/08/18 & 1.10 & 1.01 & 0.52 & 0.94 & 0.58 & 0.23 & @xmath1000.06 & @xmath512.37e+11 & @xmath1745 & 2 / 1 + 0727@xmath17115 & 2001/01/21 & 4.49 & 4.27 & 3.15 & 0.92 & 3.75 & 0.13 & 0.09 & 4.55e+12 & @xmath1797 & 7 / 3 + 0730@xmath99504 & 2003/06/15 & 1.19 & 1.36 & 1.17 & 1.10 & 1.25 & 0.21 & 0.06 & 9.38e+11 & @xmath17141 & 2 / 1 + 0735@xmath99178 & 1995/04/07 & 1.83 & 1.63 & 0.81 & 0.83 & 0.95 & 0.18 & 0.12 & @xmath512.40e+11 & 68 & 8 / 4 + 0736@xmath99017 & 2003/03/01 & 2.08 & 1.78 & 1.07 & 0.94 & 1.45 & 0.19 & @xmath1000.02 & @xmath512.51e+12 & @xmath1763 & 7 / 6 + 0738@xmath99313 & 1995/04/07 & 1.80 & 1.94 & 1.14 & 0.95 & 0.87 & 0.43 & @xmath1000.03 & @xmath516.54e+11 & 179 & 9 / 8 + 0742@xmath99103 & 1995/07/28 & 1.50 & 1.42 & 0.37 & 0.82 & 0.81 & 0.87 & 0.21 & 8.90e+10 & @xmath1710 & 7 / 2 + 0745@xmath99241 & 1997/08/18 & 0.88 & 0.95 & 0.65 & 0.87 & 0.83 & 0.22 & 0.05 & 5.74e+11 & @xmath1764 & 9 / 3 + 0748@xmath99126 & 1995/07/28 & 2.85 & 3.13 & 2.34 & 0.96 & 2.86 & 0.23 & 0.06 & 2.03e+12 & 115 & 7 / 2 + 0754@xmath99100 & 2002/11/23 & 1.96 & 1.82 & 1.39 & 0.86 & 1.42 & 0.26 & 0.05 & 7.91e+11 & 18 & 9 / 5 + 0804@xmath99499 & 1995/04/07 & 1.05 & 1.14 & 0.96 & 0.90 & 1.02 & 0.14 & 0.07 & 1.45e+12 & 128 & 5 / 2 + 0805@xmath17077 & 2002/06/15 & 1.62 & 1.57 & 1.17 & 0.97 & 1.34 & 0.27 & 0.10 & 7.80e+11 & @xmath1721 & 1 / 0 + 0808@xmath99019 & 1995/07/28 & 1.59 & 1.34 & 1.26 & 0.85 & 1.27 & 0.08 & @xmath1000.02 & @xmath519.19e+12 & @xmath17176 & 4 / 3 + 0814@xmath99425 & 2000/12/28 & 1.40 & 1.28 & 1.07 & 0.92 & 1.08 & 0.10 & 0.04 & @xmath511.40e+12 & 89 & 7 / 5 + 0821@xmath99394 & 2002/05/31 & 1.63 & 1.38 & 1.26 & 0.79 & 1.35 & 0.19 & 0.04 & 2.23e+12 & @xmath1747 & 2 / 0 + 0823@xmath99033 & 1998/10/30 & 1.60 & 1.38 & 1.14 & 0.89 & 1.10 & 0.11 & 0.05 & 1.72e+12 & 28 & 8 / 7 + 0827@xmath99243 & 2002/05/31 & 1.93 & 1.99 & 1.59 & 1.00 & 1.82 & 0.18 & 0.06 & 1.93e+12 & 114 & 2 / 0 + 0829@xmath99046 & 1995/07/28 & 1.23 & 1.35 & 0.75 & 0.90 & 0.76 & 0.14 & @xmath1000.04 & @xmath518.23e+11 & 61 & 7 / 7 + 0831@xmath99557 & 1999/01/02 & 1.66 & 0.79 & 0.09 & 0.57 & & & & & & 2 / + 0834@xmath17201 & 2002/05/29 & 3.82 & 3.25 & 1.65 & 0.83 & 3.08 & 0.27 & 0.21 & 1.10e+12 & @xmath17150 & 3 / 0 + 0836@xmath99710 & 2003/03/29 & 1.88 & 1.98 & 1.17 & 1.05 & 1.00 & 0.06 & 0.05 & 6.18e+12 & @xmath17142 & 3 / 0 + 0838@xmath99133 & 1998/09/29 & 1.05 & 0.74 & 0.38 & 0.70 & 0.45 & 0.19 & 0.10 & 2.18e+11 & 88 & 2 / 0 + 0850@xmath99581 & 2000/12/28 & 0.70 & 0.52 & 0.33 & 0.74 & 0.07 & 0.10 & @xmath1000.08 & @xmath511.09e+11 & 150 & 5 / 4 + 0851@xmath99202 & 2002/10/09 & 4.19 & 4.14 & 3.32 & 0.88 & 3.55 & 0.12 & @xmath1000.05 & @xmath513.89e+12 & @xmath17103 & 10 / 7 + 0859@xmath17140 & 1995/07/28 & 1.65 & 1.58 & 1.11 & 0.77 & 1.17 & 0.27 & @xmath1000.03 & @xmath511.58e+12 & 158 & 4 / 3 + 0859@xmath99470 & 2002/06/02 & 0.89 & 0.66 & 0.45 & 0.74 & 0.50 & 0.29 & 0.06 & 3.75e+11 & @xmath174 & 2 / 0 + 0906@xmath99015 & 2001/01/21 & 2.52 & 2.74 & 2.03 & 1.00 & 2.36 & 0.28 & @xmath1000.09 & @xmath519.83e+11 & 42 & 6 / 3 + 0917@xmath99449 & 1995/04/07 & 1.40 & 1.42 & 1.07 & 0.92 & 1.01 & 0.20 & @xmath1000.03 & @xmath512.62e+12 & 178 & 6 / 4 + 0917@xmath99624 & 2002/06/15 & 0.89 & 0.90 & 0.59 & 1.02 & 0.64 & 0.53 & 0.06 & 2.74e+11 & @xmath1734 & 2 / 0 + 0919@xmath17260 & 1998/06/05 & 1.68 & 1.33 & 0.80 & 0.79 & 1.14 & 0.24 & 0.18 & 4.71e+11 & @xmath1798 & 1 / 0 + 0923@xmath99392 & 1995/04/07 & 12.45 & 12.69 & 3.69 & 0.94 & 0.23 & 0.40 & 0.29 & 1.80e+10 & 105 & 9 / 7 + 0945@xmath99408 & 2002/10/09 & 1.77 & 1.58 & 1.04 & 0.89 & 0.99 & 0.14 & @xmath1000.03 & @xmath512.73e+12 & 116 & 7 / 3 + 0953@xmath99254 & 1995/04/07 & 1.20 & 1.31 & 0.55 & 1.01 & 0.36 & 0.23 & @xmath1000.07 & @xmath512.05e+11 & @xmath17124 & 6 / 5 + 0954@xmath99658 & 2003/03/01 & 0.51 & 0.55 & 0.46 & 1.08 & 0.45 & 0.14 & @xmath1000.03 & @xmath519.06e+11 & @xmath1732 & 1 / 1 + 0955@xmath99476 & 2002/06/15 & 1.53 & 1.76 & 1.18 & 1.03 & 1.73 & 0.28 & 0.16 & 6.01e+11 & 125 & 2 / 0 + 1012@xmath99232 & 2001/11/07 & 1.59 & 1.16 & 0.88 & 0.95 & 1.08 & 0.17 & @xmath1000.03 & @xmath511.67e+12 & 109 & 6 / 4 + 1015@xmath99359 & 1996/05/16 & 0.79 & 0.82 & 0.65 & 1.00 & 0.71 & 0.17 & 0.10 & 5.34e+11 & @xmath17170 & 7 / 1 + 1032@xmath17199 & 2002/05/29 & 0.97 & 0.96 & 0.44 & 0.96 & 0.90 & 0.67 & 0.21 & 1.12e+11 & @xmath17149 & 2 / 0 + 1034@xmath17293 & 1998/06/05 & 1.44 & 1.49 & 1.09 & 1.03 & 1.36 & 0.32 & @xmath1000.03 & @xmath519.08e+11 & 132 & 1 / 1 + 1036@xmath99054 & 2002/05/31 & 2.60 & 2.66 & 2.35 & 0.99 & 2.62 & 0.23 & 0.09 & @xmath517.12e+11 & @xmath178 & 2 / 0 + 1038@xmath99064 & 2003/05/09 & 1.67 & 1.69 & 1.34 & 1.02 & 1.40 & 0.28 & 0.04 & 1.52e+12 & 156 & 3 / 0 + 1045@xmath17188 & 2002/06/15 & 1.26 & 1.32 & 1.19 & 1.04 & 1.17 & 0.51 & @xmath1000.04 & @xmath514.43e+11 & 148 & 2 / 2 + 1049@xmath99215 & 1995/04/07 & 1.40 & 1.44 & 0.51 & 0.93 & 1.22 & 0.37 & 0.07 & 5.81e+11 & 114 & 6 / 4 + 1055@xmath99018 & 2002/05/29 & 5.67 & 5.30 & 4.28 & 0.89 & 4.93 & 0.23 & @xmath1000.02 & @xmath511.36e+13 & @xmath1750 & 8 / 5 + 1055@xmath99201 & 1999/11/06 & 0.54 & 0.38 & 0.30 & 0.65 & 0.26 & 0.27 & @xmath1000.04 & @xmath512.45e+11 & @xmath1710 & 8 / 6 + 1101@xmath99384 & 1997/03/13 & 0.69 & 0.52 & 0.39 & 0.75 & 0.45 & 0.16 & 0.07 & 2.19e+11 & @xmath1728 & 7 / 0 + 1116@xmath99128 & 1998/06/05 & 1.12 & 0.79 & 0.39 & 0.71 & 0.47 & 0.43 & 0.05 & 3.42e+11 & 5 & 1 / 0 + 1124@xmath17186 & 1998/11/01 & 2.84 & 2.82 & 2.31 & 0.99 & 2.67 & 0.18 & @xmath1000.03 & @xmath516.15e+12 & 169 & 3 / 1 + 1127@xmath17145 & 1999/11/06 & 3.75 & 3.38 & 1.91 & 0.81 & 1.95 & 0.15 & 0.06 & 2.60e+12 & 84 & 8 / 4 + 1128@xmath99385 & 1996/07/10 & 1.05 & 1.00 & 0.93 & 0.97 & 0.94 & 0.16 & @xmath1000.02 & @xmath514.94e+12 & @xmath17161 & 6 / 1 + 1144@xmath99402 & 2002/05/31 & 0.81 & 0.78 & 0.62 & 0.96 & 0.77 & 0.28 & 0.14 & 2.21e+11 & @xmath1770 & 1 / 0 + 1145@xmath17071 & 2002/04/02 & 0.86 & 0.70 & 0.27 & 0.81 & 0.64 & 0.32 & 0.19 & 1.37e+11 & @xmath1765 & 3 / 2 + 1148@xmath17001 & 1998/11/01 & 0.90 & 0.86 & 0.24 & 0.84 & 0.39 & 0.26 & @xmath1000.05 & @xmath515.41e+11 & @xmath17123 & 3 / 3 + 1150@xmath99812 & 2002/06/15 & 1.40 & 1.49 & 0.96 & 0.99 & 1.06 & 0.22 & 0.09 & 6.67e+11 & 176 & 2 / 0 + 1155@xmath99251 & 1995/04/07 & 0.40 & 0.24 & 0.14 & 0.56 & 0.12 & 0.47 & @xmath1000.09 & @xmath511.73e+10 & & 3 / 1 + 1156@xmath99295 & 1998/11/01 & 2.80 & 3.29 & 2.99 & 0.94 & 3.20 & 0.09 & 0.07 & 4.95e+12 & @xmath174 & 10 / 1 + 1213@xmath17172 & 1998/06/05 & 2.77 & 2.56 & 0.98 & 1.00 & 0.50 & 0.18 & @xmath1000.03 & @xmath515.33e+11 & 110 & 3 / 2 + 1219@xmath99044 & 2002/06/15 & 0.84 & 0.94 & 0.92 & 1.03 & 0.92 & 0.14 & @xmath1000.01 & @xmath519.73e+12 & 178 & 2 / 1 + 1219@xmath99285 & 1998/03/07 & 0.85 & 0.54 & 0.28 & 0.79 & 0.34 & 0.19 & 0.12 & 9.26e+10 & 105 & 7 / 4 + 1222@xmath99216 & 2003/05/09 & 1.08 & 0.89 & 0.70 & 0.82 & 0.71 & 0.26 & 0.05 & 4.47e+11 & @xmath176 & 2 / 0 + 1226@xmath99023 & 1999/05/21 & 35.91 & 29.12 & 7.36 & 0.74 & 1.87 & 0.13 & @xmath1000.06 & @xmath511.69e+12 & @xmath17117 & 15 / 11 + 1228@xmath99126 & 2003/02/05 & 28.10 & 2.63 & 0.73 & 0.07 & 1.39 & 0.41 & 0.27 & 6.96e+10 & @xmath1777 & 13 / 5 + 1244@xmath17255 & 1998/06/05 & 1.60 & 1.35 & 1.18 & 0.84 & 1.22 & 0.24 & 0.06 & 8.01e+11 & 143 & 1 / 0 + 1253@xmath17055 & 1996/05/16 & 22.20 & 18.49 & 11.21 & 0.91 & 15.15 & 0.30 & @xmath1000.05 & @xmath518.76e+12 & @xmath17121 & 14 / 8 + 1255@xmath17316 & 1998/06/05 & 1.90 & 0.91 & 0.38 & 0.48 & 0.85 & 0.93 & 0.20 & 7.15e+10 & 23 & 1 / 0 + 1302@xmath17102 & 1995/07/28 & 0.80 & 0.69 & 0.45 & 0.86 & 0.53 & 0.29 & @xmath1000.03 & @xmath513.72e+11 & 27 & 5 / 3 + 1308@xmath99326 & 2003/03/29 & 2.84 & 2.97 & 2.36 & 0.99 & 2.59 & 0.20 & 0.10 & 1.47e+12 & @xmath1745 & 7 / 2 + 1313@xmath17333 & 1998/06/05 & 1.30 & 1.02 & 0.57 & 0.78 & 0.71 & 0.17 & 0.09 & 5.52e+11 & @xmath1799 & 1 / 0 + 1323@xmath99321 & 1996/05/16 & 1.04 & 0.65 & 0.06 & 0.62 & & & & & & 5 / + 1324@xmath99224 & 2002/10/09 & 0.63 & 0.62 & 0.44 & 0.98 & 0.53 & 0.27 & 0.12 & 2.09e+11 & @xmath1737 & 1 / 0 + 1328@xmath99307 & 1995/04/07 & 3.44 & 1.22 & 0.20 & 0.32 & & & & & & 3 / + 1334@xmath17127 & 2001/03/15 & 7.82 & 8.88 & 7.17 & 0.98 & 7.40 & 0.16 & @xmath1000.01 & @xmath513.19e+13 & 150 & 6 / 4 + 1345@xmath99125 & 1996/04/22 & 1.40 & 0.98 & 0.07 & 0.51 & & & & & 162 & 5 / + 1354@xmath17152 & 1999/11/06 & 0.89 & 0.82 & 0.79 & 0.90 & 0.81 & 0.13 & @xmath1000.03 & @xmath512.94e+12 & 38 & 2 / 1 + 1354@xmath99195 & 2002/08/12 & 1.24 & 1.12 & 0.70 & 0.90 & 0.97 & 0.47 & 0.07 & 2.88e+11 & 145 & 3 / 1 + 1402@xmath99044 & 1998/06/05 & 0.65 & 0.56 & 0.33 & 0.86 & 0.31 & 0.22 & 0.07 & 4.40e+11 & @xmath1724 & 3 / 0 + 1404@xmath99286 & 1998/10/30 & 1.05 & 1.20 & 0.54 & 0.87 & 0.97 & 0.47 & 0.28 & 4.27e+10 & @xmath17144 & 11 / 0 + 1413@xmath99135 & 2001/01/21 & 1.48 & 1.60 & 1.34 & 0.92 & 1.42 & 0.07 & 0.03 & 4.09e+12 & @xmath17113 & 8 / 6 + 1417@xmath99385 & 2002/06/15 & 0.84 & 0.89 & 0.80 & 1.08 & 0.84 & 0.08 & 0.04 & 4.72e+12 & 162 & 2 / 0 + 1418@xmath99546 & 2003/05/09 & 1.00 & 0.92 & 0.61 & 0.99 & 0.67 & 0.32 & @xmath1000.04 & @xmath513.23e+11 & 130 & 2 / 2 + 1424@xmath99366 & 1995/04/07 & 0.60 & 0.60 & 0.50 & 0.88 & 0.59 & 0.16 & @xmath1000.04 & @xmath511.14e+12 & @xmath17117 & 5 / 2 + 1458@xmath99718 & 2001/12/30 & 1.99 & 1.47 & 0.87 & 0.63 & 0.86 & 0.26 & 0.01 & 2.81e+12 & 164 & 5 / 1 + 1502@xmath99106 & 2003/03/29 & 2.06 & 1.93 & 0.97 & 0.93 & 1.55 & 0.27 & @xmath1000.03 & @xmath513.22e+12 & 119 & 4 / 4 + 1504@xmath17166 & 2000/01/11 & 2.59 & 2.02 & 1.76 & 0.86 & 1.65 & 0.41 & 0.06 & 7.47e+11 & @xmath17172 & 3 / 1 + 1504@xmath99377 & 1997/08/28 & 0.80 & 0.71 & 0.53 & 0.77 & 0.69 & 0.48 & 0.10 & 1.26e+11 & @xmath17140 & 3 / 2 + 1508@xmath17055 & 1995/07/28 & 1.05 & 0.66 & 0.35 & 0.50 & 0.59 & 0.23 & @xmath1000.04 & @xmath517.84e+11 & 81 & 9 / 6 + 1510@xmath17089 & 2002/11/23 & 2.85 & 2.91 & 2.78 & 0.82 & 2.75 & 0.28 & @xmath1000.01 & @xmath515.60e+12 & @xmath1729 & 9 / 4 + 1511@xmath17100 & 1997/08/18 & 1.50 & 1.33 & 0.52 & 0.93 & 1.28 & 0.28 & @xmath1000.04 & @xmath511.48e+12 & 95 & 2 / 1 + 1514@xmath99004 & 1998/06/05 & 1.01 & 0.96 & 0.80 & 0.95 & 0.80 & 0.58 & @xmath1000.04 & @xmath512.19e+11 & @xmath1722 & 1 / 1 + 1514@xmath17241 & 1997/08/18 & 2.30 & 2.22 & 1.19 & 0.91 & 1.61 & 0.87 & 0.18 & 5.74e+10 & 166 & 2 / 1 + 1519@xmath17273 & 2002/04/02 & 1.90 & 1.65 & 1.11 & 0.88 & 1.57 & 0.22 & 0.17 & 5.09e+11 & @xmath1779 & 2 / 1 + 1532@xmath99016 & 1995/07/28 & 0.90 & 0.72 & 0.38 & 0.72 & 0.32 & 0.37 & 0.15 & 7.36e+10 & 128 & 6 / 1 + 1538@xmath99149 & 2002/08/12 & 1.37 & 1.06 & 0.92 & 0.71 & 0.77 & 0.31 & @xmath1000.03 & @xmath518.40e+11 & @xmath1734 & 4 / 4 + 1546@xmath99027 & 1996/10/27 & 2.50 & 2.82 & 2.45 & 0.93 & 2.76 & 0.24 & @xmath1000.03 & @xmath512.72e+12 & 173 & 8 / 4 + 1548@xmath99056 & 2003/03/01 & 3.35 & 2.90 & 1.91 & 0.91 & 0.88 & 0.22 & @xmath1000.06 & @xmath519.31e+11 & @xmath1719 & 6 / 2 + 1555@xmath99001 & 2002/08/12 & 0.72 & 0.76 & 0.40 & 1.01 & 0.72 & 0.24 & @xmath1000.05 & @xmath519.46e+11 & 97 & 3 / 2 + 1606@xmath99106 & 2003/08/28 & 2.15 & 2.26 & 1.45 & 1.03 & 1.85 & 0.23 & 0.14 & 6.71e+11 & @xmath1759 & 7 / 1 + 1607@xmath99268 & 1995/07/23 & 0.42 & 0.36 & 0.06 & 0.69 & 0.08 & 0.36 & 0.23 & 7.95e+09 & @xmath17165 & 3 / 0 + 1611@xmath99343 & 1995/04/07 & 4.19 & 4.50 & 3.03 & 0.94 & 3.37 & 0.22 & 0.09 & 2.20e+12 & 171 & 7 / 1 + 1622@xmath17253 & 1997/08/18 & 3.45 & 2.51 & 2.19 & 0.73 & 2.35 & 0.12 & @xmath1000.02 & @xmath518.47e+12 & @xmath178 & 3 / 2 + 1622@xmath17297 & 1998/06/05 & 2.00 & 1.73 & 1.04 & 0.86 & 1.15 & 0.22 & @xmath1000.04 & @xmath511.39e+12 & @xmath1766 & 1 / 1 + 1624@xmath99416 & 1999/01/02 & 0.57 & 0.49 & 0.18 & 0.86 & 0.26 & 0.21 & 0.12 & 2.07e+11 & @xmath17111 & 1 / 0 + 1633@xmath99382 & 2003/03/29 & 4.08 & 4.27 & 3.26 & 0.88 & 3.53 & 0.12 & 0.10 & 4.81e+12 & @xmath1786 & 10 / 5 + 1637@xmath99574 & 2002/05/31 & 1.81 & 1.88 & 1.55 & 1.02 & 1.74 & 0.26 & 0.07 & 8.54e+11 & @xmath17154 & 2 / 1 + 1638@xmath99398 & 1997/08/28 & 1.60 & 1.60 & 1.22 & 0.98 & 1.46 & 0.15 & 0.13 & 1.12e+12 & @xmath17137 & 7 / 1 + 1641@xmath99399 & 1999/07/19 & 9.39 & 8.73 & 4.38 & 0.86 & 5.08 & 0.20 & 0.04 & 5.42e+12 & @xmath1790 & 12 / 6 + 1642@xmath99690 & 2001/12/30 & 1.40 & 1.34 & 1.18 & 0.82 & 1.18 & 0.17 & @xmath1000.02 & @xmath513.19e+12 & @xmath17171 & 7 / 6 + 1652@xmath99398 & 1995/12/15 & 1.26 & 0.81 & 0.41 & 0.62 & 0.54 & 0.32 & 0.15 & 6.32e+10 & 150 & 7 / 0 + 1655@xmath99077 & 1997/03/13 & 2.10 & 1.72 & 1.36 & 0.97 & 1.59 & 0.52 & 0.11 & 2.55e+11 & @xmath1735 & 7 / 1 + 1656@xmath99053 & 1995/07/28 & 1.07 & 0.68 & 0.29 & 0.64 & 0.66 & 0.79 & @xmath1000.07 & @xmath511.28e+11 & 71 & 4 / 4 + 1656@xmath99477 & 1997/03/13 & 1.10 & 1.05 & 0.69 & 1.02 & 0.68 & 0.21 & 0.04 & 1.06e+12 & @xmath172 & 5 / 0 + 1726@xmath99455 & 2002/05/31 & 2.33 & 2.18 & 1.88 & 0.99 & 2.12 & 0.13 & 0.10 & 1.59e+12 & @xmath17116 & 2 / 1 + 1730@xmath17130 & 1996/10/27 & 14.31 & 10.95 & 7.49 & 0.84 & 10.58 & 0.23 & @xmath1000.03 & @xmath511.50e+13 & 12 & 7 / 5 + 1739@xmath99522 & 2003/03/29 & 1.26 & 1.43 & 1.11 & 0.99 & 0.99 & 0.11 & 0.09 & 1.32e+12 & 43 & 3 / 0 + 1741@xmath17038 & 2003/03/01 & 7.04 & 6.99 & 4.55 & 0.99 & 5.71 & 0.16 & @xmath1000.02 & @xmath512.03e+13 & @xmath17160 & 3 / 1 + 1749@xmath99096 & 1995/07/28 & 6.69 & 5.57 & 5.13 & 0.96 & 5.55 & 0.16 & @xmath1000.02 & @xmath511.34e+13 & 21 & 6 / 4 + 1749@xmath99701 & 2002/02/18 & 0.75 & 0.79 & 0.63 & 0.76 & 0.57 & 0.16 & @xmath1000.04 & @xmath518.27e+11 & @xmath1767 & 7 / 5 + 1751@xmath99288 & 2003/05/09 & 1.85 & 2.03 & 1.78 & 1.10 & 2.00 & 0.17 & 0.09 & @xmath517.07e+11 & & 2 / 0 + 1758@xmath99388 & 1996/05/16 & 1.70 & 1.75 & 1.42 & 0.99 & 1.62 & 0.20 & 0.09 & 1.47e+12 & @xmath1796 & 6 / 0 + 1800@xmath99440 & 1996/05/16 & 1.05 & 1.48 & 1.36 & 0.91 & 1.38 & 0.14 & @xmath1000.01 & @xmath519.35e+12 & @xmath17162 & 7 / 4 + 1803@xmath99784 & 1999/11/06 & 2.64 & 2.45 & 1.62 & 0.89 & 1.61 & 0.15 & 0.08 & 1.25e+12 & @xmath1792 & 8 / 0 + 1807@xmath99698 & 2001/12/30 & 1.65 & 1.35 & 0.95 & 0.82 & 0.83 & 0.31 & @xmath1000.05 & @xmath513.40e+11 & @xmath17105 & 11 / 6 + 1821@xmath99107 & 1998/09/29 & 0.48 & 0.39 & 0.24 & 0.78 & 0.29 & 0.41 & 0.13 & 6.77e+10 & @xmath1717 & 2 / 0 + 1823@xmath99568 & 1995/12/15 & 2.01 & 2.31 & 1.94 & 0.93 & 2.14 & 0.28 & 0.06 & 1.21e+12 & @xmath17161 & 9 / 6 + 1828@xmath99487 & 2003/03/29 & 3.04 & 1.89 & 1.29 & 0.61 & 1.30 & 0.20 & @xmath1000.03 & @xmath512.26e+12 & @xmath1740 & 6 / 4 + 1845@xmath99797 & 1997/08/28 & 1.70 & 0.42 & 0.28 & 0.22 & 0.30 & 0.64 & @xmath1000.03 & @xmath519.66e+10 & @xmath1738 & 11 / 3 + 1849@xmath99670 & 2003/06/15 & 1.60 & 1.74 & 1.52 & 1.09 & 1.61 & 0.20 & 0.04 & 1.76e+12 & @xmath1742 & 1 / 0 + 1901@xmath99319 & 1999/05/21 & 1.22 & 1.12 & 0.69 & 0.80 & 0.78 & 0.19 & 0.05 & 7.34e+11 & 119 & 9 / 5 + 1908@xmath17201 & 1998/06/05 & 3.30 & 2.82 & 2.20 & 0.85 & 2.60 & 0.49 & 0.12 & 5.27e+11 & 24 & 1 / 0 + 1921@xmath17293 & 2001/03/15 & 14.22 & 12.16 & 7.58 & 0.85 & 10.41 & 0.38 & 0.10 & 1.91e+12 & 20 & 5 / 0 + 1928@xmath99738 & 2002/06/15 & 3.48 & 3.90 & 2.65 & 0.81 & 2.58 & 0.20 & 0.07 & 1.37e+12 & 156 & 13 / 1 + 1936@xmath17155 & 1998/09/29 & 2.15 & 1.77 & 1.26 & 0.91 & 1.62 & 0.15 & 0.08 & 1.87e+12 & 119 & 4 / 0 + 1937@xmath17101 & 1998/09/29 & 0.42 & 0.30 & 0.16 & 0.69 & 0.21 & 0.30 & 0.12 & 1.53e+11 & 16 & 2 / 0 + 1954@xmath17388 & 1998/06/05 & 2.70 & 2.76 & 1.84 & 1.02 & 2.32 & 0.35 & 0.12 & 4.97e+11 & @xmath1761 & 1 / 0 + 1954@xmath99513 & 1999/05/21 & 0.95 & 0.94 & 0.56 & 0.99 & 0.68 & 0.21 & 0.05 & 7.51e+11 & @xmath1752 & 3 / 1 + 1957@xmath99405 & 2002/11/23 & 94.28 & 1.50 & 0.27 & 0.02 & & & & & @xmath1778 & 10 / + 1958@xmath17179 & 1998/06/05 & 2.90 & 2.67 & 2.22 & 0.91 & 2.62 & 0.16 & 0.11 & 1.28e+12 & @xmath17153 & 2 / 0 + 2000@xmath17330 & 1998/09/29 & 0.54 & 0.43 & 0.16 & 0.72 & 0.21 & 0.35 & 0.06 & 2.54e+11 & @xmath1736 & 2 / 1 + 2005@xmath99403 & 2001/03/04 & 2.83 & 2.63 & 0.59 & 0.93 & 1.80 & 0.39 & 0.31 & 2.22e+11 & 87 & 7 / 0 + 2007@xmath99777 & 1994/08/31 & 1.53 & 1.09 & 0.81 & 0.90 & 0.81 & 0.53 & @xmath1000.05 & @xmath512.34e+11 & @xmath1790 & 10 / 7 + 2008@xmath17159 & 2003/06/15 & 1.99 & 2.14 & 1.66 & 0.97 & 2.08 & 0.32 & 0.09 & 8.35e+11 & 8 & 2 / 0 + 2010@xmath99463 & 2002/11/23 & 0.33 & 0.28 & 0.23 & 0.85 & 0.25 & 0.45 & 0.12 & @xmath512.46e+10 & @xmath1741 & 1 / 0 + 2021@xmath99317 & 1995/04/07 & 2.15 & 2.13 & 0.90 & 0.86 & 1.73 & 0.65 & 0.21 & @xmath517.01e+10 & @xmath17167 & 6 / 0 + 2021@xmath99614 & 1997/08/28 & 2.55 & 2.73 & 0.81 & 0.91 & & & & & @xmath17147 & 8 / + 2029@xmath99121 & 1998/12/05 & 0.95 & 1.01 & 0.63 & 1.04 & 0.79 & 0.22 & 0.13 & 3.42e+11 & @xmath17151 & 2 / 0 + 2037@xmath99511 & 2002/10/09 & 2.70 & 2.32 & 1.56 & 0.86 & 1.79 & 0.29 & 0.11 & 8.33e+11 & @xmath17141 & 2 / 0 + 2059@xmath99034 & 1998/12/05 & 0.95 & 0.99 & 0.82 & 1.05 & 0.92 & 0.26 & 0.11 & 3.60e+11 & 29 & 2 / 0 + 2113@xmath99293 & 1995/04/07 & 0.87 & 0.93 & 0.96 & 0.91 & 0.91 & 0.30 & @xmath1000.04 & @xmath519.29e+11 & 164 & 5 / 3 + 2121@xmath99053 & 1999/11/06 & 2.30 & 2.48 & 1.98 & 1.06 & 1.98 & 0.13 & @xmath1000.03 & @xmath517.39e+12 & @xmath1787 & 4 / 3 + 2126@xmath17158 & 1998/09/29 & 1.25 & 1.23 & 0.91 & 0.95 & 1.14 & 0.23 & 0.13 & 8.55e+11 & @xmath17168 & 2 / 0 + 2128@xmath99048 & 2002/10/20 & 0.78 & 0.47 & 0.07 & 0.60 & & & & & & 3 / + 2128@xmath17123 & 1999/11/06 & 2.90 & 2.36 & 1.11 & 0.90 & 0.46 & 0.51 & @xmath1000.06 & @xmath511.29e+11 & @xmath17153 & 5 / 4 + 2131@xmath17021 & 2003/05/09 & 2.33 & 2.19 & 0.50 & 0.94 & 1.15 & 0.30 & @xmath1000.06 & @xmath517.44e+11 & 109 & 8 / 4 + 2134@xmath99004 & 1996/10/27 & 6.17 & 4.44 & 1.61 & 0.88 & 2.02 & 0.34 & @xmath1000.07 & @xmath511.31e+12 & @xmath17101 & 8 / 2 + 2136@xmath99141 & 2002/11/23 & 2.79 & 2.75 & 1.32 & 0.94 & 2.04 & 0.32 & 0.05 & 2.30e+12 & @xmath1768 & 7 / 3 + 2144@xmath99092 & 1996/10/27 & 0.75 & 0.80 & 0.60 & 0.97 & 0.55 & 0.15 & @xmath1000.04 & @xmath519.81e+11 & 81 & 6 / 3 + 2145@xmath99067 & 1999/11/06 & 10.66 & 10.33 & 6.03 & 0.97 & 7.97 & 0.20 & 0.07 & 6.01e+12 & 135 & 9 / 3 + 2155@xmath17152 & 2002/10/20 & 2.13 & 2.13 & 1.35 & 0.87 & 1.31 & 0.21 & @xmath1000.09 & @xmath516.53e+11 & @xmath17153 & 3 / 3 + 2200@xmath99420 & 1996/05/16 & 5.47 & 5.67 & 2.78 & 0.91 & 2.96 & 0.37 & @xmath1000.03 & @xmath511.45e+12 & @xmath17162 & 14 / 10 + 2201@xmath99171 & 2003/06/15 & 1.92 & 2.01 & 1.56 & 1.05 & 1.74 & 0.13 & @xmath1000.02 & @xmath519.49e+12 & 37 & 1 / 1 + 2201@xmath99315 & 2001/12/22 & 3.31 & 3.28 & 2.73 & 0.97 & 2.71 & 0.26 & @xmath1000.03 & @xmath512.77e+12 & @xmath17143 & 8 / 4 + 2209@xmath99236 & 1996/10/27 & 1.60 & 1.59 & 0.96 & 0.99 & 1.55 & 0.20 & 0.04 & 2.04e+12 & 23 & 5 / 2 + 2216@xmath17038 & 2002/05/31 & 2.65 & 2.52 & 1.65 & 0.95 & 2.02 & 0.28 & 0.11 & 7.04e+11 & @xmath17172 & 3 / 1 + 2223@xmath17052 & 2000/01/11 & 7.61 & 4.78 & 2.72 & 0.81 & 4.38 & 0.22 & @xmath1000.04 & @xmath516.53e+12 & 94 & 9 / 7 + 2227@xmath17088 & 2002/10/20 & 2.09 & 2.15 & 1.87 & 0.87 & 2.03 & 0.17 & 0.08 & 2.05e+12 & @xmath1736 & 5 / 2 + 2230@xmath99114 & 1998/03/07 & 5.17 & 4.61 & 3.11 & 0.83 & 3.52 & 0.12 & @xmath1000.03 & @xmath511.27e+13 & 142 & 9 / 5 + 2234@xmath99282 & 2001/01/21 & 1.34 & 1.44 & 0.82 & 1.00 & 0.46 & 0.41 & 0.26 & 4.20e+10 & @xmath17136 & 7 / 2 + 2243@xmath17123 & 1997/03/10 & 2.70 & 2.56 & 1.99 & 0.88 & 1.92 & 0.29 & 0.11 & 5.13e+11 & @xmath175 & 5 / 2 + 2251@xmath99158 & 1996/05/16 & 10.80 & 10.13 & 3.77 & 0.91 & 2.95 & 0.26 & 0.11 & 1.05e+12 & @xmath1788 & 11 / 4 + 2255@xmath17282 & 1997/08/18 & 6.80 & 6.79 & 5.50 & 0.99 & 6.70 & 0.13 & @xmath1000.02 & @xmath512.24e+13 & @xmath17133 & 2 / 1 + 2318@xmath99049 & 1998/09/29 & 1.19 & 1.22 & 0.94 & 1.03 & 1.20 & 0.29 & 0.15 & 2.48e+11 & @xmath1740 & 5 / 1 + 2329@xmath17162 & 1998/06/05 & 1.05 & 0.83 & 0.21 & 0.79 & 0.61 & 0.46 & 0.33 & 4.73e+10 & 84 & 1 / 0 + 2331@xmath99073 & 2002/06/15 & 1.22 & 1.24 & 0.77 & 0.99 & 0.87 & 0.12 & @xmath1000.02 & @xmath511.75e+12 & @xmath17120 & 2 / 2 + 2345@xmath17167 & 1995/07/28 & 2.65 & 2.55 & 1.29 & 0.83 & 1.46 & 0.30 & @xmath1000.04 & @xmath511.10e+12 & 128 & 6 / 4 + 2351@xmath99456 & 2002/05/31 & 2.46 & 1.79 & 1.11 & 0.83 & 1.34 & 0.19 & 0.14 & 8.53e+11 & @xmath1748 & 2 / 0 + lrllll mojave sample : bl lacs & 22 & 0.16 & 1.353 & 0.85 & 0.105 + mojave sampel : quasars & 94 & 0.16 & 1.278 & 0.85 & 0.144 + mojave sample : active galaxies & 8 & 0.46 & 1.314 & 0.57 & 0.216 + mojave sample : all & 133 & 0.18 & 1.296 & 0.83 & 0.144 + full sample : egret & 52 & 0.16 & 1.260 & 0.85 & 0.138 + full sample : non - egret & 198 & 0.26 & 1.394 & 0.76 & 0.151 + mojave sample : egret & 35 & 0.15 & 1.211 & 0.86 & 0.135 + mojave sample : non - egret & 98 & 0.20 & 1.260 & 0.81 & 0.143 + vsop sample & 116 & 0.25 & 1.314 & 0.76 & 0.147 + vsop pearson - readhead sample & 26 & 0.23 & 1.484 & 0.79 & 0.148 + lrccc full sample : high @xmath101 idv & 27 & 0.70@xmath1020.03 & 0.83@xmath1020.02 & 12.4@xmath1020.1 + full sample : low @xmath101 idv & 16 & 0.62@xmath1020.04 & 0.73@xmath1020.03 & 12.2@xmath1020.2 + full sample : non idv & 207 & 0.55@xmath1020.02 & 0.74@xmath1020.01 & 12.0@xmath1020.1 + mojave sample : high @xmath101 idv & 17 & 0.74@xmath1020.03 & 0.85@xmath1020.03 & 12.5@xmath1020.1 + mojave sample : low @xmath101 idv & 8 & 0.68@xmath1020.03 & 0.77@xmath1020.04 & 12.5@xmath1020.2 + mojave sample : non idv & 108 & 0.59@xmath1020.02 & 0.74@xmath1020.02 & 12.4@xmath1020.1 +

we have examined the compact structure in 250 flat - spectrum extragalactic radio sources using interferometric fringe visibilities obtained with the vlba at 15 ghz . with projected baselines out to 440 million wavelengths , we are able to investigate source structure on typical angular scales as small as 0.05 mas . this scale is similar to the resolution of vsop space vlbi data obtained on longer baselines at a lower frequency and with somewhat poorer accuracy . for 171 sources in our sample , more than half of the total flux density seen by the vlba remains unresolved on the longest baselines . there are 163 sources in our list with a median correlated flux density at 15 ghz in excess of 0.5 jy on the longest baselines ; these will be useful as fringe - finders for short wavelength vlba observations . the total flux densities recovered in the vlba images at 15 ghz are generally close to the values measured around the same epoch at the same frequency with the and umrao radio telescopes . we have modeled the core of each source with an elliptical gaussian component . for about 60% of the sources , we have at least one observation in which the core component appears unresolved ( generally smaller than 0.05 mas ) in one direction , usually transverse to the direction into which the jet extends . bl lac objects are on average more compact than quasars , while active galaxies are on average less compact . also , in an active galaxy the sub - milliarcsecond core component tends to be less dominant . intra - day variable ( idv ) sources typically have a more compact , more core - dominated structure on sub - milliarcsecond scales than non - idv sources , and sources with a greater amplitude of intra - day variations tend to have a greater unresolved vlba flux density . the objects known to be gev gamma - ray loud appear to have a more compact vlba structure than the other sources in our sample . this suggests that the mechanisms for the production of gamma - ray emission and for the generation of compact radio synchrotron emitting features are related . the brightness temperature estimates and lower limits for the cores in our sample typically range between @xmath0 and @xmath1k , but they extend up to @xmath2k , apparently in excess of the equipartition brightness temperature , or the inverse compton limit for stationary synchrotron sources . the largest component speeds are observed in radio sources with high observed brightness temperatures , as would be expected from relativistic beaming . longer baselines , which may be obtained by space vlbi observations , will be needed to resolve the most compact high brightness temperature regions in these sources .